2. Preliminaries

Let $(M,\xi)$ be a cooriented contact manifold and let $\lambda$ be a contact form with $\xi = \ker \lambda$ . Denote by ${\operatorname{Cont}}(M,\xi)$ (resp. ${\operatorname{Cont}}_0(M,\xi)$ ) the set of contact diffeomorphisms (resp. the identity component thereof).

Definition 2·1. For a given coorientation preserving contact diffeomorphism $\psi$ of $(M,\xi)$ we call the function g appearing in

\begin{align*}\psi^*\lambda = e^g \lambda\end{align*}

the conformal exponent for $\psi$ and denote it by $g = g_\psi$ .

Definition 2·2. A vector field X on $(M,\xi)$ is called contact if there exists a smooth function $f\;:\; M \to \mathbb R$ such that

\begin{align*}\mathcal L_X \lambda = f \lambda.\end{align*}

The associated function H defined by

(2·1) \begin{equation}H = - \lambda(X)\end{equation}

is called the contact Hamiltonian of X. We also call X the contact Hamiltonian vector field associated to H.

A straightforward calculation shows

\begin{align*}f = - R_\lambda[H].\end{align*}

Here we denote by $R_\lambda$ the Reeb vector field of $\lambda$ , i.e., the unique vector field X satisfying $X \rfloor d\lambda = 0, \, X\rfloor \lambda = 1$ . It is nothing but the Hamiltonian vector field associated to the constant function $H \equiv -1$ .

The following systematic notation will be useful to designate the Hamiltonian to a given contact isotopy as in symplectic topology [ Reference Oh and Müller39 ]. As practiced in the first-named author’s articles [ Reference Oh36, Reference Oh37 ], we adopt the following definition.

Definition 2·3 (Developing map). Let $\lambda$ be a contact form of $(M,\xi)$ . We define

(2·2) \begin{equation}{\operatorname{Dev}}_\lambda\;:\; \mathcal P({\operatorname{Cont}}(M,\xi)) \to C^\infty([0,T] \times M,\mathbb R)\end{equation}

by assigning its $\lambda$ -contact Hamiltonian functions

(2·3) \begin{equation}{\operatorname{Dev}}_\lambda(\ell)(t,x)\;:\!=\; -\lambda\left(\frac{\partial \ell}{\partial t}(t,\ell_t^{-1}(x))\right)\!.\end{equation}

Unravelling this definition, we have ${\operatorname{Dev}}_\lambda(\ell)(t,x) = H(t,x)$ if $X_t$ is the contact vector field as in Definition 2·2. We denote by $X \mapsto \psi$ if $\psi = \psi_H^1$ .

For a given general function H, the associated contact Hamiltonian vector field $X_H$ has decomposition

\begin{align*}X_H = X_H^\pi - H R_\lambda \in \xi \oplus \mathbb R \langle R_\lambda \rangle\end{align*}

from (1·2) where the projection $X_H^\pi$ to $\xi$ is uniquely determined by the equation

\begin{align*}X_H^\pi \rfloor d\lambda = dH - R_\lambda[H] \lambda\end{align*}

or equivalently

(2·4) \begin{equation}dH = X_H \rfloor d\lambda + R_\lambda[H] \lambda.\end{equation}

Example 2·4. Let $H\;:\;\mathbb R^{2n+1} \to \mathbb R$ be a smooth function on $\mathbb R^{2n+1} = J^1\mathbb R^n$ (or in Darboux coordinates). Then the contact Hamiltonian vector field $X_H$ is given by

(2·5) \begin{eqnarray}X_H & =& X_H^\pi -H R_\lambda =\sum_{i=1}^n \left(\frac{\partial H}{\partial p_i} \frac{D}{\partial q_i} - \frac{D H}{\partial q_i} \frac{\partial}{\partial p_i}\right)- H \frac{\partial}{\partial z} \\[5pt] & = & \sum_{i=1}^n \frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} -\left(\frac{\partial H}{\partial q_i} + p_i \frac{\partial H}{\partial z}\right)\frac{\partial}{\partial p_i}+ \left(\left\langle p, \frac{\partial H}{\partial p}\right \rangle - H\right)\frac{\partial}{\partial z}, \end{eqnarray}

where the set $\left\{ {D}/{\partial q_i}, {\partial}/{\partial p_i}, {\partial}/{\partial z} \right\}$ with ${D}/{\partial q_i} \;:\!=\; {\partial}/{\partial q_i} + p_i({\partial}/{\partial z})$ is the associated Darboux frame, i.e., a frame of TM which satisfies

\begin{align*}R_\lambda = \frac{\partial}{\partial z}, \quad \xi = \operatorname{span} \left\{ \frac{D}{\partial q_i}, \frac{\partial}{\partial p_i}\right\},\end{align*}

and $d\lambda({D}/{\partial q_i},{\partial}/{\partial p_i}) = \delta_{ij}$ for $1 \leq i, \, j \leq n$ . Therefore the associated contact Hamilton’s equation is given by

(2·6) \begin{equation}\begin{cases}\dot q_i = \dfrac{\partial H}{\partial p_i}, \\[9pt] \dot p_i = -\dfrac{D H}{\partial q_i} \left(= -\dfrac{\partial H}{\partial q_i} - p_i \dfrac{\partial H}{\partial z}\right)\!, \\[9pt] \dot z = \left\langle p, \dfrac{\partial H}{\partial p}\right \rangle - H. \end{cases}\end{equation}

Lemma 2·5. Solving contact Hamilton’s equation $\dot y = X_H(t,y)$ is equivalent to finding $\gamma:\mathbb R \to M$ that satisfies

(2·7) \begin{equation}(\dot \gamma - X_H(t,\gamma(t)))^\pi = 0, \quad \gamma^*\lambda + H(t,\gamma(t))dt = 0.\end{equation}

For the later purpose, we introduce the following proposition providing an explicit relationship between the contact Hamiltonians and the conformal exponents of a given contact isotopy.

Proposition 2·6 ([ Reference Oh36 , proposition 2·17]). Let $\Psi = \{\psi_t\}$ be a contact isotopy of $(M, \xi = \ker \lambda)$ with $\psi_t^* \lambda = e^{g_t}\lambda$ generated by $H = H(t,y)$ . We write $g_\Psi (t,y) = g_t(y)$ . Then

(2·8) \begin{equation}\frac{\partial g_\Psi}{\partial t}(t,y) = - R_\lambda[H](t,\psi_t(y)).\end{equation}

In particular, if $\psi_0 = {{\mathrm{Id}}}$ ,

\begin{align*}g_\Psi (t,y) = \int^t_0 - R_\lambda[H](u,\psi_u(y))\, du.\end{align*}

The same result, with different sign convention, is implicitly given in the proof of [ Reference Geiges17 , corollary 2·3·2]. (See also [ Reference Bhupal8 ] for the one-jet bundle case.)

To obtain a good variational problem for contact Hamiltonian dynamics, we use the following perturbed action functional introduced in [ Reference Oh32 ].

Definition 2·7 (Perturbed action functional). Let $H = H(t,y)$ be a contact Hamiltonian on M and $\phi_H^t$ be as in (1·4). Consider the free path space

\begin{align*}\mathcal L(M) \;:\!=\; C^{\infty}([0,1];\;M) = \{ \gamma\;:\; [0,1] \rightarrow M \}.\end{align*}

We define a functional $\mathcal A_H\;:\; \mathcal L(M) \rightarrow \mathbb R$ given by

(2·9) \begin{equation}\mathcal A_H(\gamma) \;:\!=\; \int_0^1 e^{g_{(\phi_H^t)^{-1}} (\gamma(t))} \gamma^* \lambda_H= \int_0^1 e^{g_{(\phi_H^t)^{-1}} (\gamma(t))} (\lambda(\dot \gamma (t)) + H_t (\gamma (t))) dt\end{equation}

as in (1·5). Here $\lambda_H \;:\!=\; \lambda + H dt$ with slight abuse of notation as in [ Reference Oh32 ].

The following lemma connects the perturbed action functional with the unperturbed (standard) one.

Lemma 2·8 ([ Reference Oh32 , lemma 2·2]). For given path $\gamma \in \mathcal L(M)$ , consider the path $\overline \gamma$ defined by

\begin{align*}\overline \gamma (t) \;:\!=\; (\phi_H^t)^{-1}(\gamma(t)).\end{align*}

Then we have

(2·10) \begin{equation}\mathcal A_H(\gamma) = \mathcal A_0(\overline \gamma),\end{equation}

where $\mathcal A_0$ is the standard (unperturbed) contact action functional given by $\mathcal A_0(\gamma) = \int_\gamma \lambda$ .

Following [ Reference Oh34 ] and [ Reference Oh37 ], we call the transformation $\gamma \mapsto \overline \gamma$ (resp. its inverse $\overline \gamma \to \gamma$ ) gauge transformations. Since this transformation will enter in later discussion, we formalise its definition now.

Definition 2·9 (Gauge transformation). Let $\gamma, \, \nu \in C^\infty([0,1], M)$ . We define the map $\Phi_H\;:\; \gamma \mapsto \overline \gamma$ to be

(2·11) \begin{equation}\Phi_H(\gamma)(t)\;:\!=\; \overline \gamma(t) = (\phi_H^t)^{-1}(\gamma(t))\end{equation}

and denote by $\Psi_H=(\Phi_H)^{-1}$ its inverse which is given by

(2·12) \begin{equation}\Psi_H(\nu) (t)= \phi_H^t(\nu(t)) = \psi_H^t \left((\psi_H^1)^{-1}(\nu(t))\right)\!.\end{equation}

We now quote from [ Reference Oh32 ] the following first variation formula of the action functional (1·5) on the free path space $\mathcal L(M)$ .

Proposition 2·10 ([ Reference Oh32 , proposition 2·3]). For any vector field $\eta$ along $\gamma \in \mathcal L(M)$ , we have the first variation

(2·13) \begin{eqnarray}\delta \mathcal A_H (\gamma)(\eta)&=& \int_0^1 d\lambda((d \phi_H^t)^{-1}(\eta(t)), (d \phi_H^t)^{-1}(\dot \gamma - X_H(t,\gamma(t))))dt \\& & + \lambda(\eta(1)) - e^{g_{\psi^1_H}(\gamma(0))} \lambda(\eta(0)).\end{eqnarray}

An immediate corollary of this first variation formula shows that the Legendrian boundary condition is a natural boundary condition for the action functional $\mathcal A_H$ in that it kills the boundary contribution in the first variation:

Corollary 2·12. Let R be any given Legendrian submanifold of $(M,\xi)$ . Consider the subset

\begin{align*}\mathcal L_0(M;\;R)\;:\!=\; \{ \gamma \in \mathcal L(M) \mid \gamma(0) \in R\}.\end{align*}

Then we have

\begin{align*}\delta \mathcal A_H (\gamma)(\eta)= \int_0^1 d\lambda\left(d(\phi_H^t)^{-1}(\eta(t)), d(\phi_H^t)^{-1}(\dot \gamma - X_H(t,\gamma(t)))\right)\!dt + \lambda(\eta(1))\end{align*}

on $\mathcal L_0(M;\;R)$ .

3. Carnot path space of translated horizontal paths

An immediate consequence of Corollary 2·12 is as follows. By the vanishing of $\lambda$ on the Legendrian submanifold, we have derived that $\delta \mathcal A_H(\gamma)(\eta) = 0$ for all variations $\eta$ tangent to paths satisfying Legendrian boundary conditions if and only if

(3·1) \begin{equation}((d\phi^t_H)^{-1}(\dot\gamma - X_{H_t}(\gamma(t))))^\pi = 0\end{equation}

i.e., it is equivalent to

\begin{align*}(d\phi^t_H)^{-1}(\dot \gamma - X_H(\gamma(t))) = f(t)R_\lambda(\overline{\gamma}(t)),\end{align*}

where f(t) is a smooth function.

This has led us to consider the following space

Definition 3·1 (Carnot path space). Consider the subsets of $\mathcal L(M)$ given by

\begin{align*}\mathcal L^{\text {Carnot}}(M,H)= \{ \gamma \;:\; [0,1] \rightarrow M \; \mid \; \lambda_{\gamma}(\dot \gamma) + H(\gamma) = 0 \}.\end{align*}

The defining equation $ \lambda_{\gamma}(\dot \gamma) + H(\gamma) = 0$ is equivalent to the statement

\begin{align*}\dot \gamma(t) -X_H(\gamma(t)) \in \xi_{\gamma(t)}\end{align*}

for all $t \in [0,1]$ . We note that since the distribution $\xi$ is not integrable a priori $\mathcal L ^{\text {Carnot}}(M, H)$ could be a pathological space, e.g., may not carry the structure of Fréchet manifold on which we can do some differential calculus. In this regard, we prove the following result which shows that $\mathcal L ^{\text {Carnot}} (M,H)$ carries a Fréchet manifold structure and hence we can do the calculus of constrained variation of $\mathcal A_H$ restricted thereto.

The rest of the section will be occupied by the proof of this theorem. For this purpose, we start with the following lemma.

Lemma 3·3. Let $H = H(t,y)$ be any smooth Hamiltonian on M and consider the map

\begin{align*}\Upsilon\;:\; \gamma \in \mathcal L(M) \mapsto \lambda (\dot \gamma) + H_t(\gamma) \in C^\infty ([0,1],\mathbb R).\end{align*}

Then we have its derivative

(3·2) \begin{equation}d_\gamma \Upsilon(\eta) = d\lambda(\eta,\dot \gamma-X_{H_t}(\gamma)) + \frac{d}{dt}(\lambda(\eta)) + R_\lambda[H_t] \, \lambda(\eta).\end{equation}

Proof. Note that the codomain of $\Upsilon$ is the set $C^\infty([0,1],\mathbb R)$ . Denote a variation of $\gamma$ by $\eta$ and the differential of $\Upsilon$ by

(3·3) \begin{equation}d_\gamma \Upsilon\;:\; T_\gamma \mathcal L(M) \to C^\infty([0,1], \mathbb R).\end{equation}

Then we compute the derivative

(3·4) \begin{eqnarray}d_\gamma \Upsilon(\eta) & = & \frac{d}{ds}\Big|_{s=0} \Upsilon(\gamma_s)= \frac{d}{ds}\Big|_{s=0}\Big((\lambda(\dot \gamma_s)) + H(t, \gamma_s(t))\Big)\\[5pt] & = & \frac{d}{ds}\Big|_{s=0}\left(\lambda(\dot \gamma_s)\right) + dH_t(\gamma(t))(\eta(t)).\end{eqnarray}

Utilising the identity (2·4), we evaluate

(3·5) \begin{eqnarray}dH_t(\gamma(t))(\eta(t)) & = & (X_{H_t} \mathbin{\lrcorner} d\lambda + R_\lambda[H_t] \lambda)(\eta(t)) \\[5pt] & = & d\lambda(X_{H_t}, \eta(t)) + R_\lambda[H_t]\lambda(\eta(t)).\end{eqnarray}

It remains to evaluate $\frac{d}{ds}\big|_{s=0}(\lambda(\dot \gamma_s))$ . For this, we introduce the variation function $c(s,t)\;:\!=\; \gamma_s(t)$ on $(-\epsilon,\epsilon) \times [0,1]$ for some small $\epsilon \gt 0$ that satisfies

\begin{align*}c(0,t) = \gamma(t), \quad \frac{\partial c}{\partial s}(0,t) = \eta(t).\end{align*}

Then we have

(3·6) \begin{eqnarray}\frac{d}{ds}\Big|_{s=0}(\lambda(\dot \gamma_s)) & = &\frac{d}{ds}\Big|_{s=0}(\gamma_s^*\lambda(\partial_t))= \left(\frac{d}{ds}\Big|_{s=0}\gamma_s^*\lambda\right)(\partial_t) \\[5pt] & = & \mathcal L_{\partial_s}(\gamma_s^*\lambda)|_{s=0}(\partial_t).\end{eqnarray}

By Cartan’s magic formula (applied to the Lie derivative over the map c), we have

\begin{align*}\mathcal L_{\partial_s}(\gamma_s^*\lambda)= \partial_s \rfloor d(\gamma_s^*\lambda) + d (\partial_s \rfloor \gamma_s^*\lambda),\quad \gamma_s(t) = c(s,t).\end{align*}

By evaluating this against $\partial_t$ at $s = 0$ , we obtain

\begin{eqnarray*}\mathcal L_{\partial_s}(\gamma_s^*\lambda)(\partial_t)|_{s = 0}& = & d\lambda(dc(\partial_s),dc(\partial_t))|_{s=0} + d(\lambda(dc(\partial_s)))(\partial_t)|_{s = 0} \\[5pt] & = & d\lambda(\eta,\dot \gamma) + \frac{d}{dt}(\lambda(\eta)).\end{eqnarray*}

By substituting this into (3·6), we get

\begin{align*}\frac{d}{ds}\Big|_{s=0}(\lambda(\dot \gamma_s)) =d\lambda(\eta,\dot \gamma) + \frac{d}{dt}(\lambda(\eta)).\end{align*}

Then substituting this and (3·5) into (3·4), we have derived (3·2) which completes the proof of the lemma.

By the expression of the derivative $d_\gamma \Upsilon$ in (3·2), it follows that it continuously extends to a bounded operator

(3·7) \begin{equation}d_\gamma \Upsilon\;:\; W^{1,2}(\gamma^*TM) \to L^2([0,1],\mathbb R).\end{equation}

We call this the $W^{1,2}$ -completion of the derivative (3·2). Using this, we now prove the following important property of the map $\Upsilon$ .

Proof. Let $\gamma$ be any such a curve and write $D_\gamma = d_\gamma \Upsilon$ as before. We will prove surjectivity by Fredholm alternative by considering the $L^2$ -pairing

\begin{align*}\langle\langle D_\gamma(\eta), g\rangle\rangle = \int_0^1 ( D_\gamma(\eta)\, g)dt\end{align*}

for the test function $\eta \in W^{1,2}(\gamma^*TM)$ and a function $g \in L^2([0,1],\mathbb R)$ to study.

For this purpose, we assume $\langle\langle D_\gamma(\eta), g\rangle\rangle = 0$ for all $\eta \in W^{1,2}(\gamma^*TM)$ . We would like to conclude $g = 0$ . For this purpose, it will be enough to consider smooth test functions $\eta$ WLOG. We compute

\begin{eqnarray*}0 & = & \int_0^1 ( D_\gamma(\eta)\, g)dt \\[5pt]& = & \int_0^1\left( d\lambda(\eta, \dot \gamma - X_H(t,\gamma)) + \frac{d}{dt}(\lambda(\eta)) + R_\lambda[H_t]\lambda(\eta)\right)\, g\, dt \\[5pt]& = & \int_0^1 d\lambda(\eta, \dot \gamma - X_H(t,\gamma))\, g(t) \, dt+ \int_0^1 \left(\frac{d}{dt}(\lambda(\eta)) + R_\lambda[H_t]\lambda(\eta)\right)\, g(t)dt.\end{eqnarray*}

By doing integration by parts, we rewrite the second integral into

\begin{eqnarray*}&{}& \int_0^1 \left(\frac{d}{dt}(\lambda(\eta)) + R_\lambda[H_t]\lambda(\eta)\right)\, g\, dt\\[5pt]& = & \int_0^1 \left(-\frac{dg}{dt} + R_\lambda[H_t]g\right) \lambda(\eta) dt+ g(1) \lambda(\eta(1)) - g(0)\lambda(\eta(0)).\end{eqnarray*}

Therefore we have derived

\begin{eqnarray*}0 & = & \int_0^1 d\lambda(\eta, \dot \gamma - X_H(t,\gamma))\, g(t) \, dt\\[5pt] &{}& \quad + \int_0^1 \left(-\frac{dg}{dt} + R_\lambda[H_t]g\right) \lambda(\eta) dt+ g(1) \lambda(\eta(1)) - g(0)\lambda(\eta(0))\end{eqnarray*}

for all $\eta$ . By writing $\eta = \eta^\pi + \lambda(\eta) R_\lambda$ , we can separate the two lines in our consideration.

The first line can be written as

\begin{align*}\int_0^1 d\lambda(\eta, \dot \gamma - X_H(t,\gamma))\, g(t) \, dt= \int_0^1 d\lambda(\eta^\pi, g(t) (\dot \gamma - X_H(t,\gamma))^\pi) \, dt.\end{align*}

Since $\eta^\pi \in \gamma^*\xi$ is arbitrary and by nondegeneracy of $d\lambda|_\xi$ , we have derived

(3·8) \begin{equation}g(t) (\dot \gamma - X_H(t,\gamma))^\pi = 0.\end{equation}

The second line gives rise to

(3·9) \begin{equation}-\frac{dg}{dt} + R_\lambda[H_t]g = 0, \quad g(1) = 0 = g(0)\end{equation}

since $\lambda(\eta)$ can be an arbitrary real valued function on [0,1]. Note that the equation from (3·9) is a linear homogeneous first-order ODE for g on $\mathbb R$ . Therefore, from the initial condition $g(0) = 0$ alone, we have derived $g \equiv 0$ . By the Hahn-Banach theorem, this finishes the proof of surjectivity of the operator

\begin{align*}d_\gamma \Upsilon\;:\; W^{1,2}(\gamma^*TM) \to L^2([0,1],\mathbb R).\end{align*}

We now need to derive the following smooth counterpart from this proof. The following theorem is an immediate corollary of Proposition 3·4 and the regularity theorem for the inhomogeneous first-order linear ODE

\begin{align*}\frac{df}{dt} + R_\lambda[H_t] f = h\end{align*}

on $\mathbb R$ . (See [ Reference Arnold5 , section 32] for example.)

Proof. Let $g \in C^\infty([0,1],\mathbb R)$ . By Proposition 3·4, we have an element $\eta \in W^{1,2}(\gamma^*TM)$ such that $d_\gamma \Upsilon(\eta) = g$ , i.e.,

(3·10) \begin{equation}d\lambda(\eta,\dot \gamma-X_{H_t}(\gamma)) + \frac{d}{dt}(\lambda(\eta))+ R_\lambda[H_t] \, \lambda(\eta) = g.\end{equation}

We will adjust $\eta$ to another smooth solution $\eta'$ .

Take any smooth section $\zeta$ satisfying $\zeta(0) = \eta(0)$ . We then consider the function g’ defined by

\begin{align*}g' \;:\!=\; d\lambda(\zeta, \dot \gamma - X_{H_t}(\gamma))\end{align*}

which is smooth by definition. Then we solve the equation

(3·11) \begin{equation}\begin{cases}\dfrac{df}{dt} + R_\lambda[H_t] \, f = g - g',\\[5pt] f(0) = \lambda(\eta(0))\end{cases}\end{equation}

for f. Here we remark that since $\eta \in W^{1,2}(\gamma^*TM)$ it is continuous by the Sobolev embedding theorem and hence the value $\eta(0)$ has well-defined meaning. Since $g - g'$ is smooth by hypothesis on g and by the choice of g’, the unique solution f of (3·11) is smooth by the continuity theorem of the first order ODE: for this, we observe that the directional derivative $R_\lambda[H_t]$ is a smooth function of t.

Now we denote by $\zeta^\pi$ the $\xi$ -component of $\zeta$ which is smooth. Then we consider the smooth section $\eta' \in \Gamma(\gamma^*TM)$ defined by

\begin{align*}\eta' = \zeta^\pi + f R_\lambda.\end{align*}

By the definition of $\eta'$ , it is smooth and satisfies

\begin{eqnarray*}d_\gamma\Upsilon(\eta') & = & d\lambda(\eta',\dot \gamma-X_{H_t}(\gamma)) + \frac{d}{dt}(\lambda(\eta'))+ R_\lambda[H_t] \, \lambda(\eta') \\& = & d\lambda\left(\zeta^\pi + f R_\lambda ,\dot \gamma-X_{H_t}(\gamma)\right)+ \frac{d}{dt}\left(\lambda(\zeta^\pi + f R_\lambda)\right)+ R_\lambda[H_t] \, \lambda(\zeta^\pi + f R_\lambda) \\& = & d\lambda(\zeta^\pi ,\dot \gamma-X_{H_t}(\gamma))+ \frac{d}{dt}(\lambda(f R_\lambda))+ R_\lambda[H_t] \, \lambda( f R_\lambda) \\& = & g' + \left( \frac{df}{dt}+ R_\lambda[H_t] \, f \right) = g' + (g-g') = g.\end{eqnarray*}

Furthermore, we check its initial value

\begin{align*}\eta'(0) = \zeta^\pi(0) + f(0) R_\lambda = \eta^\pi(0) + \lambda(\eta(0)) R_\lambda = \eta(0),\end{align*}

where the second equality follows from the standing requirement $\zeta(0) = \eta(0)$ imposed in the beginning and the initial condition for f imposed in (3·11). This completes the proof.

Theorem 3·5 in particular implies that $0 \in C^\infty([0,1],\mathbb R)$ is a regular value of the (Fréchet) smooth map $\Upsilon$ the linearisation of which is a linear first order ODE operator which can be seen from the formula (3·2). Recalling by definition that

\begin{align*}\mathcal L^{\text {\text {Carnot}}}(M,H) = \Upsilon^{-1}(0),\end{align*}

the implicit function theorem [ Reference Sergeraert43 ] completes the proof of Theorem 3·2.

Part I. Contact analogue to Weinstein’s canonical generating function

To motivate our discussion and for the convenience of readers, we recall Weinstein’s de-approximation [ 52 ] on the cotangent bundle in Section 4. In this de-approximation, he showed that the classical action functional

\begin{align*}\mathcal A_H^{\text{cl}}\;:\; \mathcal L_0(T^*B;\;o_{T^*B}) \to \mathbb R\end{align*}

is a generating function of the Hamiltonian isotope

\begin{align*}L = \phi_H^1(o_{T^*B}),\end{align*}

where $\phi_H^t$ is the symplectic Hamiltonian isotopy generated by H. Here the domain, denoted by $\mathcal L_0(T^*B;\;o_{T^*B})$ , of the functional is the space consisting of the paths $\gamma\;:\; [0,1] \to T^*B$ satisfying the initial boundary condition $\gamma(0) \in o_{T^*B}$ .

In Part I, we will construct the contact counterpart of this Weinstein’s canonical, but infinite dimensional generating function, on the one-jet bundle

\begin{align*}(J^1B, \lambda), \quad \lambda \;:\!=\; dz - \pi_{T^*B}^*\theta =dz - pdq,\end{align*}

where $\pi_{T^*B}\;:\; J^1B \to T^*B$ is the canonical projection in (1·14).

4. Weinstein’s de-approximation of Laudenbach–Sikorav’s construction

In this section, we explain Weinstein’s de-approximation [ 52 ] of Laudenbach–Sikorav’s ‘canonical’ generating function of the time-one image $\phi_H^1(o_{T^*B})$ in the cotangent bundle. It is a generating function, canonically arising from the classical action functional $\mathcal A_H^{\text{cl}}$ , in the sense of the Morse family on the general fiber bundle presented in Definition 1·13 whose fiber is an infinite-dimensional manifold.

A straightforward calculation gives rise to the first variation of the classical action functional

\begin{align*}\mathcal A_H^{\text{cl}}(\gamma) = \int \gamma^*\theta - H(t,\gamma(t))dt\end{align*}

on the path space $\mathcal L(T^*B)$ is given by

\begin{eqnarray*}\delta \mathcal A_H^{\text{cl}}(\gamma)(\eta) & = &\int_0^1 \omega_0(\eta(t), \dot \gamma(t) - X_{H_t}(\gamma(t)))dt+ \langle \eta(1), \theta(\gamma(1)) \rangle - \langle \eta(0), \theta(\gamma(0)) \rangle\\& = & \int_0^1 \omega_0(\eta(t), \dot \gamma(t) - X_{H_t}(\gamma(t)))dt + \langle d\pi (\eta(1)), \gamma(1) \rangle - \langle d\pi (\eta(0)), \gamma(0) \rangle. \end{eqnarray*}

(See [ Reference Arnold6, Reference Oh33, 50 ] for an explicit derivation.)

Then Weinstein directly put the action functional into the following infinite dimensional framework on the general fibration, replacing Laudenbach–Sikorav’s finite dimensional approximation of the action functional.

Proposition 4·1 (Weinstein [ 52 ]). Let $H= H(t,x)$ be a time-dependent Hamiltonian on the cotangent bundle $T^*B$ . Then the action functional $\mathcal A_H^{\text{cl}}$ restricted to the subset

\begin{align*}\mathcal L_0(T^*B,o_{T^*B}) = \{ \gamma\;:\; [0,1] \to T^*B \mid \gamma(0) \in o_{T^*B}\}\end{align*}

of the full path space is a generating function of the time-one image $\phi_H^1(o_{T^*B})$ of the zero section under the Hamiltonian flow of H.

Proof. We consider the path space $\mathcal L_0(T^*B,o_{T^*B})$ and the diagram

(4·1)

where the fibration $\Pi\;:\; \mathcal L_0(T^*B,o_{T^*B}) \to B$ is defined by $\Pi\;:\!=\; \pi \circ \operatorname{ev}_1$ and $\operatorname{ev}_1\;:\; \mathcal L_0(T^*B;\;o_{T^*B}) \to T^*B$ is the evaluation map

\begin{align*}\operatorname{ev}_1(\gamma) \;:\!=\; \gamma(1), \quad \text{for } \, \gamma \in \mathcal L_0(T^*B;\;o_{T^*B}).\end{align*}

The above first variation formula for $\eta$ tangent to $\mathcal L_0(T^*B,o_{T^*B})$ satisfies $p(\gamma(0))= 0$ and hence we have

\begin{align*}\delta \mathcal A_H^{\text{cl}}(\gamma)(\eta) =\int_0^1 \omega_0(\eta(t), \dot \gamma(t) - X_{H_t}(\gamma(t)))dt + \langle d\pi (\eta(1)), \gamma(1) \rangle.\end{align*}

It follows from the definition of the fibration $\Pi$ the vertical derivative of $\mathcal A_H$ for $\Pi$ is nothing but

\begin{align*}\delta^{\text{vert}}\mathcal A_H^{\text{cl}}(\gamma)(\eta)\;:\!=\; \int_0^1 \omega_0(\eta(t),\dot \gamma(t) - X_{H_t}(\gamma(t)))dt\end{align*}

and hence the vertical critical point $\gamma$ is nothing but the Hamiltonian trajectory satisfying

\begin{align*}\dot \gamma(t) = X_H(t,\gamma(t)), \qquad \gamma(0) \in o_{T^*B}.\end{align*}

In other words, we have

\begin{align*}\Sigma_{\mathcal A_H^{\text{cl}}} = \{ \gamma \in \mathcal L(T^*B) \mid \dot \gamma(t) = X_H(t,\gamma(t)), \, \gamma(0) \in o_{T^*B}\}.\end{align*}

Then we obtain the horizontal derivative

\begin{align*}\delta^{\text{hor}}\mathcal A_H^{\text{cl}}(\gamma) = \gamma(1).\end{align*}

Since $\gamma(t)$ is a Hamiltonian trajectory whose final point lies in $\phi_H^1(o_{T^*B})$ , we can write

\begin{align*}\gamma(t) = \phi_H^t(\phi_H^1)^{-1}(x), \, \gamma(1) = x \in \phi_H^1(o_{T^*B}).\end{align*}

This proves that $\mathcal A_H^{\text{cl}}|_{\mathcal L_0(T^*B;\;o_{T^*B})}$ is a generating function of $L\;:\!=\; \phi_H^1(o_{T^*B})$ .

The content and its proof of the above Weinstein’s formulation of the action functional as a generating function can be succinctly summarised as follows:

1. (1) first solve the fiberwise (or vertical) critical point equation. This provides the equation of motion, Hamilton’s equation $\dot x = X_H(t,x)$ ;

2. (2) then push forward the image of the differential $\delta\mathcal A_H^{\text{cl}}$ by the map $\pi \circ \operatorname{ev}_1$ to the cotangent bundle $T^*B$ . This provides the final location of the particle;

3. (3) the resulting push-forward image is precisely $\phi_H^1(o_{T^*B})$ . This provides the final momentum of the particle.

The rest of the paper will be occupied by our construction of the contact/Legendrian counterpart of Laudenbach–Sikorav’s construction which we apply to our effective contact action functional $\mathcal A _H^{CC}$ on the one-jet bundle $J^1B$ .

5. Perturbed action functional as a contact generating function

In this section, we assume that H is a compactly supported Hamiltonian function on $J^1B$ . By specialising Definition 3·1 to the case $M = J^1B$ , we consider the subset

\begin{align*}\mathcal L ^{\text {Carnot}}_{0}(J^1B, H;\; R) \;:\!=\; \mathcal L^{\text {Carnot}} (J^1B, H) \cap \mathcal L_{0}(J^1B;\;R)\end{align*}

by considering the Legendrian boundary condition at $t = 0$ . More explicitly, we have

(5·1) \begin{equation}\mathcal L ^{\text {Carnot}}_{0}(J^1B, H;\; R) = \{ \gamma \in \mathcal L (J^1B) \mid \lambda_{\gamma}(\dot \gamma) + H(\gamma) = 0,\, \gamma(0) \in R \}.\end{equation}

To construct a contact analogue to the above mentioned Weinstein’s de-approximation [ 52 ] of the case of the 1-jet bundles $J^1B$ , we need to consider the following action functional modified from $\mathcal A_H$ . One reason for this modification is that by definition we have

\begin{align*}\mathcal A_H(\gamma) = 0\end{align*}

for any contact Hamiltonian trajectory $\gamma$ . We now introduce the aforementioned effective action functional.

Definition 5·1 (Effective action functional). We define $\widetilde{\mathcal{A}}_H\;:\; \mathcal L(J^1B)\to \mathbb R$ to be

\begin{eqnarray*}\widetilde{\mathcal{A}}_H(\gamma) & = & - \mathcal A_H (\gamma) +z(\gamma(1)) \\& = & - \int_0^1 e^{g_{(\phi_H^t)^{-1}}(\gamma (t))} (\lambda_{\gamma(t)} (\dot \gamma (t)) + H(\gamma(t))) dt + z(\gamma(1)).\end{eqnarray*}

For later purposes, we write this functional in terms of the gauge-transformed path $\overline \gamma$ defined in (2·11).

Proposition 5·3. Let $\overline \gamma$ be the curve given by $\overline \gamma(t) = (\phi_H^t)^{-1}(\gamma(t))$ . Then

\begin{align*}\widetilde{\mathcal{A}} _H (\gamma) = - \int {\overline \gamma}^*\lambda + z(\overline \gamma (1)).\end{align*}

We will consider the variational study of the critical point problem of $\widetilde{\mathcal{A}}_H$ as a constrained variational problem. In this regard, we restrict $\widetilde{\mathcal{A}}_H$ to the Carnot path space $\mathcal L ^{\text {Carnot}}_{0}(J^1B, H;\; o_{J^1B})$ to make Hamilton’s equation $\dot y = X_H(t,y)$ as the critical point equation or as the equation of motion in the physics term.

Definition 5·4. We define the functional

\begin{align*}\mathcal A _H^{CC} \;:\; \mathcal L_0 ^{\text {Carnot}} (J^1B,H;\;o_{J^1B}) \to \mathbb R\end{align*}

to be the restriction of $\widetilde{\mathcal{A}} _H$ to $\mathcal L_0 ^{\text {Carnot}} (J^1B,H;\;o_{J^1B})$ . (Here ’CC’ stands for Carnot and Carathéodory.)

6. Vertical critical point analysis

We now specialise the discussion around (1·15) in Subsection 1·3 of the introduction to the function

\begin{align*}\mathcal A_H^{CC}\;:\;\mathcal L_0^{\text {Carnot}}(J^1B,H;\;o_{J^1B}) \to \mathbb R\end{align*}

of the following diagram

(6·1)

for the setting of generating functions: the vertical arrow map

\begin{align*}\Pi \;:\!=\; \pi_B \circ {ev}_{1} \;:\; \mathcal L_{0}^{\text {Carnot}}(J^1B, H;\; o_{J^1B}) \rightarrow B, \qquad \gamma \mapsto\pi_B\circ\gamma(1)\end{align*}

is a fibration and the horizontal arrow map is the real valued functional $\mathcal A_H^{CC}$ . We want to regard this diagram as the contact counterpart of the diagram associated to Weinstein’s de-approximation.

6·1. Vertical critical points: equation of motion

We first need to determine the vertical critical points of the functional $\mathcal A _H^{CC}$ of the fibration $\Pi$ above. By definition, we have

\begin{align*}VT_\gamma \mathcal E = T_\gamma(\Pi^{-1}(q))\end{align*}

for all elements $\gamma \in \mathcal L_0^{\text {Carnot}}(J^1B,H;\;o_{J^1B})$ satisfying the boundary condition

\begin{align*}\pi_B\circ\gamma(1) = q\end{align*}

at each given point $q \in B$ . Note that the fiber $\Pi^{-1}(q)$ at $q \in B$ is expressed by

(6·2) \begin{eqnarray}\Pi^{-1}(q) & = & \{ \gamma \in \mathcal L ^{\text {Carnot}}_{0}(J^1B, H;\; o_{J^1B}) \; \mid \; \gamma(1) \in J^1_q B \}\\& \;=\!:\; & \mathcal L ^{\text {Carnot}} (J^1B,H;\; o_{J^1B}, J^1_q B).\end{eqnarray}

Remark 6·1. We would like to attract the reader’s attention that while $o_{J^1B}$ is a Legendrian submanifold, the fiber $J_q^1B$ is a co-Legendrian submanifold of dimension $\dim B+1$ . We refer to [ Reference Oh37 ] for the definition.

Note that the first variation formula (2·13) gives the vertical derivative

(6·3) \begin{eqnarray}\delta^{\text {vert}} \mathcal A _H^{CC} (\gamma) (\eta)&=& -\int_0^1 d\lambda ((d\phi_H^t)^{-1} \eta(t), (d\phi_H^t)^{-1} (\dot \gamma (t) - X_H (\gamma (t)))) dt\\&{}& + e^{g_{(\psi^1_H)^{-1}}(\gamma(0))} \lambda_{\gamma(0)} (\eta(0)) - \lambda_{\gamma(1)} (\eta(1))+ dz_{\gamma(1)} (\eta(1)) \end{eqnarray}

for a variation $\eta \in T_\gamma \Pi^{-1}(q)$ . Note that

\begin{eqnarray*}- \lambda_{\gamma(1)} (\eta(1)) + dz_{\gamma(1)} (\eta(1))& = & \theta(\pi_{T^*B}(\gamma(1))) (d\pi_{T^*B}(\eta(1))\\& = & \pi_{T^*B}(\gamma(1))(d\pi_B(\eta(1))= \langle p(\gamma(1)), d\pi_B(\eta(1)) \rangle.\end{eqnarray*}

On the other hand we have

(6·4) \begin{equation}\eta (0) \in To_{J^1B} \subset \xi, \qquad \eta (1) \in T(J_q^1B)\end{equation}

by definition and so we obtain

\begin{align*}\lambda (\eta (0)) = 0, \qquad \langle p(\gamma(1)), d\pi_B(\eta(1)) \rangle = 0\end{align*}

for all $\eta \in T\Pi^{-1}(q)$ .

Combining the above discussion, the first variation formula is reduced to

\begin{align*}\delta^{\text {vert}} \mathcal A _H^{CC} (\gamma) (\eta)= -\int_0^1 d\lambda\left((d\phi_H^t)^{-1} \eta(t), (d\phi_H^t)^{-1} (\dot \gamma (t) - X_H (\gamma (t)))\right) dt.\end{align*}

It means that

(6·5) \begin{eqnarray} \delta^{\text {vert}} \mathcal A _H^{CC} (\gamma) \equiv 0&\Longleftrightarrow& ((d\phi_H^t)^{-1}(\dot \gamma (t) - X_H (\gamma(t)) ) )^\pi = 0 \\&\Longleftrightarrow& (d\phi^t_H)^{-1}(\dot \gamma (t) - X_H (\gamma(t)))= f(t) R_\lambda (\overline{\gamma} (t))\end{eqnarray}

for some function $f=f(t)$ . On the other hand, by definition any element $\gamma \in \mathcal L_{0}^{\text {Carnot}}(J^1B, H;\; o_{J^1B})$ satisfies

(6·6) \begin{equation}0 = \lambda \left(\dot \gamma (t)) + H_t (\gamma (t)\right)\!.\end{equation}

Furthermore, we also have

\begin{align*}\lambda(\dot \gamma (t)) + H_t (\gamma (t)) = \lambda\left(\dot \gamma (t) - X_H (\gamma (t))\right)\end{align*}

since $\lambda(X_H) = - H$ by definition of $X_H$ . By combining the last three equalities, we obtain

\begin{align*}0= \lambda\left(\dot \gamma (t) - X_H (\gamma (t))\right) = \lambda \left(f(t) d\phi^t_H(R_\lambda(\overline{\gamma}(t)))\right) = f(t) e^{g_{\phi_H^t} (\overline \gamma (t))} \end{align*}

and hence $ f(t) \equiv 0 $ .

We summarise the above discussion into the following.

Proposition 6·2 (Equation of motion). An element $\gamma \in \mathcal L ^{\text {Carnot}}_{0} (J^1B,H;\; o_{J^1B})$ is a vertical critical point of $\mathcal A _H^{CC}$ if and only if it is a contact Hamiltonian chord, that is, $\gamma$ satisfies

\begin{align*}\dot \gamma (t) = X_H(\gamma (t)), \quad \gamma(0) \in o_{J^1B}.\end{align*}

6·2. Vertical critical point set and horizontal derivative

We now consider the vertical critical point set of the diagram (6·1), which we denote by

\begin{eqnarray*}\Sigma_{\mathcal A _H^{CC}} & = &\left \{ \gamma \in \mathcal L ^{\text {Carnot}}_{0} (J^1B,H;\; o_{J^1B}) \; \mid \;\delta^{\text {vert}} \mathcal A _H^{CC} (\gamma) = 0 \right \}\\& = & \{ \gamma \in \mathcal L (J^1B) \; \mid \; \dot \gamma = X_H (\gamma), \;\gamma(0) \in o_{J^1B} \}.\end{eqnarray*}

We consider the restriction of $\Pi$ in (6·1) to the subset $\Sigma_{\mathcal A _H^{CC}}$ .

We write the fibration $\Pi$ given in (6·1) as $\Pi\;:\;\mathcal E \to B$ with

\begin{align*}\mathcal E\;:\!=\; \mathcal L^{\text {Carnot}}_0(J^1B,H;\;o_{J^1B})\end{align*}

and take an Ehresmann connection

\begin{align*}T\mathcal E = VT \mathcal E \oplus HT \mathcal E,\end{align*}

where the vertical tangent space of $\Pi$ at $\gamma$ is nothing but $\ker d_\gamma \Pi$ , and $HT_\gamma \mathcal E \cong T_{\pi_B(\gamma(1))}B$ . We have the associated splitting of the cotangent bundle

\begin{align*}T^*\mathcal E = \left(HT\mathcal E\right)^\perp \oplus \left(VT\mathcal E\right)^{\perp}\!,\end{align*}

where we denote by

\begin{align*}\left(HT_\gamma\mathcal E\right)^\perp, \, \, \left(VT_\gamma\mathcal E\right)^\perp \subset T_\gamma^*\mathcal E\end{align*}

the annihilators of $HT_\gamma\mathcal E, \, \, VT_\gamma\mathcal E \subset T_\gamma \mathcal E$ respectively. We mention that via the map $d(\pi_B \circ \operatorname{ev}_1)$ , $\left(HT_\gamma\mathcal E\right)^\perp$ can be identified with the vertical cotangent space, at least formally.

Definition 6·3 (The vertical cotangent bundle $\mathcal V \to \mathcal E$ ). We form the union

\begin{align*}\mathcal V = \bigcup_{\gamma \in \mathcal E} \{\gamma\} \times \mathcal V_\gamma\end{align*}

and regard it as a vector bundle over $\mathcal E$ with its fiber given by the vertical cotangent space identified with the fiber

\begin{align*}\mathcal V_\gamma \;:\!=\;\left(VT_\gamma\mathcal L^{\text {Carnot}}_0(J^1B,H;\;o_{J^1B})\right)^*.\end{align*}

We regard the map

\begin{align*}\gamma \mapsto \delta^{\text {vert}} \mathcal A _H^{CC}(\gamma)\end{align*}

as a section of the (infinite dimensional) vector bundle $\mathcal V \to \mathcal E$ . Obviously if the map $\delta^{\text {vert}} \mathcal A _H^{CC}\;:\; \mathcal E \to \mathcal V$ is transversal to the zero section, then $\Sigma_{\mathcal A _H^{\text{CC}}}$ becomes a smooth embedded submanifold of dimension n.

Even without the transversality hypothesis, we can still say something about the image by the $\delta \mathcal A_H^{\text{CC}}$ of $\Sigma_{\mathcal A_H^{\text{CC}}}$ as follows. Note that at each $\gamma \in \Sigma_{\mathcal A _H^{CC}}$ , we derive

(6·7) \begin{eqnarray}\delta (\mathcal A _H^{CC} |_{\Sigma_{\mathcal A _H^{CC}}}) (\gamma) (\eta)&=& p (\gamma(1)) dq_{\gamma(1)} (\eta(1)) \end{eqnarray}

(6·8) \begin{eqnarray} \mathcal A _H^{CC} (\gamma) & = & - \mathcal A _H (\gamma) + z(\gamma(1)) = z(\gamma(1))\end{eqnarray}

from the first variation formula (2·13) and the definition of $\mathcal A_H^{\text{CC}}$ . Therefore the map

\begin{align*}\iota_{\mathcal A _H^{CC}} \;:\; \Sigma_{\mathcal A _H^{CC}} \rightarrow J^1B\end{align*}

is well-defined and written as

\begin{eqnarray*}\iota_{\mathcal A _H^{CC}}(\gamma) & = & \left(\Pi (\gamma), \delta \mathcal A _H^{CC} (\gamma), \mathcal A _H^{CC} (\gamma)\right)\\& = & (q(\gamma (1)), p(\gamma (1)), z(\gamma (1))) = \gamma (1) \in \psi^1_H (o_{J^1B}).\end{eqnarray*}

Since any point $y \in \psi^1_H (o_{J^1B})$ can be written as $y = \gamma(1)$ by definition, we have proved the following general proposition without knowing that $\Sigma_{\mathcal A _H^{CC}}$ is a smooth manifold.

Proposition 6·4. For any H, the map $\iota_{\mathcal A _H^{CC}}$ is a bijective continuous map such that

\begin{align*}\operatorname{Image}\left(\iota_{\mathcal A _H^{CC}}\right) = \psi^1_H (o_{J^1B})\end{align*}

as a set.

7. Generating function transversality and the second variation

Although the map $\iota_{\mathcal A _H^{CC}}$ is a bijective continuous map, we cannot talk about its smoothness as a map unless we have some transversality result in advance so that $\Sigma_{\mathcal A _H^{CC}}$ carries a smooth structure. This proof of smoothness is now in order.

7·1. Carnot path space and gauge transformation

To study the aforementioned transversality, we need to consider the second variation of the action functional $\mathcal A_H^{\text{CC}}$ . We first recall the definition of the gauge transformation $\Psi_H=(\Phi_H)^{-1}$ given in Definition 2·9, and the identity (2·10)

\begin{align*}\mathcal A_H(\gamma) = \mathcal A_0(\overline \gamma)\end{align*}

for a general path $\gamma\;:\;[0,1] \to J^1B$ . Therefore we will work with the expression $\mathcal A_0(\overline \gamma)$ for our derivation of the second variation.

Lemma 7·1. A path $\gamma$ belongs to $\mathcal L^{\text {Carnot}}(J^1B,H)$ if and only if the gauge-transformed curve

\begin{align*}\overline \gamma(t) = (\phi_H^t)^{-1}(\gamma(t)) \, \left(= \Phi_H(\gamma)(t)\right)\end{align*}

is horizontal, i.e., $\dot{\overline \gamma}(t) \in \xi_{\overline \gamma(t)}$ .

Proof. A direct computation shows

\begin{align*}\frac{d}{dt} \overline \gamma(t) = d(\phi_H^t)^{-1}\left(\dot \gamma(t) - X_{H_t}(\gamma(t))\right)\!.\end{align*}

By evaluating $\lambda$ against this vector, we obtain

\begin{align*}\lambda(d(\phi_H^t)^{-1}(\dot \gamma(t) - X_{H_t}(\gamma(t))))= e^{g_{(\phi_H^t)^{-1}}(\gamma(t))}\left(\lambda(\dot \gamma) + H_t(\gamma)\right)\!.\end{align*}

Therefore $\lambda(\dot{\overline \gamma}) = 0 $ if and only if $\lambda(\dot \gamma) + H_t(\gamma) = 0$ . This finishes the proof.

Motivated by this lemma, we consider the set of horizontal curves

(7·1) \begin{equation}\mathcal L^\pi_0 (J^1B;\;\psi_H(o_{J^1B})) \;:\!=\; \{ \nu \mid \dot \nu \in \xi, \; \nu (0) \in \psi_H (o_{J^1B}) \}\end{equation}

so that the curves of the type $\nu = \overline \gamma$ is contained therein. The gauge transformation $\Psi_H$ and its inverse $\Phi_H$ relate the two path spaces

(7·2) \begin{equation}\mathcal L^\pi_0 (J^1B;\;\psi_H(o_{J^1B})) \stackrel{ \Psi_H}{\longrightarrow}\mathcal L^{\text {Carnot}}_0 (J^1B,H ;o_{J^1B})\end{equation}

in one-to-one correspondence, which is a Fréchet diffeomorphism with respect to the smooth structures given by Theorem 3·2.

7·2. Constrained second variation

In the rest of this section, we will mainly work with the space $\mathcal L^\pi_0 \left(J^1B;\;\psi_H(o_{J^1B})\right)$ of horizontal paths utilising the one-to-one correspondence (7·2). We define

(7·3) \begin{equation}\mathcal A^\pi(\nu) \;:\!=\; \mathcal A_H^{\text{CC}}\circ \Psi_H(\nu) = - \mathcal A_0 (\nu) + z(\nu(1)) = z(\nu(1)).\end{equation}

Recalling $\phi_H^1 = {{\mathrm{Id}}}$ , the vertical tangent vector $\eta \in VT_\nu (\mathcal L^\pi_0(J^1B;\;\psi_H(o_{J^1B})))$ satisfies the same boundary condition tangent to the fiber,

(7·4) \begin{equation}\eta(1) \in T_{\nu(1)}(J^1_qB),\end{equation}

where $q = \pi(\nu(1))$ .

To take the second variation of $\mathcal A^\pi$ , we take an Ehresmann connection as follows. Following [ Reference Le, Oh, Tortorella and Vitagliano26 , section 6], on a one-jet bundle $J^1B$ , we introduce a set of horizontal vector fields

\begin{align*}\frac{D}{\partial q_i} \;:\!=\; \frac{\partial}{\partial q_i} + p_i \frac{\partial}{\partial z}\end{align*}

which are tangent to $\xi$ . Together with ${\partial}/{\partial p_i}$ , the set $\left\{ {D}/{\partial q_i}, {\partial}/{\partial p_i} \right\}$ provides a Darboux frame of the contact distribution $\xi$ , i.e., a frame of $\xi$ satisfying $d\lambda( {D}/{\partial q_i}, {\partial}/{\partial p_j}) = \delta_{ij}$ on the chart of canonical coordinates $(q_i,p_i,z)$ . (In [ Reference Le, Oh, Tortorella and Vitagliano26 ], ${D}/{\partial q_i}$ is denoted by $D_i$ and there is a sign difference in the choice of canonical symplectic form on $T^*B$ .)

After gauge transformation, the diagram (6·1) is transformed to

(7·5)

Choice 7·2. (Choice of Ehresmann connection of $\Pi\circ \Psi_H$ ). We take the Ehresmann connection of $\Pi\circ \Psi_H$ in the diagram (7·5) so that the horizontal tangent space of its associated splitting

\begin{align*}T(\mathcal L^\pi_0(J^1B;\;\psi_H^1(o_{J^1B}))) = VT(\mathcal L^\pi_0(J^1B;\;\psi_H^1(o_{J^1B}))) \oplus HT(\mathcal L^\pi_0(J^1B;\;\psi_H^1(o_{J^1B})))\end{align*}

is given by

\begin{align*}HT_\nu (\mathcal L^\pi_0(J^1B;\;\psi_H(o_{J^1B}))) \;:\!=\;(d\Pi_\nu|_{HT_\nu \mathcal E})^{-1}\left( \operatorname{span} \left\{ \frac{D}{\partial q_j} \right\} \right)\!,\end{align*}

where we have

\begin{align*}\operatorname{span} \left\{ \frac{D}{\partial q_j} \right\} \subset \xi_{\nu(1)}, \; q=\pi_B(\nu(1))\end{align*}

locally (over the canonical Darboux chart at (q,p,z)), and then glue them together over B. See Subsection 1·2, especially the definition (1·11), for a relevant discussion on this description of horizontal space.

For the calculation of the covariant derivatives, we fix the triad metric defined by

\begin{align*}g(\cdot,\cdot) = d\lambda(\cdot, J\cdot) + \lambda \otimes \lambda,\end{align*}

associated to a contact triad $(J^1B,\lambda, J)$ , and use its associated Levi-Civita connection $\nabla$ .

Then the variation $\eta$ satisfies the final boundary condition

(7·6) \begin{equation}\eta(1) \in \operatorname{span} \left\{ \frac{D}{\partial q_j} \right\}\Big|_{\nu(1)} \cong HT_\nu (\mathcal L^\pi_0(J^1B;\;\psi_H(o_{J^1B})))\end{equation}

in terms of the aforementioned local Darboux local frame. We mention the isomorphism $HT_\nu (\mathcal L^\pi_0(J^1B;\;\psi_H(o_{J^1B}))) \cong T_{\pi_B(\nu(1))}B$ is induced by the parallel transport map followed by

\begin{align*}\frac{D}{\partial q_j} \mapsto \frac{\partial}{\partial q_j}\end{align*}

by considering the final value problem

(7·7) \begin{equation}\nabla_t\eta=0,\quad \eta(1)=v \in \operatorname{span} \left\{ \frac{D}{\partial q_j} \right\}\Big|_{\nu(1)}.\end{equation}

To study the aforementioned transversality condition, we need to study the second partial variation $\delta^{hor}\delta^{\text {vert}}\mathcal A^\pi(\nu)$ , first in the vertical and then in the horizontal direction, of $\mathcal A^\pi$ at $\nu$ .

We start with the following formula which is a special case of the formula (3·2).

Lemma 7·3. Let $\nu$ be a Reeb chord, and consider the tangent vector

\begin{align*} \eta \in T_u \mathcal L^\pi(J^1B;\;\psi_H(o_{J^1B})). \end{align*}

Then $\eta$ satisfies

(7·8) \begin{equation}d\lambda(\eta(t),\dot u (t)) + \frac{d}{dt}(\lambda(\eta(t))) = 0.\end{equation}

Proof. This follows from the formula (3·2) of the covariant variation with $H=0$ , which in turn follows from the defining equation of the space $\mathcal L^\pi(J^1B;\;\psi_H(o_{J^1B}))$ consisting of horizontal paths.

For the calculation of the mixed second derivative

\begin{align*}\delta^{hor}\delta^{\text {vert}}\mathcal A^\pi(\nu),\end{align*}

we first translate Proposition 6·2 into the following after gauge transformation.

Corollary 7·4. We have

\begin{align*}{\operatorname{Crit}} \delta^{\text {vert}}\mathcal A^\pi = \{\nu \mid \nu(t) \equiv \text{constant}\}.\end{align*}

Proposition 7·5. Let $q \in B$ be a given point and let $\gamma \in\mathcal L_0^{\text {Carnot}}(J^1B,H;\;o_{J^1B})$ and $\nu = \overline \gamma$ so that $\delta^{\text {vert}}\mathcal A^\pi(\nu)=0$ , i.e., $\nu$ is a horizontal Reeb chord starting from $\psi_H^1(o_{J^1B})$ satisfying $\nu(1) \in J^1_q B$ . Then we have

(7·9) \begin{equation}\delta^{hor} \delta^{\text {vert}} \mathcal A^\pi(\nu) (\eta_1, \eta_2)= \int_0^1 d\lambda\left(\nabla_t \eta_1, \eta_2 \right)\!dt \end{equation}

for any $\eta_1 \in VT_{\nu} \mathcal L^\pi_0 (J^1B;\;\psi_H(o_{J^1B})), \, \eta_2 \in HT_{\nu} \mathcal L^\pi_0 (J^1B;\;\psi_H(o_{J^1B}))$ .

Proof. Consider two variations of the types

(7·10) \begin{equation}\eta_1 \in VT_{\nu} \mathcal L^\pi_0 (J^1B;\;\psi_H(o_{J^1B})),\end{equation}

(7·11) \begin{equation}\eta_2 \in HT_{\nu} \mathcal L^\pi_0 (J^1B;\;\psi_H(o_{J^1B}))\end{equation}

and represent the latter $\eta_2$ by the variation satisfying $\nabla_t\eta_2=0$ . Since $\nu$ is a constant curve by Corollary 7·4, we derive

\begin{align*} \eta_2(t)\in \xi_y \end{align*}

from the final boundary condition $\eta(1)\in \xi_y$ for $y=\nu(1)\equiv \nu(t)$ . For we have

\begin{align*}\frac{d}{dt}\lambda_{\nu(t)}(\eta_2(t))=\frac{d}{dt}\lambda_y(\eta_2(t)) = \lambda_y(\nabla_t\eta_2)=0\end{align*}

which implies $\lambda_y(\eta_2(t))\equiv 0$ and hence follows the claim.

We take a two-parameter variation $\Gamma\;:\; (-\delta,\delta)^2 \to \mathcal L^\pi_0(J^1B;\;\psi_H(o_{J^1B}))$ with $\Gamma = \Gamma(s,u)$ such that

\begin{align*}\Gamma (0,0) = u, \quad\left. \frac{\partial \Gamma}{\partial s}\right|_{(s,u) = (0,0)} = \eta_1, \quad\left. \frac{\partial \Gamma}{\partial u}\right|_{(s,u) = (0,0)} = \eta_2.\end{align*}

For convenience, we also write $\Gamma(s,u)(t)$ as $\Gamma(s,u,t)$ . Then using the definition (7·3) we compute

(7·12) \begin{equation}\frac{\partial}{\partial s} \mathcal A^\pi (\Gamma(s,0))= dz\left(\frac{\partial \Gamma}{\partial s}(s,0,1)\right)= \lambda\left(\frac{\partial \Gamma}{\partial s}(s,0,1)\right)\!.\end{equation}

Here the second equality follows from the condition that $({\partial \Gamma}/{\partial s})(s,0,1)$ be a vertical variation which implies it satisfies the boundary condition (7·4). Then using (7·12), we compute the second derivative

(7·13) \begin{equation}\left. \frac{\partial^2}{\partial u\partial s}\right|_{(s,u) = (0,0)} \mathcal A^\pi(\Gamma(s,u))= \left. \frac{\partial}{\partial u} \left[ \lambda\left(\frac{\partial \Gamma}{\partial s}(s,u,1)\right)\right] \right|_{(s,u)=(0,0)}.\end{equation}

From (7·8), we derive

\begin{eqnarray*} \lambda \left(\frac{\partial \Gamma}{\partial s}(s,u,1)\right) \Big|_{(s,u)=(0,0)}& = & \int_0^1 \frac{\partial}{\partial t} \left(\lambda \left(\frac{\partial \Gamma}{\partial s}(s,u,t)\right) \right)\!dt\Big|_{(s,u)=(0,0)}\\[5pt] & = & \int_0^1 d\lambda \left( \frac{\partial \Gamma}{\partial t}(s,u,t),\frac{\partial \Gamma}{\partial s}(s,u,t) \right) dt \Big|_{(s,u)=(0,0)},\end{eqnarray*}

where for the first equality we also used the vanishing

\begin{align*} \lambda \left(\frac{\partial \Gamma}{\partial s}(s,u,0)\right) \Big|_{(s,u)=(0,0)} = \lambda(\eta_1(0)) = 0\end{align*}

by the boundary condition (6·4). Therefore by swapping u and s in (7·13), we can rewrite

\begin{eqnarray*}\left. \frac{\partial^2}{\partial u\partial s}\right|_{(s,u) = (0,0)} \mathcal A^\pi(\Gamma(s,u)) & = &\left. \frac{\partial^2}{\partial s\partial u }\right|_{(s,u) = (0,0)} \mathcal A^\pi(\Gamma(s,u))\\[5pt] & = & \left. \frac{\partial}{\partial s} \int_0^1 d\lambda\left( \frac{\partial \Gamma}{\partial t}(s,u,t), \frac{\partial \Gamma}{\partial u}(s,u,t) \right) dt \right|_{(s,u)=(0,0)}.\end{eqnarray*}

We compute

\begin{eqnarray*}&& \left. \frac{\partial}{\partial s} \int_0^1 d\lambda\left( \frac{\partial \Gamma}{\partial t}(s,u,t), \frac{\partial \Gamma}{\partial u}(s,u,t) \right) dt \right|_{(s,u)=(0,0)}\\[5pt] &= & \left. \int_0^1 (\nabla_s d\lambda)\left( \frac{\partial \Gamma}{\partial t}(s,u,t), \frac{\partial \Gamma}{\partial u}(s,u,t) \right) dt \right|_{(s,u)=(0,0)} \\[5pt] & & + \left. \int_0^1 d\lambda\left( \nabla_s \frac{\partial \Gamma}{\partial t}(s,u,t), \frac{\partial \Gamma}{\partial u}(s,u,t) \right) dt \right|_{(s,u)=(0,0)} \\[5pt] & & + \left. \int_0^1 d\lambda\left( \frac{\partial \Gamma}{\partial t}(s,u,t), \nabla_s \frac{\partial \Gamma}{\partial u}(s,u,t) \right) dt \right|_{(s,u)=(0,0)} \\[5pt] &=& \left. \int_0^1 d\lambda\left( \nabla_s \frac{\partial \Gamma}{\partial t}(s,u,t), \frac{\partial \Gamma}{\partial u}(s,u,t) \right) dt \right|_{(s,u)=(0,0)} \\[5pt] &=& \int_0^1 d\lambda\left( \left. \nabla_s \frac{\partial \Gamma}{\partial t}(s,u,t) \right|_{(s,u)=(0,0)}, \eta_2(t) \right) dt.\end{eqnarray*}

Here the second equality follows from the fact that $\left. ({\partial \Gamma}/{\partial t})\right|_{(s,u)=(0,0)} = \dot u$ , and Corollary 7·4 which implies that $\nu$ is a constant curve valued in $\psi^1_H(o_{J^1B}) \cap J^1_q B$ . Moreover, using the torsion freeness of $\nabla$ , we derive

\begin{align*}\nabla_s \frac{\partial \Gamma}{\partial t} \Big|_{(u,s) = (0,0)} = \nabla_t \eta_1.\end{align*}

Therefore this completes the proof of (7·9).

To verify our desired transversality, it is sufficient to prove surjectivity of the partial Hessian

\begin{align*}\delta^{hor} \delta^{\text {vert}} \mathcal A^\pi (\nu),\end{align*}

since $\delta^{hor} \delta^{\text {vert}} \mathcal A^\pi(\nu)$ represents its horizontal part. This is verified in the following lemma.

Lemma 7·6. $\operatorname{Coker} \delta^{hor} \delta^{\text {vert}} \mathcal A^\pi (\nu) = 0.$

Proof. Recall $\nu(t) \equiv y$ for all $t \in [0,1]$ . Suppose $\eta_2 \in HT_u \mathcal L^\pi_0 (J^1B;\; \psi_H(o_{J^1B}))$ satisfies that

\begin{align*}\delta^{hor} \delta^{\text {vert}} \mathcal A^\pi (\eta_1,\eta_2) = 0\end{align*}

for all $\eta_1 \in VT_u \mathcal L^\pi_0 (J^1B;\; \psi_H(o_{J^1B}))$ . Then we have

(7·14) \begin{equation}\int_0^1 d\lambda\left(\nabla_t \eta_1, \eta_2 \right)\!dt = 0\end{equation}

for all $\eta_1\in VT_u \mathcal L^\pi_0 (J^1B;\; \psi_H(o_{J^1B}))$ . Using $\nabla_t \eta_2 = 0$ , we rewrite

\begin{eqnarray*}0 & = &\int_0^1 d\lambda\left(\nabla_t \eta_1, \eta_2 \right)\!dt =\int_0^1 \frac{d}{dt}\left(d\lambda(\eta_1, \eta_2) \right)\!dt\\[5pt] & = & d\lambda\left(\eta_1(1), \eta_2(1) \right) - d\lambda\left(\eta_1(0), \eta_2(0) \right)\!.\end{eqnarray*}

Therefore we obtain

(7·15) \begin{equation} d\lambda\left(\eta_1(1), \eta_2(1) \right) = d\lambda\left(\eta_1(0), \eta_2(0) \right)\end{equation}

for all variation $\eta_1 \in VT_u \mathcal L^\pi_0 (J^1B;\; \psi_H(o_{J^1B}))$ . By definition, $\eta_i$ $i=1,2$ satisfy

\begin{align*}\eta_i(0) \in T_{\nu(0)}\psi_H^1(o_{J^1B})=T_y\psi_H^1(o_{J^1B})\end{align*}

we have

\begin{align*}d\lambda(\eta_1(0),\eta_2(0))=0\end{align*}

since $\psi_H^1(o_{J^1B})$ is Legendrian. Therefore it follows from (7·15) and the vanishing $R_\lambda \mathbin{\lrcorner} d\lambda = 0$ that

\begin{align*}d\lambda(\eta_1(1),\eta_2(1))=0\end{align*}

for all $\eta_1(1) \in \xi_y$ . By the nondegeneracy of $d\lambda$ on $\xi_y$ , this proves $\eta_2^\pi(1)=0$ . Since $\eta_2$ is assumed to be in $HT_u \mathcal L^\pi_0 (J^1B;\; \psi_H(o_{J^1B}))$ , $\eta_2(1) = 0$ . Since $\eta_2$ is parallel, it implies $\eta_2= 0$ . This finishes the proof of the lemma.

Corollary 7·7. The transversality of $\delta^{\text {vert}}\mathcal A^{CC}_H$ to the zero section always holds. In particular, $\Sigma_{\mathcal A^{CC}_H}$ is a smooth embedded submanifold of dimension n.

Part II. Contact analogue of Laudenbach–Sikorav’s generating function

In this part, we give a construction of ‘canonical’ ${{\mathrm{GFQI}}}$ of any Hamiltonian isotope of the zero section $o_{J^1B}$ of the one-jet bundle $J^1B$ as a finite dimensional approximation of contact Hamiltonian trajectories as the contact counterpart of Laudenbach–Sikorav’s broken Hamiltonian trajectory approximation given in [ Reference Laudenbach and Sikorav27 ]. We will imitate the construction from [ Reference Laudenbach and Sikorav27 , section 2] by suitably adapting it to the current contact setting of the one-jet bundle $J^1B \cong T^*B \times \mathbb R$ .

Since the general procedure of the two constructions is similar [ Reference Laudenbach and Sikorav27 , section 2], we will focus on the different points and modifications we need to handle the current Legendrian case.

8. Generating functions of Lagrangian submanifolds: review

In this section we review basic results on the generating functions quadratic at infinity (GFQI) of Legendrian submanifolds from [ Reference Sikorav44, Reference Théret46, Reference Viterbo48 ] with applications to spectral invariants of the Viterbo-type in our mind via the stable Morse theory. GFQI is a particular type of a Morse family in Definition 1·13 defined on a vector bundle which enables one to carry out the Morse theory via iterative handle attaching operations [ Reference Chaperon9, Reference Laudenbach and Sikorav27, Reference Sikorav44, Reference Théret46, Reference Traynor47, Reference Viterbo48 ] or via the Morse homology construction as in [ Reference Milinković28, Reference Milinković29 ].

8·1. Summary of GFQI construction

Let $\pi_E\;:\; E \to B$ be a vector bundle and $S\;:\; E \to \mathbb R$ be a function that is quadratic at infinity, i.e., there exists a fiberwise nondegenerate quadratic function $Q\;:\;E \to \mathbb R$ such that $S = Q$ outside a compact subset of E. We define the subset of $\mathbb{R}$ ,

(8·1) \begin{equation}\operatorname{Spec}(S) = \{S(e) \in \mathbb R \mid dS(e) = 0 \}\end{equation}

and call it the spectrum of S. Since B is assumed to be compact and S is quadratic at infinity, $\operatorname{Spec}(S) \subset \mathbb R$ is a compact subset of measure zero in general.

Consider Legendrian submanifolds R in the 1-jet bundle $J^1B$ of a smooth manifold B. Denote by $\mathfrak{Leg}(J^1B)$ (resp. $\mathfrak{Leg}_0(J^1B)$ ) the set of Legendrian submanifolds of $J^1B$ (resp. those Legendrian isotopic to the zero section $R_0 = o_{J^1B}$ ).

A generic R transversely intersects the ‘zero wall’

\begin{align*}Z\;:\!=\; o_{T^*B} \times \mathbb R \subset J^1B = T^*B\times \mathbb R.\end{align*}

Let $\pi_R\;:\; R \to T^*B$ be the natural Lagrangian projection which is the restriction to R of $\pi\;:\;J^1B \to T^*B$ . The map $\pi|_R\;:\; R \to T^*B$ is an exact Lagrangian immersion. We denote by

\begin{align*}\mathfrak{Leg}^{\text{nd}}_\star(J^1B)\end{align*}

the set consisting of R’s intersecting the zero wall $Z = o_{T^*B} \times \mathbb R$ transversely.

The following definition is standard in the 1-jet bundle and its generic structure is well-known.

Definition 8·1. The wave front of $R \subset J^1B$ is the image of the front projection $ R \subset J^1B \to B \times \mathbb R$ .

Now let $S\;:\; E\to \mathbb R$ be a smooth function. We denote by $d^vS\;:\; E \to E^*$ the fiber derivative of S, i.e., the map whose value at $e \in E$ is defined by

(8·2) \begin{equation}d^vS(e)(\xi) = \frac{d}{ds}\Big|_{s=0} S(e + s \xi)\end{equation}

for $\xi \in E_{\pi(e)}$ . This defines nonlinear bundle map from E to $E^*$ .

With respect to an affine connection of E, we have the splitting

\begin{align*}TE = VTE \oplus HTE.\end{align*}

Under the canonical identification $VT_e E \cong E_{\pi(e)}$ , the value $d^vS(e)$ of the vertical derivative $d^vS$ at $e \in E$ can be canonically identified with the fiber derivative $dS(e)|_{VT_e E}$ . We define

\begin{align*}\Sigma_S \;:\!=\; (d^{v}S)^{-1}(0)\end{align*}

and call it the fiber-critical set of S. Under the transversality condition

(8·3) \begin{equation}d^vS \pitchfork o_{E^*} \Longleftrightarrow dS|_{VTE} \pitchfork o_{(VT)^*E},\end{equation}

$\Sigma_S$ is a smooth manifold of dimension equal to $\dim B$ . We recall the horizontal derivative map $\iota_S$ from (1·11)

\begin{align*}\iota_{S}(e) = \left(\pi(e), D^hS(e) \circ (d\pi_e|_{HT_eE})^{-1}, S(e)\right)\end{align*}

which does not depend on the connection as a linear map

(8·4) \begin{equation}D^h S(e) = dS(e)|_{HT_eE} \;:\; HT_eE \to \mathbb R\end{equation}

at any zero point of $d^vS$ , i.e, at $e \in \Sigma_S$ . Composing this with the natural isomorphism $HT_e E \cong T_{\pi(e)}B$ , $D^h S(e)$ can be naturally identified with an element in $T_{\pi(e)}^*B$ . We denote this map by

\begin{align*}\iota_S \;:\; \Sigma_S \subset E \to J^1B\end{align*}

(8·5) \begin{equation}\iota_S(e) = \left(\pi(e), dS(e) \circ (d\pi|_{HT_eE})^{-1}, S(e)\right)\!.\end{equation}

We emphasise that $\iota_S$ is still well-defined as a map without assuming transversality condition given in (8·3). Under the transversality condition, the map $\iota_S$ becomes an immersion which is also Legendrian.

Remark 8·2. We remark that the map $\iota_S$ is not necessarily an embedding but only an immersion in general for the symplectic case of cotangent bundles even when S is nondegenerate. However by the dimensional reason, the set of S whose generating Legendrian submanifolds are embedded can be shown to be open and residual in the set of GFQI. (See [ Reference Oh and Park40 ] for the proof of this last statement.)

Definition 8·3. Let $S\;:\; E \to \mathbb R$ be a smooth function. We say S generates a Legendrian submanifold $R \subset J^1B$ , if R is the image of a Legendrian immersion $\iota_S$ defined above, and call S a generating function of R. If, in addition, S is quadratic at infinity, we call it a generating function quadratic at infinity ( ${{\mathrm{GFQI}}}$ ).

The following existence theorem for ${{\mathrm{GFQI}}}$ is proved by Laudenbach–Sikorav [ Reference Laudenbach and Sikorav27, Reference Sikorav44 ] for the symplectic case (see also [ Reference Théret46 ]), and generalised to the contact case by Chaperon [ Reference Chaperon9 ] and Chekanov [ Reference Chekanov10 ].

Proposition 8·4 (Existence). Let $R \in \mathfrak{Leg}_0(J^1B)$ . Then there exists a generating function $S \;:\; E \to \mathbb R$ that generates R in the sense that

\begin{align*}R = \iota_S( \Sigma_S )\end{align*}

as explained above. Furthermore, since R is compact, we may choose S so that it is quadratic at infinity.

It is also proved by Viterbo and Théret in [ Reference Théret46, Reference Viterbo48 ] that the following uniqueness result holds:

Proposition 8·5 (Uniqueness). Let B and R be as above in Proposition 8·4. Suppose $S\;:\;E \to \mathbb R$ and $S'\;:\;E' \to \mathbb R$ are GFQI of R. Then S’ can be obtained from S by applying a sequence of the following operations:

1. (i) (Fiber-preserving diffeomorphism) $S' = S \circ \Phi$ for some fiber-preserving diffeomorphism $\Phi\;:\;E' \to E$ ;

2. (ii) (Stabilisations) $S' = S + Q\;:\; E' = E \oplus F \to \mathbb R$ , where $F \to B$ is a vector bundle and $Q\;:\; F \to \mathbb R$ is a fiberwise nondegenerate quadratic form.

Definition 8·6. We say two ${{\mathrm{GFQI}}}$ $S,\, S'$ are equivalent if S’ is obtained from S by applying the operations of (i) and (ii) a finite number of times.

Remark 8·7. We would like to recall that there is another operation, addition by a constant, $S' = S +c$ for some $c \in \mathbb R$ . In the symplectic-Lagrangian context, this operation should be also contained in the definition of equivalence because it does not change the generated Lagrangian submanifold in that context. However it changes the generated Legendrian submanifold in the contact-Legendrian context. It will be important to keep track of this operation of adding a constant in the study of spectral invariant of Legendrian submanifolds as that of [ Reference Oh and Yu41 ] and [ Reference Bhupal8, Reference Sandon42 ].

9. Broken trajectory approximation space E of $\mathcal L^{\text {Carnot}}(J^1B,H;\;o_{J^1B})$

Assume that B is a closed manifold and consider its one-jet bundle $M = J^1B$ equipped with the canonical contact form $\lambda = dz - pdq$ . For the simplicity of notation, we often simply write the base projection of $J^1B$ and of $T^*B$ by

\begin{align*}\pi_B(y)\;=\!:\; q(y) \quad \text{and }\, \pi_B(x) = q(x)\end{align*}

respectively. We fix a Riemannian metric g on B and its associated Levi-Civita connection. All norms on TB and $T^*B$ will be computed in terms of this fixed metric g, and the exponential map on B is in terms of the Levi-Civita connection.

For given integer $N \geq 1$ , we consider partitions of the interval [0,1] into N subintervals

\begin{align*}0 = t_0 \lt t_1 \lt \cdots \lt t_N =1, \quad t_i = \frac{i}{N}.\end{align*}

Let $\gamma \in \mathcal L^{\text {Carnot}}(J^1B,H;\;o_{J^1B})$ . By definition, we have

(9·1) \begin{equation}0 = \lambda(\dot \gamma) + H_t(\gamma(t)) = \big(\dot z - p(\gamma)(\dot q) + H_t\big)(\gamma(t)).\end{equation}

When $\gamma$ is a Hamiltonian trajectory, this is reduced to (5·2) after integration over $t \in [0,1]$ .

Remark 9·1. Consider a Hamiltonian trajectory $\gamma$ from $y_0$ to $y_1$ . Then we can express $\gamma$ in two ways:

\begin{align*}\gamma(t) = \psi^t_H(y_0) = \phi^t_H(y_1).\end{align*}

Unlike before, we use the symbol $\psi$ in this section because we frequently use Hamiltonian trajectories starting from some given points denoted by $y^+$ .

For each $i=1,\ldots,N$ , we define a Hamiltonian isotopy over the interval $[t_{i-1},t_i]$ by

(9·2) \begin{equation}\psi_{H;i}^t\;:\!=\;\psi_H^t(\psi_H^{t_{i-1}})^{-1}\end{equation}

in accordance with the convention explained in the remark above. We mention $\psi_{H;i}^{t_{i-1}} = {{\mathrm{Id}}}$ for all i. Therefore the reparameterized path

(9·3) \begin{equation}t \in [0,1] \mapsto \psi_{H;i}^{t_{i-1} + \frac{t}{N}}\end{equation}

can be made uniformly as close to ${{\mathrm{Id}}}$ as possible on [0,1] for all $i = 1, \cdots, N$ by letting $N \to \infty$ . Starting from $y_0\;:\!=\; (q_0,0) = \gamma_1(0) \in o_{J^1B}$ , we will concatenate $\{\gamma_i\}$ into a broken trajectory with jumps at $t_i$ ’s. For the simplicity of notation in our later calculations, we set

(9·4) \begin{equation}\psi_i\;:\!=\;\psi_{H;i}^{t_i}.\end{equation}

Once such N is chosen, we fix it for the rest of the construction, while allowing further enlargement of N later if necessary.

As in [ Reference Laudenbach and Sikorav27 ], we now consider the vector bundle

(9·5) \begin{equation}E = (TB)^{\oplus (N-1)} \oplus (T^*B)^{\oplus (N-1)} \to B.\end{equation}

We denote a point $e \in E$ by

\begin{align*}e = (q_0, X, P),\end{align*}

where $q_0 \in B$ ,

(9·6) \begin{equation}X \in (T_{q_0}B)^{\oplus(N-1)}, \quad X = (X_1, \ldots, X_{N-1}),\end{equation}

and

(9·7) \begin{equation}P \in (T^*_{q_0} B)^{\oplus(N-1)}, \quad P = (P_1, \ldots, P_{N-1}).\end{equation}

We put the initial condition of each broken trajectory $\gamma$ with

\begin{align*}\gamma(0) = (q_0, 0, 0) \;=\!:\; y_0 \in o_{J^1B}\subset J^1B.\end{align*}

To associate a broken trajectory to each element $(q_0,X,P)$ of E, we will use the exponential map associated with the Riemannian metric g. For this reason, we need to ensure that each $X_k \in T_{q_0}B$ has its norm less than the injectivity radius at $q_0$ . We define the $L_\infty$ -norm of $X=(X_1,\ldots,X_{N-1})$ by

\begin{align*}\|X\|_\infty \;:\!=\; \max\{\|X_k\|_g\mid 1\leq k \leq N-1\},\end{align*}

where $\|X_k\|_g$ denotes the norm induced by the metric g. Let $\iota_g$ be the injectivity radius of (B, g). Since B is compact, $\iota_g \gt 0$ . We fix such a constant $\epsilon_0 \gt 0$ such that $\epsilon_0 \lt \iota_g$ , and define a neighborhood

(9·8) \begin{equation}U \;:\!=\; \{(q_0,X,P)\in E \mid \|X\|_\infty \lt\epsilon_0\}\end{equation}

of the zero section of E. We mention that there is no constraint on our choice co-vectors P. We will further specify the constant $\epsilon_0$ later so that Proposition 13·5 below holds.

For the clarity of our later exposition we will adopt the different expression

\begin{align*}e = (q_0, \widetilde{X}, \widetilde{P}) \in U\end{align*}

for the element e from U, while the letters without ‘tilde’ will be reserved for those from the whole E, not restricted to U.

Step 1. Recall that $[t_0,t_1]=[0,1/N]$ . We define the reparameterization

(9·9) \begin{equation}\gamma_1(t)\;:\!=\; (q_1(t),p_1(t),z_1(t)) \;:\!=\; \psi_H^{t_0 + \frac{t}{N}}(q_0,0,0) \in J^1B,\qquad t\in[0,1], \quad t_0 = 0\end{equation}

of the segment of the Hamilonian trajectory $t \mapsto \psi_H^t(q_0,0,0)$ restricted to the first subinterval $[t_0,t_1]$ . We denote the initial condition by $y_0\;:\!=\;(q_0,0,0)\in o_{J^1B}$ , and express the endpoint of the trajectory (recalling $t_0 = 0$ ) as

\begin{eqnarray*}y_0^+ \;:\!=\; \psi_H^{t_0}(y_0) = \gamma_1(0) = (q_0,0,0) & \;=\!:\; &(q_0^+,p_0^+, z_0^+) \\[5pt] y_1^- \;:\!=\; \psi_H^{t_1}(y_0) = \gamma_1(1) & \;=\!:\; & (q_1^-, p_1^-,z_1^-)\end{eqnarray*}

componentwise.

Let $\widetilde{x}_1$ and $\widetilde{p}_1$ be the parallel transports of $\widetilde{X}_1 \in T_{q_0}B$ and $\widetilde{P}_1 \in T_{q_0}^*B$ , respectively, along the base path $q_1(t)$ with respect to the Levi-Civita connection. We will extensively use the parallel transports along the short geodesics in the rest of the paper. For this purpose, we assume that $d(q_0,q_1) \lt \iota_g$ where $\iota_g$ is the injectivity radius of the metric g. We denote by $\exp_q\;:\; T_qB \to B$ the exponential map issued at q, and denote by

\begin{align*}\operatorname{Exp}\;:\; TB \to B\end{align*}

the collective exponential map

(9·10) \begin{equation}\operatorname{Exp}(q,v)\;:\!=\; \exp_q(v)\end{equation}

and its fiberwise inverse

(9·11) \begin{equation}\text{E}(q,q')\;:\!=\; \exp_q^{-1}(q')\end{equation}

for any pair (q, q’) with $d(q,q') \lt \iota_g$ .

We will also utilise the following standard exponential estimates. (See [ Reference Karcher21 ], for example.)

Lemma 9·2. Assume B is compact or the metric g is of bounded geometry, i.e., bounded curvature and injectivity radius $\iota_g \gt 0$ . Denote by $D_1{\operatorname{Exp}}(x,v), \, d_2{\operatorname{Exp}}(x,v)$ the first and second partial derivatives of ${\operatorname{Exp}}$ at $(x,v) \in TB$ . Then there exists $C \gt 0$ depending only on g and $\delta \gt 0$ below such that

(9·12) \begin{eqnarray} |D_1{\operatorname{Exp}}(x, v)(u)| &\leq& C|v|\, |u|, \end{eqnarray}

(9·13) \begin{eqnarray} |d_2{\operatorname{Exp}}(x,v)(u)| & \leq & C|v|\, |u|, \end{eqnarray}

and similarly for the map E, provided $|v| \leq \iota_g - \delta$ with $0 \lt \delta \lt \frac12 \iota_g$ .

Now we define a path $c_1 \;=\!:\;c_{(\widetilde X_1,\widetilde P_1)}\;:\;[0,1]\to T^*B$ by

(9·14) \begin{equation}c_1(t) = \left(\exp_{q_1^-}(t\widetilde{x}_1),((d\exp_{q_1^-}(t\widetilde{x}_1)^*)^{-1}(\widetilde{p}_1)\right)\end{equation}

and consider its horizontal lift $\mu_1(t)\;:\;[0,1]\to J^1B$ with respect to $\xi$ , i.e., $\dot{\mu}_1(t) \in \xi_{\mu_1(t)}$ for each $t \in [0,1]$ . This lift is given by

(9·15) \begin{equation}\mu_1(t) = (c_1(t),z_1(t)), \quad z_1(t) = z_1^- + \int_{c_1|_{[0,t]} }p\, dq.\end{equation}

In particular, we have the formulae for its end points,

\begin{align*}\mu_1(0)=(q_1^-,\widetilde{p}_1,z_1^-),\qquad \mu_1(1)=(q_1^+,p_1^+,z_1^+),\end{align*}

where

(9·16) \begin{equation}\begin{cases} q_1^+ = \exp_{q_1^-}(\widetilde{x}_1), \\p_1^+ = ((d\exp_{q_1^-}(\widetilde{x}_1))^*)^{-1}(\widetilde{p}_1),\\z_1^+ = z_1^- + \int_{c_1} p\, dq.\end{cases}\end{equation}

We ‘concatenate’ them and define this concatenation, denoted by $\gamma_1 \# \mu_1$ , to be the piecewise continuous curve

(9·17) \begin{equation}\gamma_1 \# \mu_1(t) =\begin{cases} \gamma_1(2t) \quad & 0 \leq t \lt \dfrac12\\[5pt]\mu_1(2t-1) \quad & \dfrac12 \leq t \leq 1.\end{cases}\end{equation}

This concatenation may have a jump-discontinuity at $t= 1/2$ . We denote this concatenated curve by

(9·18) \begin{equation}\widetilde \gamma_{(\widetilde X_1,\widetilde P_1)}\;:\!=\; \gamma_1 \# \mu_1\end{equation}

to emphasise its dependence on $(\widetilde X_1,\widetilde P_1)$ . We emphasise that

\begin{align*}\gamma_{(\widetilde X_1,\widetilde P_1)}(0) = (\psi_H^{t_0})^{-1}(\gamma_1(0)) = (q_0,0,0) \in o_{J^1B}\end{align*}

(recalling $t_0 = 0$ ). We denote by $d_1$ the curve defined by

(9·19) \begin{equation}d_1(t)\;:\!=\; \pi_{T^*B}(\gamma_1(t)).\end{equation}

Then

(9·20) \begin{equation}\Pi_{q(d_1)}^{-1}(\widetilde x_1,\widetilde p_1) = (\widetilde X_1,\widetilde P_1)\in T_{q_0}B \oplus T_{q_0}^*B\end{equation}

by the definition of $(\widetilde x_1,\widetilde p_1)$ . This is the end of the first step.

In summary, we collect the four key points in Step 1 here for later reference:

(9·21) \begin{equation}\begin{cases}\gamma_1(0) = (q_0^+, p_0^+,z_0^+) , \quad &\gamma_1(1) = (q_1^-, p_1^-,z_1^-), \\[5pt]\mu_1(0) = (q_1^-, \widetilde p_1, z_1^-),\quad &\mu_1(1) = (q_1^+,p_1^+,z_1^+).\end{cases}\end{equation}

Remark 9·4. Note that the initial point of $\mu_1$ is given by $\mu_1(0)=(q_1^-,\widetilde{p}_1,z_1^-)$ , and the end point of the Hamiltonian trajectory $\gamma_1$ is given by

\begin{align*} \gamma_1(1)=(q_1^-,p_1^-,z_1^-).\end{align*}

The two points share the same q- and z-coordinates but generally differ in p-coordinate. As a result, the two curves $\gamma_1$ and $\mu_1$ are not continuously concatenated in general as a single trajectory in $J^1B$ . However, we will see later similarly as in Laudenbach–Sikorav’s case [ Reference Laudenbach and Sikorav27 ] that these two curves will be continuously joined together forming a single Hamiltonian trajectory for each vertical critical element of the broken-trajectory approximation we define below.

Then the concatenation $d_1 \# c_1\;:\;[0,1] \to T^*B$ has the formula

\begin{align*}d_1\# c_1(t) = \pi_{T^*B} \left(\widetilde \gamma_{(\widetilde X_1, \widetilde P_1)}(t)\right)\end{align*}

by the definitions (9·14), (9·19) and (9·18).

Step 2. As in Step 1, we now consider the second segment $[t_1,t_2] = [1/N,2/N]$ and define the reparameterization

(9·22) \begin{equation}\gamma_2(t) \;:\!=\; (q_2(t), p_2(t), z_2(t)) \;:\!=\; \psi_{H;2}^{t_1 + \frac{t}{N}}(y_1^+) \in J^1B, \qquad t\in[0,1].\end{equation}

We denote its initial point by

\begin{align*}y_1^+ \;:\!=\; (q_1^+, p_1^+, z_1^+) = \gamma_2(0).\end{align*}

Recall that $\psi_{H;2}^t=\psi_H^t\circ(\psi_H^{t_1})^{-1}$ . We write the endpoint of the trajectory as

\begin{align*}y_2^- \;:\!=\; \gamma_2(1) = \psi_H^{t_2} (\psi_H^{t_1})^{-1} (y_1^+) \;=\!:\; (q_2^-, p_2^-, z_2^-)\end{align*}

componentwise. We consider the parallel transport from $t=0$ to $t=1$ along the curve $q((d_1 \# c_1) \# d_2)$ . Then we define $\widetilde{x}_2$ and $\widetilde{p}_2$ by the equation

(9·23) \begin{equation}(\widetilde x_2, \widetilde p_2) = \Pi_{q((d_1 \# c_1) \# d_2)}(\widetilde X_2, \widetilde P_2),\end{equation}

with the parallel transports of $\widetilde{X}_2 \in T_{q_0}B$ and $\widetilde{P}_2 \in T_{q_0}^*B$ to $q_2^-$ respectively, along the concatenated base path consisting of $q_1(t)$ , the projection of $c_1(t)$ and $q_2(t)$ in B. As before, we define the path $c_2$ in $T^*B$ using the exponential map at $q_2^-$ , and its horizontal lift $\mu_2$ in $J^1B$ given by

\begin{eqnarray*}\mu_2(t) \;:\!=\; (c_2(t),z_2(t)), \quad z_2(t) = z_2^- + \int_{c_2|_{[0,t]} }p\, dq, \\c_2(t) = \left(\exp_{q_2^-}(t\widetilde{x}_2),((d\exp_{q_2^-}(t\widetilde{x}_2))^*)^{-1}(\widetilde{p}_2)\right) \;=\!:\; c_{(\widetilde X_2,\widetilde P_2)}(t).\end{eqnarray*}

In particular, we have

\begin{align*}\mu_2(0)=(q_2^-,\widetilde{p}_2,z_2^-),\qquad \mu_2(1)=(q_2^+,p_2^+,z_2^+),\end{align*}

where

\begin{align*}\begin{cases} q_2^+ = \exp_{q_2^-}(\widetilde{x}_2), \\[5pt] p_2^+ = ((d\exp_{q_2^-}(\widetilde{x}_2))^*)^{-1}(\widetilde{p}_2),\\[5pt] z_2^+ = z_2^- + \int_{c_2} p\, dq.\end{cases}\end{align*}

Then we define

(9·24) \begin{equation}\widetilde \gamma_2(t) = \widetilde \gamma_{(\widetilde X_2,\widetilde P_2)}(t)\;:\!=\; \gamma_2 \# \mu_2(t).\end{equation}

In summary, we collect the four key points in $J^1B$ in Step 2 here:

(9·25) \begin{equation}\begin{cases}\gamma_2(0) = \mu_1(1) = (q_1^+,p_1^+,z_1^+), \quad &\gamma_2(1) = (q_2^-, p_2^-,z_2^-), \\[5pt] \mu_2(0) = (q_2^-, \widetilde p_2, z_2^-),\quad & \mu_2(1) = (q_2^+,p_2^+,z_2^+).\end{cases}\end{equation}

Step 3 (Induction).

We now construct a sequence of contact Hamiltonian trajectories and horizontal curves over the interval $[t_{k-1},t_k]$ , for each $k = 3, \ldots, N$ , generalising the construction of previous steps.

Suppose we have defined $\widetilde x_i$ , $\widetilde p_i$ and the paths $c_i$ , $z_i$ and $\mu_i = (c_i,z_i)$ for $i = 1, \cdots, k-1$ . Then we will construct all of them for $i = k$ .

Repeating the operation performed in Step 2 iteratively, we denote by $d_i$ the curve defined by $d_i(t)\;:\!=\; \pi_{T^*B}(\gamma_i(t))$ and consider the concatenation $d_i \# c_i\;:\;[0,1] \to T^*B$ . Then we consider the concatenation

(9·26) \begin{equation}(d_1\# c_1) \# (d_2 \# c_2) \# \cdots \# (d_{k-1} \# c_{k-1}) \#d_k \;=\!:\; (d \# c)_k\end{equation}

and consider the parallel transport $\Pi_{q((d \# c)_k)}$ from $t=0$ to $t=1$ defined similarly as in Step 2. We alert readers that our notation $(d \# c)_k$ should not be confused with $d_k \# c_k$ .

We define the Hamiltonian trajectory $\gamma_k\;:\;[0,1] \to J^1B$ by

(9·27) \begin{equation}\gamma_k(t)\;:\!=\; \psi_{H;k}^{t_{k-1} + \frac{t}{N}} (y_{k-1}^+),\quad t\in[0,1],\end{equation}

and denote its endpoint by

(9·28) \begin{equation}y_k^- \;:\!=\; \gamma_k(1).\end{equation}

We also write $\gamma_k(t) = (q_k(t), p_k(t), z_k(t))$ and $y_k^-= (q_k^-,p_k^-,z_k^-)$ respectively componentwise.

Definition 9·5 (Coordinates $\widetilde x_k$ and $\widetilde p_k$ ). We define $\widetilde{x}_k$ and $\widetilde{p}_k$ are the parallel transports of $\widetilde{X}_k$ and $\widetilde{P}_k$

(9·29) \begin{equation}(\widetilde x_k,\widetilde p_k) =\Pi_{q((d \# c)_k)}(\widetilde X_k,\widetilde P_k)\end{equation}

along the base path $q((d\#c)_k)$ of the concatenation $(d\#c)_k$ .

Then we define $c_k\;:\;[0,1]\to T^*B$ to be the path defined by

(9·30) \begin{equation}c_k(t) = \left(\exp_{q_k^-}(t\widetilde{x}_k),((d\exp_{q_k^-}(t\widetilde{x}_k))^*)^{-1}(\widetilde{p}_k)\right)\end{equation}

and compute its horizontal lift $\mu_k\;:\;[0,1]\to J^1B$ by

(9·31) \begin{equation}\mu_k(t) = (c_k(t),z_k(t)), \quad z_k(t) = z_k^- + \int_{c_k|_{[0,t]} }p\, dq.\end{equation}

We set the endpoint $y_k^+\;:\!=\;\mu_k(1)=(q_k^+,p_k^+,z_k^+)$ , where explicitly:

(9·32) \begin{equation}\begin{cases}q_k^+ = \exp _{q_k^-} (\widetilde{x}_k), \\[5pt] p_k^+ = ((d \exp _{q_k^-} (\widetilde{x}_k))^*)^{-1}(\widetilde{p}_k), \\[5pt] z_k^+ = z_k^- + \int_{c_k} p\, dq.\end{cases}\end{equation}

We define

(9·33) \begin{equation}\widetilde \gamma_k = \gamma_k \# \mu_k \;=\!:\; \widetilde \gamma_{(\widetilde X_k,\widetilde P_k)}.\end{equation}

Putting together all the Hamiltonian trajectories and horizontal lifts constructed above, we obtain a collection of curves

\begin{align*}\{\gamma_1,\mu_1,\ldots,\gamma_{N-1},\mu_{N-1},\gamma_N\}\end{align*}

that form a broken trajectory in $J^1B$ , depending on the data

\begin{align*} (\widetilde{X}_j,\widetilde{P}_j), \quad j =1,\ldots,N-1.\end{align*}

This construction is illustrated schematically in Figure 1.

This broken trajectory can be represented as a piecewise smooth curve

(9·34) \begin{eqnarray}\widetilde \gamma_{(q_0,\widetilde X,\widetilde P)} & \;:\!=\;& (\widetilde \gamma_1 \#\widetilde \gamma_2 \# \cdots \#\widetilde \gamma_{N-1}) \# \gamma_N \\& = & \left(\gamma_1 \# \mu_1\right) \# \left(\gamma_2 \# \mu_2\right) \# \cdots \#\left(\gamma_{N-1} \# \mu_{N-1}\right) \# \gamma_N\end{eqnarray}

whose derivative can make a jump at each operation $\#$ inside the parenthesis depending on the datum $(\widetilde X_j,\widetilde P_j)$ from E for $i = 1, \ldots N-1$ .

In summary, we collect the four key points in $J^1B$ in Step $N-1$ , the last iteration, here:

(9·35) \begin{equation}\begin{cases}\gamma_{N-1}(0) = \mu_{N-2}(1) = (q_{N-2}^+,p_{N-2}^+,z_{N-2}^+), \quad &\gamma_{N-1}(1) = (q_{N-1}^-, p_{N-1}^-,z_{N-1}^-), \\[5pt] \mu_{N-1}(0) = (q_{N-1}^-, \widetilde p_{N-1}, z_{N-1}^-),\quad & \mu_{N-1}(1)= (q_{N-1}^+,p_{N-1}^+,z_{N-1}^+).\end{cases}\end{equation}

Step 4. Coordinate change. We now mention that by construction all $\mu_k$ , $\gamma_k$ , $q_k^{\pm}$ , $p_k^{\pm}$ , etc., are functions of $(q_0,\widetilde X,\widetilde P)$ for $k = 0, \ldots, N-1$ . The above construction iterates up to Step $N\!-\!1$ and then ends with $\gamma_N$ , the N-th $\gamma$ -segment: there exists no Nth $\mu$ -segment $\mu_N$ .

We will view all $\gamma$ -segments as functions of $(q_0,\widetilde X, \widetilde P)$ . For example, we have

\begin{align*} q_{k+1}^- = q(\gamma_{k+1}(1)) = q(\psi_{k+1}(y_k^+)), \end{align*}

where the map

\begin{align*} \psi_{k+1} = \psi_{H;k+1}^{t_{k+1}} \;:\!=\; \psi_H^{t_{k+1}}\circ(\psi_H^{t_k})^{-1} \end{align*}

is given in (9·4) and (9·2). The assignment $y_k^+ \mapsto q_{k+1}^-$ is a smooth function of $y_k^+ = (q_k^+,p_k^+,z_k^+)$ which is determined by the rescaled Hamiltonian trajectory of H along the segment $\gamma_{k+1}$ . We will next show that the map

(9·36) \begin{equation}(q_0,\widetilde{X},\widetilde{P}) \mapsto \left(q_1^-, \widetilde p_1, \ldots, q_{N-1}^-, \widetilde p_{N-1}, q_N^-\right)\!.\end{equation}

defines a coordinate change by proving invertibility of its Jacobian.

10. Jacobian of coordinate change

We need to study the derivatives of the map (9·36) with respect to $(q_0,\widetilde X,\widetilde P)$ or equivalently with respect to its parallel transport

\begin{align*}(q_0,\widetilde x,\widetilde p) = \left(q_0,(\widetilde x_1,\widetilde p_1), (\widetilde x_2,\widetilde p_2),\cdots, (\widetilde x_{N-1},\widetilde p_{N-1})\right)\!.\end{align*}

To avoid confusion and for the clarity of exposition, we henceforth denote the derivatives thereof by

\begin{align*}\delta \mu_k, \,\,\delta \gamma_k, \, \, \delta q_k^{\pm}, \, \,\delta p_k^{\pm}\end{align*}

etc., respectively. Then we can write its variation as

(10·1) \begin{equation}\delta q^-_{k+1} = D_1(q\circ \psi_{k+1})\delta q^+_k + d_2(q\circ \psi_{k+1})\delta p^+_k+ d_3(q\circ \psi_{k+1})\delta z^+_k. \end{equation}

Here, we use the notation $D_1,d_2,d_3$ to denote partial derivatives with respect to the first, the second, and the third variables of a given map, respectively. (See the remarks right after Lemma 9·2 for relevant remarks.)

For the simplicity of exposition, we consider the linear map from $T_{q_k^-}B$ to $T_{q_{k+1}^-}B$ defined by

(10·2) \begin{equation}D^k \;:\!=\; D_1(q\circ \psi_{k+1})d\exp_{q^-_k}(\widetilde{x}_k)+ d_3(q\circ \psi_{k+1})\langle \widetilde{p}_k, \cdot \rangle.\end{equation}

The following is a key element of the proof of Proposition 13·5.

Proof. It follows from the compact support assumption on H that for a sufficiently large N, the $C^1$ distance between the time-one map $\psi_{k+1}$ and the identity map is of order $1/N$ for each k: This can be easily seen by considering $\psi_{k+1}$ as the time-one map of the reparameterization

(10·3) \begin{equation}t \in [0,1] \mapsto \psi_{H;k+1}^{t_k + \frac{t}{N}}\end{equation}

of the isotopy

\begin{align*}t \in [t_k, t_{k+1}] \mapsto \psi_{H;k+1}^t=\psi_H^t \circ (\psi_H^{t_k})^{-1}.\end{align*}

Note that the rescaled isotopy (10·3) is generated by the Hamiltonian

(10·4) \begin{equation} {\operatorname{Dev}}_\lambda \left(t \mapsto \psi_{H;k+1}^{t_k + \frac{t}{N}}\right) = \frac1N H_{(k+t)/N}. \end{equation}

(See Definition 2·3 for the definition of ${\operatorname{Dev}}_\lambda$ .) Then it follows from (2·6) that

(10·5) \begin{equation}\|D_1(q\circ \psi_{k+1}) -\Pi\| \leq \frac{C_1}{N}, \, \quad \|d_3(q\circ \psi_{k+1})\| \leq \frac{C_2}{N}\end{equation}

for some universal constant $C_1, \, C_2 \gt 0$ : Here $\Pi$ is the parallel transport along the unique short geodesic from $q_k^-$ to $q_{k+1}^-$ . Uniqueness of the short geodesic is guaranteed if the distance between them is smaller than the injectivity radius $\iota_g$ , which holds when N is large enough.

On the other hand, it also follows from the compact support hypothesis of H that there exists a sufficiently large $R_0\gt 0$ such that

\begin{align*}\operatorname{supp} H \subset D_{R_0}(J^1B),\end{align*}

where we put

(10·6) \begin{equation}D_{R}(J^1B)\;:\!=\; D_{R}(T^*B) \times [0,R],\end{equation}

where $D_{R}(T^*B)$ is the disc-bundle of radius R.

When $\| \widetilde P_k \| \gt R_0$ , we have

(10·7) \begin{equation}d_3(q\circ \psi_{k+1}) = 0.\end{equation}

By using the fact that $d\exp_{q_k^-}(0) = {{\mathrm{Id}}}$ and by the standard exponential estimate arising from Lemma 9·2, we obtain

\begin{align*}\|d\exp_{q^-_k}(\widetilde x_k) - \Pi \| \leq C\|\widetilde x_k\| \leq C \epsilon_0,\end{align*}

for some $C\gt0$ where $\Pi$ is a similar parallel transport from $q_k^-$ to $\exp_{q_k^-}(\widetilde x_k)$ . In particular,

(10·8) \begin{equation}\|d\exp_{q^-_k}(\widetilde x_k) \| \geq (1 - C \epsilon_0)\end{equation}

for any $\widetilde x_k$ satisfying $\|\widetilde x_k\| \lt \epsilon_0$ .

SubLemma 10·2. We have

\begin{align*}\|D^k - \Pi\| \leq \frac12\end{align*}

for all $k = 1, 2, \ldots$ , provided $0 \lt \epsilon_0 \lt 1/{4C}$ and $N \gt 4(2C_1 + C_2 R_0)$ .

Proof. It follows from (10·2) that

\begin{align*}\|D^k - \Pi\| \leq \|D_1(q\circ \psi_{k+1}) \cdot d\exp_{q^-_k}(\widetilde x_k) - \Pi\|+ \|d_3(q\circ \psi_{k+1})\langle \widetilde p_k, \cdot \rangle \|.\end{align*}

By (10·7), we obtain

\begin{align*}\| d_3(q\circ \psi_{k+1})\langle \widetilde p_k, \cdot \rangle \| \leq \frac{C_2}{N} \cdot R_0.\end{align*}

By the inequalities (10·5) and (10·8), we can write

\begin{align*} D_1(q\circ \psi_{k+1}) = \Pi + B_1, \quad d\exp_{q^-_k}(\widetilde x_k) = {{\mathrm{Id}}} + B_2 \end{align*}

with $\|B_1\| \leq {C_1}/{N}$ and $\|B_2\| \leq C \epsilon_0$ . Then we expand

\begin{align*}D_1(q\circ \psi_{k+1}) \cdot d\exp_{q^-_k}(\widetilde x_k) - \Pi = (\Pi + B_1)\cdot ({{\mathrm{Id}}} + B_2) - \Pi= B_1 + \Pi \cdot B_2 + B_1 \cdot B_2.\end{align*}

Therefore we have

\begin{eqnarray*}\|D_1(q\circ \psi_{k+1}) \cdot d\exp_{q^-_k}(\widetilde x_k) - \Pi\|& \leq & (\|(B_1 \| + \|\Pi \cdot B_2\| + \|B_1 \cdot B_2\|)\\& \leq & C \epsilon_0 + \left(\frac{C_1}{N} \cdot C \epsilon_0\right) \\& \leq & 2C \epsilon_0 + \frac{2C_1}{N},\end{eqnarray*}

where the last inequality follows whenever $\epsilon_0\lt 1, \, N \gt 1$ . Combining the above two inequalities, we have derived

\begin{align*}\|D^k - \Pi\| \leq 2C \epsilon_0 + \frac{2C_1}{N} + \frac{C_2}{N} \cdot R_0 = 2C \epsilon_0 + \frac{2C_1 + C_2 R_0}{N}.\end{align*}

Therefore if we first choose $0 \lt \epsilon_0 \lt 1/{4C}$ and $N \gt 4(2C_1 + C_2 R_0)$ , then $\|D^k - \Pi\| \leq \frac12$ which finishes the proof of the sublemma.

As a corollary, we have shown that $D^k$ is invertible, if we choose

(10·9) \begin{equation} 0 \lt \epsilon_0 \lt \frac1{4C}, \quad N \gt 4(2C_1 + C_2 R_0).\end{equation}

This finishes the proof of Lemma 10·1.

11. A fiberwise re-scaling transformation on $E \to B$

In the next section, we will define a finite dimensional approximation $S \;:\; E \to \mathbb R$ of the action functional $\mathcal A_H^{\text{CC}}$ . Since the broken trajectories are well-defined only in the open neighborhood $U \subset E$ , we need to find a way of defining them globally on E, without affecting the vertical critical point set.

We fix $0 \lt \epsilon_0 \lt \iota_g$ that was chosen in Section 9. Then for each $0\lt \delta \ll \epsilon_0$ , we choose a smooth function $\rho_{\delta,\epsilon_0}\;:\; \mathbb R_{\geq 0} \to (0,\infty)$ of the form

(11·1) \begin{equation}\begin{cases}\rho_{\delta,\epsilon_0} (r) = 1 \quad & {for } \quad 0\leq r \leq \delta, \\0 \lt r \rho_{\delta,\epsilon_0} (r) \lt \epsilon_0\quad & {for } \quad 0 \leq r \lt \infty,\\r \rho_{\delta,\epsilon_0}'(r) + \rho_{\delta,\epsilon_0}(r) \gt 0 \quad & {for} \quad 0 \leq r \lt \infty.\end{cases}\end{equation}

Proof. Instead of constructing $\rho_{\delta,\epsilon_0}$ directly, we first construct a smooth strictly increasing function

\begin{align*}f_{\delta,\epsilon_0}(r) \;:\!=\; r\rho_{\delta,\epsilon_0}(r).\end{align*}

The conditions in (11·1) are equivalent to requiring

\begin{align*}f_{\delta,\epsilon_0}(r) = r \quad \text{for } 0\le r\le \delta,\qquad 0 \lt f_{\delta,\epsilon_0}(r) \lt \epsilon_0 \quad \text{for } r\ge0,\qquad f'_{\delta,\epsilon_0}(r) \gt0.\end{align*}

To construct such a function, consider

(11·2) \begin{equation}f(r)\;:\!=\; \begin{cases}r \quad & r \in [0,\delta],\\A_{\epsilon_0} \arctan(r) \quad & r \in [\epsilon_0,\infty)\end{cases}\end{equation}

with a choice of $A_{\epsilon_0} = {2 \epsilon_0}/{\pi}$ . Then for all $r\ge0$ we have

\begin{align*}0 \le f(r) \lt \epsilon_0.\end{align*}

We compute

\begin{align*}f'(r) = \frac{A_{\epsilon_0}}{1+r^2} \gt0\end{align*}

on $[\epsilon_0,\infty)$ . We also note

\begin{align*}f(\epsilon_0) = \frac{2\epsilon_0}{\pi}\arctan(\epsilon_0).\end{align*}

Since $f(\delta)=\delta$ , we can make $\delta \lt f(\epsilon_0)$ by taking $\delta$ sufficiently small. Therefore any smooth monotone interpolation of r and $A_{\epsilon_0}\arctan(r)$ on the intermediate region $[\delta,\epsilon_0]$ produces a smooth strictly increasing function $f_{\delta,\epsilon_0}$ satisfying the above conditions. Finally we define

\begin{align*}\rho_{\delta,\epsilon_0}(r) =\begin{cases}1 & r=0,\\ \dfrac{f_{\delta,\epsilon_0}(r)}{r} & r\gt0.\end{cases}\end{align*}

Then $\rho_{\delta,\epsilon_0}$ is smooth on $[0,\infty)$ and satisfies (11·1). This completes the proof.

We consider a family of maps

\begin{align*}\psi_{\delta,\epsilon_0}\;:\; TB \oplus T^*B \to D_{\epsilon_0}(TB) \times_B T^*B\end{align*}

(11·3) \begin{equation}\psi_{\delta,\epsilon_0}(q,v,\beta) = \left(q, \rho_{\delta,\epsilon_0}(|v|) \, v,\left(\rho_{\delta,\epsilon_0}(|v|)\right)^{-1} \beta\right)\end{equation}

which is a fiberwise re-scaling diffeomorphism and preserves the natural pairing

(11·4) \begin{equation}\langle \cdot, \cdot \rangle\;:\; TB \oplus T^*B \to \mathbb R.\end{equation}

We will first prove that the Jacobian of $\psi_{\delta,\epsilon_0}$ is invertible and hence it is a local diffeomorphism. Here $\times_B$ is the fiber product of $\pi_B\;:\; TB \to B$ and $\pi_B^*\;:\; T^*B \to B$ .

We denote by (X, P) and $(\widetilde X,\widetilde P)$ an element of the domain $TB\oplus T^*B$ and $D_{\epsilon_0}(TB) \times_B T^*B$ respectively. Recall that we cannot control the size of $\widetilde P$ where these variables are defined. Note that as $\delta \to 0$ , $\|\widetilde P\|$ can be arbitrarily large, while we will make the vectors $\widetilde{X}$ satisfy

(11·5) \begin{equation}\|\widetilde{X}\| = \rho_{\delta,\epsilon_0}(\|X\|) \|X\| \lt \epsilon_0.\end{equation}

We will fix a sufficiently small $\delta \gt 0$ and the associated function $\rho_{\delta,\epsilon_0}$ above so that the inequality (11·5) holds once and for all.

For the later purpose, we state the following obvious derivative formula of the map $\psi_{\delta,\epsilon_0}$ .

Lemma 11·2. Let $q \in B$ and take a local frame $\{e_i\}_{i=1}^n$ of TB and its dual frame $\{f_j\}_{j=1}^n$ of $T^*B$ near a given point $q \in B$ . We regard X, P (resp. $\widetilde X$ and $\widetilde P$ ) as vectors of $\mathbb R^n$ and its dual space. Put $r\;:\!=\; \|X\|$ . Then we have:

(11·6) \begin{equation}\begin{cases}\dfrac{\partial X \circ \psi_{\delta,\epsilon_0}}{\partial X} = \dfrac{ \rho_{\delta,\epsilon_0}'(r)}{r}\, X_jX_k \,|e_j\rangle \langle e_k|+ \rho_{\delta,\epsilon_0}(r)\, \delta_{jk}\,|e_j\rangle \langle e_k|, \quad \dfrac{\partial X\circ \psi_{\delta,\epsilon_0}}{\partial P} = 0, \\[12pt] \dfrac{\partial P \circ \psi_{\delta,\epsilon_0}}{\partial X} = - \dfrac{\rho_{\delta,\epsilon_0}'(r)}{ \rho_{\delta,\epsilon_0}^2(r)r}\,X_jX_k\, |e_j\rangle\langle f_k|, \quad \dfrac{\partial P \circ \psi_{\delta,\epsilon_0}}{\partial P} = \dfrac{1}{\rho_{\delta,\epsilon_0}(r)}\, \delta_{jk}\, |f_j\rangle \langle f_k|,\end{cases}\end{equation}

where we use Dirac’s bra-ket notation and Einstein’s summation convention.

Writing $\psi_{\delta,\epsilon_0}(X,P) \;=\!:\; (\widetilde X,\widetilde P)$ , we note that the Jacobian matrix

(11·7) \begin{equation}(D\psi_{\delta,\epsilon_0}) = \left (\begin{matrix} \dfrac{\partial \widetilde X}{\partial X} & \dfrac{\partial \widetilde X}{\partial P} \\[10pt] \dfrac{\partial \widetilde P}{\partial X} & \dfrac{\partial \widetilde P}{\partial P} \end{matrix}\right) = \left (\begin{matrix} \dfrac{\partial \widetilde X}{\partial X} &0 \\[10pt] \dfrac{\partial \widetilde P}{\partial X} & \dfrac{\partial \widetilde P}{\partial P} \end{matrix}\right)\end{equation}

is a triangular matrix.

With this preparation, we are ready to prove the invertibility of the map $\psi_{\delta,\epsilon_0}$ .

Proposition 11·3. The fiberwise map

\begin{align*}\psi_{\delta,\epsilon_0}\;:\; TB \oplus T^*B \to D_{\epsilon_0}(TB) \times_B T^* B\end{align*}

is a diffeomorphism onto its image provided $\epsilon_0$ is sufficiently small.

Proof. In (11·6), we have computed the Jacobian of $\psi_{\delta,\epsilon_0}$ . It follows that the matrix

\begin{align*}\frac{\partial P \circ \psi_{\delta,\epsilon_0}}{\partial P} = \frac{1}{\rho_{\delta,\epsilon_0}(r)}\, \delta_{jk}|f_j\rangle \langle f_k|= \frac{1}{\rho_{\delta,\epsilon_0}(r)}\, {{\mathrm{Id}}}\end{align*}

is invertible. We also obtain

\begin{eqnarray*}\frac{\partial X \circ \psi_{\delta,\epsilon_0}}{\partial X} & = & \frac{ \rho_{\delta,\epsilon_0}'(r)}{r}\, X_jX_k \,|e_j\rangle \langle e_k|+ \rho_{\delta,\epsilon_0}(r)\, \delta_{jk}\, |e_j\rangle \langle e_k|\\& = & \frac{ \rho_{\delta,\epsilon_0}'(r)}{r}\, |X\rangle \langle X|+ \rho_{\delta,\epsilon_0}(r)\, {{\mathrm{Id}}}\end{eqnarray*}

therefrom. We will prove this matrix is also invertible. For this purpose, we will show that its kernel is zero. To prove this, let v satisfy $({\partial X \circ \psi_{\delta,\epsilon_0}}/{\partial X})(v) = 0$ , i.e.,

\begin{equation*}0 = \frac{ \rho_{\delta,\epsilon_0}'(r)}{r}\, |X\rangle \langle X|v \rangle+ \rho_{\delta,\epsilon_0}(r) |v \rangle. \end{equation*}

Therefore we obtain

\begin{align*}v = - \frac{ \rho_{\delta,\epsilon_0}'(r)}{r\rho_{\delta,\epsilon_0}(r)}\,\langle X|v \rangle X.\end{align*}

In particular v is parallel to X. Therefore by taking the inner product of the last equation with X, we obtain

\begin{align*}\langle X|v \rangle = - \frac{ \rho_{\delta,\epsilon_0}'(r)}{r\rho_{\delta,\epsilon_0}(r)}\,\langle X|v \rangle \|X\|^2.\end{align*}

If $v eq 0$ , then we cancel $\langle X|v \rangle$ away and obtain

\begin{align*}0 = \frac{ \rho_{\delta,\epsilon_0}'(r)}{r}\, r^2+ \rho_{\delta,\epsilon_0}(r) =r\rho_{\delta,\epsilon_0}'(r)+\rho_{\delta,\epsilon_0}(r) .\end{align*}

This contradicts to the inequality $r \rho_{\delta,\epsilon_0}'(r) + \rho_{\delta,\epsilon_0}(r) \gt 0$ from (11·1). This proves that the matrix ${\partial X \circ \psi_{\delta,\epsilon_0}}/{\partial X}$ is invertible. Combining the above with the triangular form of the Jacobian (11·7), we have shown that the Jacobian of $\psi_{\delta,\epsilon_0}$ is invertible and so it is a local diffeomorphism.

It remains to prove that the map $\psi_{\delta,\epsilon_0}$ is one to one. Suppose $\psi_{\delta,\epsilon_0}(q, X,P) = \psi_{\delta,\epsilon_0}(q',X',P')$ . Since $\pi_B \circ \psi_{\delta,\epsilon_0} = {{\mathrm{Id}}}$ , we have $q' = q$ . Then we have

(11·8) \begin{equation}\rho_{\delta,\epsilon_0}(\|X\|)X = \rho_{\delta,\epsilon_0}(\|X'\|)X', \quad \rho_{\delta,\epsilon_0}(\|X\|)^{-1} P = \rho_{\delta,\epsilon_0}(\|X'\|)^{-1} P'\end{equation}

so that X and X’ are parallel, more specifically, we have

\begin{align*}X' = \frac{\rho_{\delta,\epsilon_0}(\|X\|)}{\rho_{\delta,\epsilon_0}(\|X'\|)} X \;=\!:\; c X\end{align*}

for some constant c. We have now only to prove $c = 1$ . By taking the norm of the above, we derive

\begin{align*}\frac{r'}{r} =\frac{\rho_{\delta,\epsilon_0}(r)}{\rho_{\delta,\epsilon_0}(r')}, \quad r = \|X\|, \, r'= \|X'\|.\end{align*}

Since $r\rho_{\delta,\epsilon_0}(r)$ is monotonically increasing, this equality implies $r = r'$ , i.e., $c = 1$ which in turn implies that $X = X'$ and $\rho_{\delta,\epsilon_0}(\|X\|) = \rho_{\delta,\epsilon_0}(\|X'\|)$ . Finally this together with the second equation of the standing hypothesis (11·8) also implies $P = P'$ .

Therefore combining them all, $\psi_{\delta,\epsilon_0}$ is one to one and so concludes the proof of Proposition 11·3.