2.1. Spin groups

For $n \geq 2$, the universal (double) cover of $\mathrm{SO}_{n+1}$ is the spin group $\mathrm{Spin}_{n+1}$; let $\Pi_{n+1} : \mathrm{Spin}_{n+1} \rightarrow \mathrm{SO}_{n+1} $ be the natural projection. The group $\mathrm{Spin}_{n+1}$ is therefore a simply connected Lie group, which is also compact, and it has the same Lie algebra (and hence the same dimension) as $\mathrm{SO}_{n+1}$. Let us denote by $\mathbf 1$ the identity element in $\mathrm{Spin}_{n+1}$, and by −1 the unique non-trivial element in $\mathrm{Spin}_{n+1}$ such that $\Pi_{n+1}(-\mathbf 1)=I$.

We will give a brief description of $\mathrm{Spin}_{n+1}$ in the cases n = 2 and n = 3 since it will be fundamental in this work. It is well known that $\mathrm{Spin}_3 \simeq \mathbb S^3$ and $\mathrm{Spin}_4 \simeq \mathbb S^3 \times \mathbb S^3$.

First, let us recall the definition of the algebra of quaternions:

\begin{equation*} \mathbb{H}:=\{a\mathbf 1+b\mathbf i+c\mathbf j+d\mathbf k \; | \; (a,b,c,d) \in \mathbb{R}^4\}, \end{equation*}

where $\mathbf 1 \in \mathbb{H}$ is the multiplicative unit, and $\mathbf i,\mathbf j,\mathbf k$ satisfy the product rules $ \mathbf i^2=\mathbf j^2=\mathbf k^2=\mathbf i\mathbf j\mathbf k=-1$. As a real vector space, $ \mathbb{H}$ is isomorphic to $\mathbb{R}^4$, hence one can define a Euclidean norm on $\mathbb{H}$ and the set of quaternions with unit norm, $U(\mathbb{H})$, can be naturally identified with $\mathbb S^3$. The space of imaginary quaternions (i.e. of real part 0) is naturally identified with $\mathbb{R}^3$.

The natural projection $\Pi_3 : \mathrm{Spin}_3 \rightarrow \mathrm{SO}_3$ is given by $\Pi_3(z)h = zh\bar{z}$, for any $h \in \mathbb{R}^3$. The natural projection $\Pi_4 : \mathrm{Spin}_4 \rightarrow \mathrm{SO}_4$ is given by $\Pi_3(z_l,z_r)q = z_lq\bar{z_r}$, for any $q \in \mathbb{R}^4$. For a matrix-notation approach of these objects we refer the reader to [Reference Alves and Saldanha2, section 2.1].

2.2. Bruhat cells and the Coxeter–Weyl group

We denote by $\mathrm{Up}^+_{n+1}$ the group of upper triangular matrices with positive diagonal entries and by $\mathrm{Up}^1_{n+1} \subset \mathrm{Up}^+_{n+1}$ the subgroup of upper triangular matrices with diagonal entries equal to one.

The map $B : \mathrm{Up}^1_{n+1} \times \mathrm{SO}_{n+1} \mapsto \mathrm{SO}_{n+1}$ defined by $ B(U,Q)=UQU' $ where Uʹ is the unique matrix in $\mathrm{Up}^+_{n+1}$ such that $UQU' \in \mathrm{SO}_{n+1}$ is called the Bruhat action. The Bruhat action is clearly a group action of $\mathrm{Up}^1_{n+1}$ on $\mathrm{SO}_{n+1}$, and we call its finitely many orbits the (signed) Bruhat cells. It turns out that, two matrices $Q \in \mathrm{SO}_{n+1}$ and $Q' \in \mathrm{SO}_{n+1}$ belong to the same Bruhat cell if and only if there exist U and Uʹ in $\mathrm{Up}^+_{n+1}$ such that $Q'=UQU'$. In other words, given $Q \in \mathrm{SO}_{n+1}$ the Bruhat cell of Q is the set of matrices $UQU' \in \mathrm{SO}_{n+1}$, where U and Uʹ belong to $\mathrm{Up}^+_{n+1}$. We denote by $\mathrm{Bru}_Q$ the (signed) Bruhat cell of $Q \in \mathrm{SO}_{n+1}$.

Let $\mathrm{B}_{n+1} \subset \mathrm{O}_{n+1}$ the Coxeter–Weyl group of signed permutations matrices, so $|\mathrm{B}_{n+1}| = 2^{n+1}(n+1)!$. Let $\mathrm{B}_{n+1}^+ = \mathrm{B}_{n+1} \cap \mathrm{SO}_{n+1}$ the group of signed permutations matrices of determinant one, so $|\mathrm{B}_{n+1}^+| = 2^{n}(n+1)!$. Given a signed permutation matrix in $\mathrm{B}_{n+1}^+$, we can drop its entries signs and define a homomorphism from $\mathrm{B}_{n+1}^+$ to the symmetric group $S_{n+1}$. Each Bruhat cell contains a unique signed permutation matrix $P \in \mathrm{B}_{n+1}^+$; it follows that, given any two Bruhat cells, associated with two distinct signed-permutation matrices, they are disjoint. Therefore, we have the Bruhat decomposition of $\mathrm{SO}_{n+1}$:

\begin{equation*} \mathrm{SO}_{n+1}=\bigsqcup_{P \in \mathrm{B}_{n+1}^+}\mathrm{Bru}_P, \end{equation*}

and there are $2^n(n+1)!$ different Bruhat cells.

Since the group $\mathrm{Up}_{n+1}^1$ is contractible, its Bruhat action on $\mathrm{SO}_{n+1}$ lifts to a Bruhat action on $\mathrm{Spin}_{n+1}$ that, for simplicity, we still denote by $B : \mathrm{Up}_{n+1}^1 \times \mathrm{Spin}_{n+1} \rightarrow \mathrm{Spin}_{n+1}$. As before, the Bruhat cells on $\mathrm{Spin}_{n+1}$ are the orbits of the Bruhat action.

Let us denote by $\tilde{\mathrm{B}}_{n+1}^+ = \Pi_{n+1}^{-1}(\mathrm{B}_{n+1}^+) \subset \mathrm{Spin}_{n+1}$, so $|\tilde{\mathrm{B}}_{n+1}^+| = 2^{n+1}(n+1)!$. Let $z \in \mathrm{Spin_{n+1}}$, we define the Bruhat cell $\mathrm{Bru}_z$ as the connected component of $\Pi_{n+1}^{-1}(\mathrm{Bru}_{\Pi_{n+1}(z)})$ which contains z. Obviously $\Pi_{n+1}^{-1}(\mathrm{Bru}_{\Pi_{n+1}(z)}) = \mathrm{Bru}_z \sqcup \mathrm{Bru}_{-z}$, where each set $\mathrm{Bru}_{z}$, $\mathrm{Bru}_{-z}$ is contractible and non-empty.

Since the group $\mathrm{Up}_{n+1}^1$ is contractible, its Bruhat action on $\mathrm{SO}_{n+1}$ lifts to a Bruhat action on $\mathrm{Spin}_{n+1}$ that, for simplicity, we still denote by $B : \mathrm{Up}_{n+1}^1 \times \mathrm{Spin}_{n+1} \rightarrow \mathrm{Spin}_{n+1}$. As before, the Bruhat cells on $\mathrm{Spin}_{n+1}$ are the orbits of the Bruhat action.

Therefore, the Bruhat decomposition of $\mathrm{SO}_{n+1}$ can be lifted to the universal cover $\Pi_{n+1}: \mathrm{Spin}_{n+1} \rightarrow \mathrm{SO}_{n+1}$ and we have the Bruhat decomposition for $\mathrm{Spin}_{n+1}$:

\begin{equation*} \mathrm{Spin}_{n+1}=\bigsqcup_{\tilde{P} \in \tilde{B}_{n+1}^+}\mathrm{Bru}_{\tilde{P}}, \end{equation*}

and there are $2^{n+1}(n+1)!$ disjoint Bruhat cells in $\mathrm{Spin}_{n+1}$.

3.1. Frenet frame curves and generic curves

Consider a locally convex curve $\gamma:[0,1] \rightarrow \mathbb S^n$. Applying Gram-Schmidt orthonormalization to the $(n+1)$-vectors $(\gamma(t),\gamma'(t),\dots,\gamma^{(n)}(t))$, there exists a unique $\mathcal{F}_\gamma(t) \in \mathrm{SO}_{n+1}$ and $R_\gamma(t) \in \mathrm{Up}_{n+1}^+$ such that

(1)\begin{equation} (\gamma(t),\gamma'(t),\dots,\gamma^{(n)}(t))=\mathcal{F_{\gamma}}(t) R_\gamma(t), \end{equation}

where we recall that $\mathrm{Up}_{n+1}^+$ is the space of upper triangular matrices with positive diagonal entries and real coefficients. The curve $\mathcal{F_{\gamma}}:[0,1] \rightarrow \mathrm{SO}_{n+1}$ defined by (1) is called the Frenet frame curve of the locally convex curve $\gamma:[0,1] \rightarrow \mathbb S^n$. For applications of Frenet frame curves on higher-dimensional spheres, we refer the reader to [Reference Goulart and Saldanha16, Reference Goulart and Saldanha17, Reference Saldanha and Shapiro27].

Notice that the definition of Frenet frame curve associated with a locally convex curve γ on the n-sphere does not depend on the choice of a positive (or orientation preserving) reparametrization: indeed a computation using the chain rule shows that

\begin{equation*} (\gamma \circ \phi(t),(\gamma \circ \phi)'(t),\dots,(\gamma \circ \phi)^{(n)}(t))=(\gamma(\tau),\gamma'(\tau),\dots,\gamma^{(n)}(\tau))U, \end{equation*}

where $\tau=\phi(t)$ is a positive reparametrization (i.e. the sign of $\phi'(t)$ is positive), and where U is an upper triangular matrix whose diagonal is given by $(1,\phi'(t), \dots,(\phi'(t))^n)$. So $U \in \mathrm{Up}_{n+1}^+$, which implies that $\mathcal{F_{\gamma \circ \phi}}(t)=\mathcal{F_{\gamma}}(\tau)$.

We denote by ${\mathcal{L}\mathbb S^{n}}$, the set of all locally convex curves $\gamma : [0,1] \rightarrow \mathbb S^{n}$ such that $\mathcal{F}_\gamma(0)=I$. Obviously, ${\mathcal{L}\mathbb S^{n}}(Q) \subset {\mathcal{L}\mathbb S^{n}}$.

Many authors already discussed the topological structures of the spaces $\mathcal{L}\mathbb S^n(Q)$. It is well known that different topological structures give different spaces which are homotopically equivalent. Therefore, we will consider that our curves are smooth. We notice, however, that even if juxtaposition of curves jeopardizes smoothness, there is no loss of generality in assuming so. For more about this discussion we refer to the reader [Reference Saldanha26, Reference Saldanha and Shapiro27] or [Reference Alves1].

Even though we will be mainly interested in locally convex curves, it will be useful in the sequel to consider a larger space of curves.

A curve $\gamma: [0,1] \rightarrow \mathbb S^n$ of class C^k ( $k \geq n$) is called generic if the vectors $\gamma(t), \gamma'(t),\gamma''(t),\dots,\gamma^{(n-1)}(t)$ are linearly independent for all $t \in [0,1]$. Given γ a generic curve on the n-sphere we can still define its Frenet frame curve. In fact, by applying Gram-Schmidt orthonormalization to the linearly independent n-vectors $\gamma(t),\gamma'(t),\dots,\gamma^{(n-1)}(t)$ we obtain n orthonormal vectors $u_0(t),u_1(t), \dots, u_{n-1}(t)$ and then, there is a unique vector $u_n(t)$ for which $u_0(t), u_1(t), \dots, u_{n-1}(t), u_n(t)$ is a positive orthonormal basis. We may thus set

(2)\begin{equation} \mathcal{F_{\gamma}}(t)=(u_0(t),u_1(t), \dots, u_{n-1}(t), u_n(t)) \in \mathrm{SO}_{n+1} \end{equation}

and make the following more general definition. The curve $\mathcal{F_{\gamma}}:[0,1] \rightarrow \mathrm{SO}_{n+1}$ defined by (2) is called the Frenet frame curve of the generic curve $\gamma:[0,1] \rightarrow \mathbb S^n$. Clearly, the latter definition coincides with the former when γ is locally convex.

Given $Q \in \mathrm{SO}_{n+1}$, we denote by $\mathcal{G}\mathbb S^n(Q)$ the set of all generic curves $\gamma:[0,1] \rightarrow \mathbb S^{n}$ such that $\mathcal{F_{\gamma}}(0)=I$ and $\mathcal{F_{\gamma}}(1)=Q$. For $z \in \mathrm{Spin}_{n+1}$, we define ${\mathcal{G}\mathbb S^{n}}(z)$ as the subset of ${\mathcal{G}\mathbb S^{n}}(\Pi_{n+1}(z))$ for which $\tilde{\mathcal{F}_{\gamma}}(1)=z$. Obviously, $\mathcal{L}\mathbb S^n(Q) \subset \mathcal{G}\mathbb S^n(Q)$ and $\mathcal{L}\mathbb S^n(z) \subset \mathcal{G}\mathbb S^n(z)$.

The homotopy type of the spaces $\mathcal{G}\mathbb S^{n}(z)$, $z \in \mathrm{Spin}_{n+1}$, is well-known. Let us define $\Omega \mathrm{Spin}_{n+1}(z)$ to be the space of all continuous curves $\alpha : [0,1] \rightarrow \mathrm{Spin}_{n+1}$ with $\alpha(0)=\mathbf{1}$ and $\alpha(1)=z$. It is well understood that different values of $z \in \mathrm{Spin}_{n+1}$ does not change the space $\Omega \mathrm{Spin}_{n+1}(z)$ up to homeomorphism, therefore we will usually drop z from the notation and write $\Omega \mathrm{Spin}_{n+1}$ instead of $\Omega \mathrm{Spin}_{n+1}(z)$. Follows from the works of Hirsch and Smale ([Reference Hirsch18, Reference Smale35]) that the Frenet frame injection $\tilde{\mathcal{F}}: \mathcal{G}\mathbb S^{n}(z) \rightarrow \Omega \mathrm{Spin}_{n+1}$ defined by $(\tilde{\mathcal{F}}(\gamma))(t) = \tilde{\mathcal{F}}_\gamma(t)$ is a homotopy equivalence (see section 5.2 of [Reference Alves and Saldanha2] for more on this).

Let us look at the special case where γ is a generic curve on the 2-sphere, i.e. γ is an immersion. Let us denote by $\mathbf{t}_\gamma(t)$ the unit tangent vector of γ at the point $\gamma(t)$, that is $\mathbf{t}_\gamma(t):=\frac{\gamma'(t)}{||\gamma'(t)||} \in \mathbb S^2,$ and by $\mathbf{n}_\gamma(t)$ be the unit normal vector of γ at the point $\gamma(t)$, that is $ \mathbf{n}_\gamma(t):= \gamma(t) \times \mathbf{t}_\gamma(t) $ where × is the cross-product in $\mathbb{R}^3$. We then have $ \mathcal{F_{\gamma}}(t)=(\gamma(t),\mathbf{t}_\gamma(t),\mathbf{n}_\gamma(t)) \in \mathrm{SO}_3 $ where $\mathbf{t}_\gamma(t)$ is the unit tangent and $\mathbf{n}_\gamma(t)$ the unit normal we defined above. The geodesic curvature $\kappa_\gamma(t)$ is by definition $ \kappa_\gamma(t):=\mathbf{t}_\gamma'(t)\cdot\mathbf{n}_\gamma(t) $ where ⋅ is the Euclidean inner product. Here’s a geometric definition of locally convex curves in $\mathbb S^2$.

Let us now consider a generic curve γ on the 3-sphere and let $e_1,e_2,e_3,e_4$ be the canonical basis of $\mathbb{R}^4$. It is clear that $ \mathcal{F}_\gamma(t)e_1=\gamma(t), \quad \mathcal{F}_\gamma(t)e_2=\mathbf{t}_\gamma(t)=\frac{\gamma'(t)}{||\gamma'(t)||}. $ We define the unit normal $\mathbf{n}_\gamma(t)$ and binormal $\mathbf{b}_\gamma(t)$ by the formulas $ \mathbf{n}_\gamma(t)=\mathcal{F}_\gamma(t)e_3, \, \mathbf{b}_\gamma(t)=\mathcal{F}_\gamma(t)e_4 $ so that

\begin{equation*} \mathcal{F_{\gamma}}(t)=(\gamma(t),\mathbf{t}_\gamma(t),\mathbf{n}_\gamma(t),\mathbf{b}_\gamma(t)) \in \mathrm{SO}_4. \end{equation*}

The geodesic curvature $\kappa_\gamma(t)$ is still defined by $ \kappa_\gamma(t):=\mathbf{t}_\gamma'(t)\cdot\mathbf{n}_\gamma(t) $ but we further define the geodesic torsion $\tau_{\gamma}(t)$ by $ \tau_\gamma(t):=-\mathbf{b}_\gamma'(t)\cdot\mathbf{n}_\gamma(t). $ It is clear that the geodesic curvature is never zero. We can then characterize locally convex curves in $\mathbb S^3$.

Proof. Without loss of generality, we may assume that γ is parametrized by arc-length. Then, as before, $\mathbf{t}_\gamma(t)=\gamma'(t)$ and $\kappa_\gamma(t)=\gamma''(t)\cdot\mathbf{n}_\gamma(t)$. Also $ \gamma''(t)=-\gamma(t)+\kappa_\gamma(t)\mathbf{n}_\gamma(t), $ and hence $ \gamma'''(t)=-\gamma'(t)+\kappa_\gamma(t)'\mathbf{n}_\gamma(t)+\kappa_\gamma(t)\mathbf{n}_\gamma'(t). $ Since $\mathbf{b}_\gamma(t)\cdot \mathbf{n}_\gamma(t)=0$, we have $ \tau_\gamma(t)=-\mathbf{b}_\gamma'(t)\cdot \mathbf{n}_\gamma(t)=\mathbf{b}_\gamma(t)\cdot \mathbf{n}'_\gamma(t). $ One then easily computes

\begin{equation*} \gamma'''(t)\cdot\gamma(t)=0, \end{equation*}

\begin{equation*} \gamma'''(t)\cdot\gamma'(t)=-1-\kappa_\gamma(t)^2, \end{equation*}

\begin{equation*} \gamma'''(t)\cdot\mathbf{n}_\gamma(t)=\kappa_\gamma'(t), \end{equation*}

\begin{equation*} \gamma'''(t)\cdot\mathbf{b}_\gamma(t)=\kappa_\gamma(t)\tau_\gamma(t). \end{equation*}

So we have the equality $ (\gamma(t),\gamma'(t),\gamma''(t),\gamma'''(t))=\mathcal{F}_\gamma(t)R_\gamma(t) $ with

\begin{equation*} R_\gamma(t)= \begin{pmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1-\kappa_\gamma(t)^2\\ 0 & 0 & \kappa_\gamma(t) & \kappa_\gamma'(t) \\ 0 & 0 & 0 & \kappa_\gamma(t)\tau_\gamma(t) \end{pmatrix}. \end{equation*}

Since $\mathcal{F}_\gamma(t)$ has determinant 1 and $\kappa_\gamma(t)$ is never zero, we have

\begin{equation*} \mathrm{det}(\gamma(t),\gamma'(t),\gamma''(t),\gamma'''(t))=\mathrm{det}R_\gamma(t)=\kappa_\gamma(t)^2\tau_\gamma(t), \end{equation*}

and this proves the statement.

We continue collecting preliminary notions and auxiliary results instrumental to our arguments. Let $ \Gamma : [0,1] \rightarrow \mathrm{SO}_{n+1} $, the logarithmic derivative of the curve Γ is defined as $ \Lambda(t)=(\Gamma(t))^{-1}\Gamma'(t) $, that is, $\Gamma'(t)=\Gamma(t)\Lambda(t). $ Notice that Λ belongs to the Lie algebra of $\mathrm{SO}_{n+1}$. Therefore $\Lambda(t)$ is automatically a skew-symmetric matrix for all $t \in [0,1]$.

Let $\mathfrak{J} \subset \mathfrak{so}_{n+1}$ be the set of Jacobi matrices, i.e. $\mathfrak{J}$ is the set of tridiagonal skew-symmetric matrices with positive subdiagonal entries, that is, matrices of the form

\begin{equation*} \begin{pmatrix} 0 & -c_1 & 0 & \ldots & 0 \\ c_1 & 0 & -c_2 & & 0 \\ & \ddots & \ddots & \ddots & \\ 0 & & c_{n-1} & 0 & -c_n \\ 0 & & 0 & c_n & 0 \end{pmatrix}, \quad c_1 \gt 0, \dots, c_n \gt 0. \end{equation*}

We are interested in the following definition in order to characterize the Frenet frame curves associated with a locally convex curves. Let us call a curve $\Gamma : [0,1] \rightarrow \mathrm{SO}_{n+1}$ Jacobian if its logarithmic derivative $\Lambda(t)=(\Gamma(t))^{-1}\Gamma'(t)$ belongs to $\mathfrak{J}$ for all $t \in [0,1]$.

In [Reference Saldanha and Shapiro27] was introduced a homeomorphism between locally convex curves in $\mathcal{L}\mathbb S^n$ and Jacobian curves starting at the identity, more precisely:

The Lemma 2.1 in [Reference Saldanha and Shapiro27] gives us a correspondence as follows: given Γ a Jacobian curve with $\Gamma(0)=I$, then if we define $\gamma_\Gamma$ by setting $\gamma_\Gamma(t)=\Gamma(t)e_1$ then $\gamma_\Gamma \in \mathcal{L}\mathbb S^n$ and conversely, given $\gamma \in \mathcal{L}\mathbb S^n$, its Frenet frame curve is a Jacobian curve. Notice that the Frenet frame curve $\mathcal{F}_\gamma$ uniquely determines the curve γ.

For instance, let us characterize Frenet Frame curves of locally convex curves on the 3-sphere that will be useful in this work. If $\gamma : [0,1] \rightarrow \mathbb S^3$ is locally convex, then $ \mathcal{F_{\gamma}}(t)=(\gamma(t),\mathbf{t}_\gamma(t),\mathbf{n}_\gamma(t),\mathbf{b}_\gamma(t)) \in \mathrm{SO}_4 $ and one gets

(3)\begin{equation} \Lambda_\gamma(t)= ||\gamma'(t)|| \begin{pmatrix} 0 & $-$1 & 0 & 0 \\ 1 & 0 & $-$\kappa_\gamma(t) & 0 \\ 0 & \kappa_\gamma(t) & 0 & $-$\tau_\gamma(t) \\ 0 & 0 & \tau_\gamma(t) & 0 \end{pmatrix}. \end{equation}

3.2. Convex curves

Next we introduce a special class of locally convex curves of fundamental importance in the study of the space of locally convex curves.

Consider $\gamma : [0,1] \rightarrow \mathbb S^n$, a smooth curve, and let $H \subseteq \mathbb{R}^{n+1}$ be a hyperplane. Let us call γ a globally convex curve if the image of γ intersects H, counting with multiplicity, in at most n points. This definition requires us to clarify the notion of multiplicity. First, endpoints of the curve are not counted as intersections. So, if there exists $t \in (0,1)$ such that $\gamma(t) \in H$, then the multiplicity of the intersection point $\gamma(t)$ is the smallest integer $k \geq 1$ such that $ \gamma^{(j)}(t) \in H, \quad 0 \leq j \leq k-1. $ Therefore, a multiplicity is one if $\gamma(t) \in H$ but $\gamma'(t) \notin H$, it is two if $\gamma(t) \in H$, $\gamma'(t) \in H$ but $\gamma''(t) \notin H$, and so on.

From the definition it is easy to prove that every globally convex curve is locally convex. For ease of presentation, we refer to globally convex curves as convex curves.

Given any $n \geq 2$ and $z \in \mathrm{Spin}_{n+1}$ recall that $\mathcal{L}\mathbb S^n(z)$ is the space of locally convex curves in $\mathbb S^n$ whose initial lifted Frenet frame is $\mathbf 1$ and the final lifted Frenet Frame is z.

Convexity is strongly related with the number of connected components of the spaces $\mathcal{L}\mathbb S^n(z)$, $z \in \mathrm{Spin}_{n+1}$. Let us recall also the following result.

This result highlights the importance of identifying the existence of convex curves in the spaces of locally convex curves. It will be critical in what follows.

5.1. Proof of theorem 1.1

In this subsection, we give the proof of Theorem 1.1 which characterizes convexity in the spaces $\mathcal{L}\mathbb S^3(-\mathbf 1,\mathbf k)$ by merely considering the left part of the curve.

Proof of theorem 1.1

Recall that we want to prove that a curve $\gamma \in \mathcal{L}\mathbb S^3(-\mathbf 1,\mathbf k)$ is convex if and only its left part $\gamma_l \in \mathcal{L}\mathbb S^2(-\mathbf 1)$ is convex.

It is clear that $\mathcal{L}\mathbb S^2(-\mathbf 1)$ contains convex curves; the curve σ_c, for $0 \lt c \lt 2\pi$ defined in $\S$ 4 is convex, since it intersects any hyperplane of $\mathbb{R}^3$ (or equivalently any great circle) in exactly two points. Using Theorem 3.4, the space $\mathcal{L}\mathbb S^2(-\mathbf{1})$ has therefore 2 connected components,

\begin{equation*} \mathcal{L}\mathbb S^2(-\mathbf{1})=\mathcal{L}\mathbb S^2(-\mathbf{1})_c \sqcup \mathcal{L}\mathbb S^2(-\mathbf{1})_n, \end{equation*}

where $\mathcal{L}\mathbb S^2(-\mathbf{1})_c$ is the component associated with convex curves and $\mathcal{L}\mathbb S^2(-\mathbf{1})_n$ the component associated with non-convex curves.

The space $\mathcal{L}\mathbb S^3(-\mathbf 1,\mathbf k)$ also contains convex curves. Indeed, we prove in the proposition 4.3 that the curve $\gamma_{1}^1 \in \mathcal{L}\mathbb S^3(-\mathbf 1,\mathbf k)$ is convex. Therefore, again using the Theorem 3.4, the space $\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)$ has therefore 2 connected components,

\begin{equation*} \mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)=\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c \sqcup \mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_n, \end{equation*}

where $\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c$ is the component associated with convex curves and $\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_n$ the component associated with non-convex curves.

Then we can use Theorem 4.1 to define a continuous map

\begin{equation*} L : \mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k) \rightarrow \mathcal{L}\mathbb S^2(-\mathbf{1}) ,\end{equation*}

by setting $L(\gamma)=\gamma_l$, where $(\gamma_l,\gamma_r)$ is the pair of curves associated with γ. Since L is continuous and $\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c$ is connected, its image by L is also connected. Moreover, we know $\gamma_1^1 \in \mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c$, and that $L(\gamma_1^1)=\sigma_1 \in \mathcal{L}\mathbb S^2(-\mathbf{1})_c$, therefore the image of $\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c$ by L intersects $\mathcal{L}\mathbb S^2(-\mathbf{1})_c$; since the latter is connected we must have the inclusion

\begin{equation*} L\left(\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c\right) \subset \mathcal{L}\mathbb S^2(-\mathbf{1})_c. \end{equation*}

This proves one part of the statement, namely that if $\gamma \in \mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_c$, then its left part $\gamma_l=L(\gamma) \in \mathcal{L}\mathbb S^2(-\mathbf{1})_c$. To prove the other part, it is enough to verify that

\begin{equation*} L\left(\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_n\right) \subset \mathcal{L}\mathbb S^2(-\mathbf{1})_n. \end{equation*}

To show this inclusion, using continuity and connectedness arguments as before, it is enough to find one element in $\mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_n$ whose image by L belongs to $\mathcal{L}\mathbb S^2(-\mathbf{1})_n$. We claim that the curve γ ₁⁵ from example 4.2 does the job. To see that $\gamma_1^5 \in \mathcal{L}\mathbb S^3(-\mathbf{1},\mathbf k)_n$, one can easily check that if we define the plane

\begin{equation*} H=\{(x_1,0,0,x_4) \in \mathbb{R}^4 \; | \; x_1 \in \mathbb{R}, \; x_4 \in \mathbb{R}\} \end{equation*}

then

\begin{equation*} \gamma_1^5(t_i) \in H, \quad t_i=\frac{i}{5}, \quad 1 \leq i \leq 4. \end{equation*}

Hence γ ₁⁵ has at least 4 points of intersection with H; this shows that γ ₅ is not convex. To conclude, it is clear that $L(\gamma_1^5)=\sigma_5 \in \mathcal{L}\mathbb S^2(-\mathbf{1})_n$. Hence this proves the desired inclusion and concludes the proof.

5.2. Proof of Theorem  1.2

Now we detail the proof of Theorem 1.2, which gives a necessary condition for a curve in $\mathcal{L}\mathbb S^3(\mathbf 1,-\mathbf 1)$ to be locally convex by merely considering its left part. First we need to recall some basic definition and properties.

An open hemisphere H in $\mathbb S^2$ is a subset of $\mathbb S^2$ of the form $ H_h=\{x \in \mathbb S^2 \; | \; h\cdot x \gt 0\} $ for some $h \in \mathbb S^2$. A closed hemisphere is the closure $\bar{H}$ of an open hemisphere; that is, it has the form $ \bar{H}_h=\{x \in \mathbb S^2 \; | \; h\cdot x \geq 0\}. $ We can make the following definition. A closed curve $\gamma : [0,1] \rightarrow \mathbb S^2$ is hemispherical if its image is contained in an open hemisphere of $\mathbb S^2$. It is borderline hemispherical if it is contained in a closed hemisphere but not contained in any open hemisphere.

Following [Reference Zühlke38], we define a rotation number for any closed curve γ in $\mathbb S^2$ contained in a closed hemisphere (such a curve is either hemispherical or borderline hemispherical). To such a closed curve γ contained in a closed hemisphere, there is a distinguished choice of hemisphere h_γ containing the image of γ (this hemisphere h_γ is the barycenter of the set of all closed hemisphere containing the image of γ, the latter being geodesically convex, see [Reference Zühlke38] for further details). Let $\Pi_{h_\gamma} : \mathbb S^2 \rightarrow \mathbb{R}^2$ be the stereographic projection from $-h_\gamma$, and $\eta_\gamma= \Pi_{h_\gamma} \circ \gamma$. The curve η_γ is now a closed curve in the plane $\mathbb{R}^2$, and it is an immersion. The definition of its rotation number $\mathrm{rot}(\eta_\gamma) \in \mathbb{Z}$ is now classical: for instance, it can be defined to be the degree of the map

\begin{equation*} t \in \mathbb S^1 \mapsto \frac{\eta_\gamma'(t)}{||\eta_\gamma'(t)||} \in \mathbb S^1. \end{equation*}

Given a closed curve contained in a closed hemisphere in $\mathbb S^2$, its rotation number $\mathrm{rot}(\gamma)$ is defined by $ \mathrm{rot}(\gamma):=-\mathrm{rot}(\eta_\gamma) \in \mathbb{Z}. $

The proof of Theorem 1.2 will be based on two lemmas. The first lemma is a well-known property. For ease of presentation we omit the proof.

The next lemma will be proven below.

Now those lemmas build upon the setup of the problem to yield the proof of Theorem 1.2.

To conclude, it remains to prove Lemma 5.2.

Proof of Lemma 5.2

We argue by contradiction, and assume that there exists a curve $\beta \in \mathcal{L}\mathbb S^3(\mathbf 1,-\mathbf 1)_c$ (that is a convex curve $\beta \in \mathcal{L}\mathbb S^3(\mathbf 1,-\mathbf 1)$) such that its left part β_l is borderline hemispherical curve with rotation number equal to 2.

First we use our assumption that β is convex, which implies that $\mathcal{F}_{\beta}(t)$ belongs to the Bruhat cell of $A ^\top$ for all time $t \in [0,1]$ (see Proposition 64 in [Reference Alves1] or Theorem 3 in [Reference Goulart and Saldanha16]), where

\begin{equation*} A ^\top= \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}. \end{equation*}

Therefore, by definition, there exist matrices $U_1(t) \in \mathrm{Up}_4^+$, $U_2(t) \in \mathrm{Up}_4^+$ (recall that $\mathrm{Up}_4^+$ is the group of upper triangular $4 \times 4$ matrices with positive diagonal entries) such that

\begin{equation*} \mathcal{F}_{\beta}(t)=U_1(t){} A ^\top U_2(t).\end{equation*}

This condition can also be written as

\begin{equation*} \mathcal{F}_{\beta}(t)={} A ^\top L_1(t)U_2(t), \quad L_1(t):=AU_1(t){}A ^\top,\end{equation*}

with $L_1(t) \in \mathrm{Lo}_4^+$, where $\mathrm{Lo}_4^+$ is the group of lower triangular $4 \times 4$ matrices with positive diagonal entries. Such a decomposition is not unique, but there exists a unique decomposition

(4)\begin{equation} \mathcal{F}_{\beta}(t)={} A ^\top L(t)U(t), \end{equation}

where $U(t) \in \mathrm{Up}_4^+$. Now $L(t)\in \mathrm{Lo}_4^1$, where $\mathrm{Lo}_4^1$ is the group of lower triangular $4 \times 4$ matrices with diagonal entries equal to one. Using the fact that $\mathcal{F}_{\beta}(t)^{-1}\mathcal{F}_{\beta}'(t)$ belongs to $\mathfrak{J}$ (because β is in particular locally convex), it is easy to see, by a simple computation, that the matrix L(t) in (4) is such that $L(t)^{-1}L'(t)$ has positive subdiagonal entries and all other entries are zero. That is, we can write

(5)\begin{equation} L(t)^{-1}L'(t)= \begin{pmatrix} 0 & 0 & 0 & 0 \\ + & 0 & 0 & 0 \\ 0 & + & 0 & 0 \\ 0 & 0 & + & 0 \end{pmatrix} , \quad t \in [0,1]. \end{equation}

Then we use our assumption that the left part β_l is a borderline hemispherical curve with rotation number equal to 2.

This implies (see Figure 6, where the dotted circle represents the equator of the sphere) that there exist times t ₁ and t ₂ and reals θ ₁ and θ ₂ such that

\begin{equation*} \tilde{\mathcal{F}}_{\beta_l}(t_1)=\exp(\theta_1\mathbf k) \in \mathbb S^3, \quad \tilde{\mathcal{F}}_{\beta_l}(t_2)=\exp(\theta_2\mathbf k) \in \mathbb S^3.\end{equation*}

Figure 6.

The curve β_l.

Consequently, for β, we have

(6)\begin{equation} \begin{cases} \tilde{\mathcal{F}}_{\beta}(t_1)=(\exp(\theta_1\mathbf k),z_r(t_1)) \in \mathbb S^3 \times \mathbb S^3, \\ \tilde{\mathcal{F}}_{\beta}(t_2)=(\exp(\theta_2\mathbf k),z_r(t_2)) \in \mathbb S^3 \times \mathbb S^3. \end{cases} \end{equation}

Following subsection 4.1 in [Reference Alves and Saldanha2], let us denote by $\mathbf k_l$ the matrix in $\mathfrak{so}_4$ that corresponds to the left multiplication by $\mathbf k \in \mathbb{H}$. This matrix is given by

\begin{equation*} \mathbf k_l= \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & +1 & 0 & 0 \\ +1 & 0 & 0 & 0 \end{pmatrix}. \end{equation*}

Recalling that $\Pi_4 : \mathbb S^3 \times \mathbb S^3 \rightarrow \mathrm{SO}_4$ is the canonical projection, it follows from (6) that $\mathcal{F}_{\beta}(t_1)=\Pi_4(\tilde{\mathcal{F}}_{\beta}(t_1))$ and $\mathcal{F}_{\beta}(t_2)=\Pi_4(\tilde{\mathcal{F}}_{\beta}(t_2))$ belong to the subgroup H of matrices in $\mathrm{SO}_4$ that commutes with the matrix $\mathbf k_l$. Clearly, this subgroup H consists of matrices of the form

\begin{equation*} \begin{pmatrix} q_{11} & q_{12} & -q_{42} & -q_{41} \\ q_{21} & q_{22} & -q_{32} & -q_{31} \\ q_{31} & q_{32} & q_{22} & q_{21} \\ q_{41} & q_{42} & q_{12} & q_{11} \end{pmatrix} \in \mathrm{SO}_4. \end{equation*}

Using this explicit form of H and the fact that $\mathcal{F}_{\beta}(t_1) \in H$ and $\mathcal{F}_{\beta}(t_2) \in H$, one finds, after a direct computation, that the matrix

(7)\begin{equation} L(t)= \begin{pmatrix} 1 & 0 & 0 & 0 \\ l_{21}(t) & 1 & 0 & 0 \\ l_{31}(t) & l_{32}(t) & 1 & 0 \\ l_{41}(t) & l_{42}(t) & l_{43}(t) & 1 \end{pmatrix}, \end{equation}

defined in (4) satisfies, at $t=t_1$ and $t=t_2$, the conditions

(8)\begin{equation} l_{21}(t_1)=-l_{43}(t_1), \quad l_{21}(t_2)=-l_{43}(t_2). \end{equation}

But clearly, (8) is not compatible with (5), and this gives the desired contradiction.