2.1 Conventions

We regard vector fields $X\in \mathfrak{X}(M)$ as derivations $f\mapsto {\mathcal{L}}_{X}f\equiv X(f)$ of the algebra $C^{\infty }(M)$. We denote by $\unicode[STIX]{x1D711}_{s}^{X}$, $s\in \mathbb{R}$, the (local) flow, with the sign convention ${\mathcal{L}}_{X}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}s)|_{s=0}(\unicode[STIX]{x1D711}_{-s}^{X})^{\ast }$ as operators on functions. For example, the flow of $X=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ on $\mathbb{R}$ is given by $\unicode[STIX]{x1D711}_{s}^{X}(t)=t-s$. Given a vector bundle $E\rightarrow M$, we denote by $\mathfrak{aut}_{{\mathcal{V}}{\mathcal{B}}}(E)$ the infinitesimal vector bundle automorphisms, consisting of vector fields $\widetilde{X}\in \mathfrak{X}(E)$ whose (local) flow is by vector bundle automorphisms. By [Reference Grabowski and RotkiewiczGR09], these are exactly the vector fields homogeneous of degree zero under scalar multiplication. As a consequence of the homogeneity, the domain of definition of $\unicode[STIX]{x1D711}_{s}^{\widetilde{X}}$ for a given $s$ is simply the domain of definition of $\unicode[STIX]{x1D711}_{s}^{X}$, where $X$ is the restriction of $\widetilde{X}$ to the zero section. In particular, $\widetilde{X}$ is complete if and only if $X$ is complete.

A vector bundle automorphism $\widetilde{\unicode[STIX]{x1D711}}$ of $E$, covering $\unicode[STIX]{x1D711}:M\rightarrow M$, acts on the space of sections $\unicode[STIX]{x1D6E4}(E)$ by $\widetilde{\unicode[STIX]{x1D711}}\cdot \unicode[STIX]{x1D70E}=\widetilde{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D70E}\circ \unicode[STIX]{x1D711}^{-1}$. We can use this action to see that infinitesimal automorphisms $\widetilde{X}\in \mathfrak{aut}_{{\mathcal{V}}{\mathcal{B}}}(E)$ are equivalently expressed as linear operators $D$ on $\unicode[STIX]{x1D6E4}(E)$ for which there exists $X\in \mathfrak{X}(M)$ with

(1)$$\begin{eqnarray}D(f\unicode[STIX]{x1D70E})=fD(\unicode[STIX]{x1D70E})+({\mathcal{L}}_{X}f)\,\unicode[STIX]{x1D70E},\end{eqnarray}$$

for all $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(E)$ and $f\in C^{\infty }(M)$; here $X$, called the symbol of $D$, is uniquely determined (as the restriction of the corresponding $\widetilde{X}\in \mathfrak{aut}_{{\mathcal{V}}{\mathcal{B}}}(E)$ to the zero section). The relation between $\widetilde{X}$ and $D$ is $D(\unicode[STIX]{x1D70E})=(d/ds)|_{s=0}\unicode[STIX]{x1D711}_{s}^{\widetilde{X}}\cdot \unicode[STIX]{x1D70E}$.

2.2 Deformation spaces

In the $C^{\infty }$-category, the deformation space of a manifold $M$ with respect to a submanifold $N$ (also called the deformation to the normal cone) was introduced by Hilsum and Skandalis in [Reference Hilsum and SkandalisHS87, § 3.1] as a generalization of Connes’ tangent groupoid [Reference ConnesCon94]. The analogous construction in algebraic geometry goes back to work by MacPherson and Verdier in the 1970s; see the historical comments in [Reference FultonFul84, ch. 4]. Other references for the material in this section include [Reference Haj Saeedi Sadegh and HigsonHH18, Reference HigsonHig10, Reference Kashiwara and SchapiraKS94, Reference WeinsteinWei89].

For any pair $(M,N)$ consisting of a manifold $M$ and a submanifold $i:N{\hookrightarrow}M$, let $\unicode[STIX]{x1D708}(M,N)=TM|_{N}/TN$ be the normal bundle:

(2)

We denote by ${\mathcal{E}}\in \mathfrak{X}(\unicode[STIX]{x1D708}(M,N))$ the Euler vector field on $\unicode[STIX]{x1D708}(M,N)$, characterized as the unique vector field such that ${\mathcal{L}}_{{\mathcal{E}}}f=f$ for all linear functions $f\in C^{\infty }(\unicode[STIX]{x1D708}(M,N))$.

The deformation space for $(M,N)$ is a manifold ${\mathcal{D}}(M,N)$, with a decomposition into two submanifolds:

(3)$$\begin{eqnarray}{\mathcal{D}}(M,N)=\unicode[STIX]{x1D708}(M,N)\sqcup (M\times \mathbb{R}^{\times }).\end{eqnarray}$$

The manifold structure on ${\mathcal{D}}(M,N)$ is uniquely determined by the following properties.

1. (i) The map

   $$\begin{eqnarray}\unicode[STIX]{x1D70B}:{\mathcal{D}}(M,N)\rightarrow \mathbb{R},\end{eqnarray}$$
   given by the constant map to $0$ on $\unicode[STIX]{x1D708}(M,N)$ and by $(m,t)\mapsto t$ on $M\times \mathbb{R}^{\times }$, is a smooth submersion.

2. (ii) The map

   $$\begin{eqnarray}\unicode[STIX]{x1D705}:{\mathcal{D}}(M,N)\rightarrow M,\end{eqnarray}$$
   given on $M\times \mathbb{R}^{\times }$ by $(m,t)\mapsto m$ and on $\unicode[STIX]{x1D708}(M,N)$ by $i\circ p$, is smooth.

3. (iii) For any function $f\in C^{\infty }(M)$ with $f|_{N}=0$, the function $\widetilde{f}:{\mathcal{D}}(M,N)\rightarrow \mathbb{R}$, given on $M\times \mathbb{R}^{\times }$ by $(m,t)\mapsto (1/t)f(m)$ and on $\unicode[STIX]{x1D708}(M,N)$ by $(v\!\hspace{0.6em}{\rm mod}\hspace{0.2em}TN)\mapsto v(f)$, is smooth.

Intuitively, one may think of the fibers $\unicode[STIX]{x1D70B}^{-1}(t)\cong M$ for $t\neq 0$ as being ‘stretched in directions normal to $N$’ as $t\rightarrow 0$, so that the fiber over $0$ is the normal bundle itself. By abuse of notation, we will denote the map $\unicode[STIX]{x1D70B}$, regarded as an element of $C^{\infty }({\mathcal{D}}(M,N))$, simply by ‘$t$’. The algebra of smooth functions of ${\mathcal{D}}(M,N)$ is generated by this function $t$, together with functions of the form $\unicode[STIX]{x1D705}^{\ast }f$ for $f\in C^{\infty }(M)$ and functions $\widetilde{f}$ for $f|_{N}=0$. One can use these three types of functions to define local coordinates on the deformation space; see Appendix A. We will use the notation

(4)$$\begin{eqnarray}j_{t}:\unicode[STIX]{x1D70B}^{-1}(t)\rightarrow {\mathcal{D}}(M,N)\end{eqnarray}$$

for the inclusions of the fibers of $\unicode[STIX]{x1D70B}$; for $t\neq 0$ this may be viewed as an inclusion of $M$. For $t=0$ we will write

$$\begin{eqnarray}j=j_{0}:\unicode[STIX]{x1D708}(M,N)\rightarrow {\mathcal{D}}(M,N).\end{eqnarray}$$

We list some properties of the deformation space, which are easily verified using the definition, or using local coordinates (see Appendix A).

1. (i) For any morphism $\unicode[STIX]{x1D711}:(M_{1},N_{1})\rightarrow (M_{2},N_{2})$ (i.e. a smooth map $\unicode[STIX]{x1D711}:M_{1}\rightarrow M_{2}$ taking $N_{1}$ into $N_{2}$) the map $\unicode[STIX]{x1D711}\times \operatorname{id}:M_{1}\times \mathbb{R}^{\times }\rightarrow M_{2}\times \mathbb{R}^{\times }$ extends to a smooth map

   $$\begin{eqnarray}{\mathcal{D}}(\unicode[STIX]{x1D711}):{\mathcal{D}}(M_{1},N_{1})\rightarrow {\mathcal{D}}(M_{2},N_{2}).\end{eqnarray}$$
   Its restriction to the zero fiber is the natural map $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D711}):\unicode[STIX]{x1D708}(M_{1},N_{1})\rightarrow \unicode[STIX]{x1D708}(M_{2},N_{2})$.

2. (ii) For a vector bundle $V\rightarrow N$, the map $V\times \mathbb{R}^{\times }\rightarrow V\times \mathbb{R}$, $(v,t)\mapsto ((1/t)v,t)$, extends to a global diffeomorphism

   $$\begin{eqnarray}{\mathcal{D}}(V,N)\rightarrow V\times \mathbb{R}.\end{eqnarray}$$
   Its restriction to the zero fiber is the standard identification $\unicode[STIX]{x1D708}(V,N)\rightarrow V$.

3. (iii) There is a unique vector field $\unicode[STIX]{x1D6E9}\in \mathfrak{X}({\mathcal{D}}(M,N))$ such that

   $$\begin{eqnarray}{\mathcal{L}}_{\unicode[STIX]{x1D6E9}}t=t,\quad {\mathcal{L}}_{\unicode[STIX]{x1D6E9}}\unicode[STIX]{x1D705}^{\ast }f=0,\quad {\mathcal{L}}_{\unicode[STIX]{x1D6E9}}\widetilde{f}=-\widetilde{f}\quad (\text{for }f|_{N}=0).\end{eqnarray}$$
   This vector field is tangent to the zero fiber, with $-{\mathcal{E}}{\sim}_{j}\unicode[STIX]{x1D6E9}$, while its restriction to $M\times \mathbb{R}^{\times }$ is the vector field $t(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t)$. It generates the $\mathbb{R}^{\times }$-action on ${\mathcal{D}}(M,N)$, where $a\in \mathbb{R}^{\times }$ acts on $M\times \mathbb{R}^{\times }$ by $a\cdot (m,t)=(m,at)$ and on $\unicode[STIX]{x1D708}(M,N)$ by scalar multiplication by $a^{-1}$.

4. (iv) Every vector field $Y\in \mathfrak{X}(M)$ determines a vector field $\widehat{Y}$ on ${\mathcal{D}}(M,N)$ with the properties

   $$\begin{eqnarray}{\mathcal{L}}_{\widehat{Y}}t=0,\quad {\mathcal{L}}_{\widehat{Y}}\unicode[STIX]{x1D705}^{\ast }(f)=t\unicode[STIX]{x1D705}^{\ast }({\mathcal{L}}_{Y}f),\quad {\mathcal{L}}_{\widehat{Y}}\widetilde{f}=\unicode[STIX]{x1D705}^{\ast }{\mathcal{L}}_{Y}f\quad (\text{for }f|_{N}=0).\end{eqnarray}$$
   This vector field is vertical for the submersion $\unicode[STIX]{x1D70B}$; its restriction to the zero fiber is the equivalence class $[Y|_{N}]\in \unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D708}(M,N))$, regarded as a fiberwise constant vector field on $\unicode[STIX]{x1D708}(M,N)$, while the restriction to $\unicode[STIX]{x1D70B}^{-1}(t)=M$ for $t\neq 0$ is $tY$.

5. (v) If $Y\in \mathfrak{X}(M)$ is tangent to $N$, then the vector field in (iv) vanishes along the zero fiber, and hence is divisible by $t$. The vector field ${\mathcal{D}}(Y)=t^{-1}\widehat{Y}\in \mathfrak{X}({\mathcal{D}}(M,N))$ is characterized by

   $$\begin{eqnarray}{\mathcal{L}}_{{\mathcal{D}}(Y)}t=0,\quad {\mathcal{L}}_{{\mathcal{D}}(Y)}\unicode[STIX]{x1D705}^{\ast }f=\unicode[STIX]{x1D705}^{\ast }{\mathcal{L}}_{Y}f,\quad {\mathcal{L}}_{{\mathcal{D}}(Y)}\widetilde{f}=\widetilde{{\mathcal{L}}_{Y}f}\quad (\text{for }f|_{N}=0).\end{eqnarray}$$
   Its restriction to the zero fiber is the linear approximation $\unicode[STIX]{x1D708}(Y)$, while the restriction to $M\times \mathbb{R}^{\times }$ is simply $Y\times 0$. (Equivalently, $\unicode[STIX]{x1D708}(Y)$ is obtained by applying the normal bundle functor to $Y:(M,N)\rightarrow (TM,TN)$; see [Reference Bursztyn, Lima and MeinrenkenBLM19, § 2.2].)

6. (vi) If a differential form $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}^{k}(M)$ vanishes along $N$, then $(1/t)\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}^{k}(M\times \mathbb{R}^{\times })$ extends smoothly to a form ${\mathcal{D}}(\unicode[STIX]{x1D6FC})\in \unicode[STIX]{x1D6FA}^{k}({\mathcal{D}}(M,N))$ (see Appendix A). Its restriction to the zero fiber is a linear $k$-form on $\unicode[STIX]{x1D708}(M,N)$, the linear approximation $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})$ of $\unicode[STIX]{x1D6FC}$. For $k=0$, that is, $\unicode[STIX]{x1D6FC}=f\in C^{\infty }(M)$, we have ${\mathcal{D}}(f)=\widetilde{f}$.

2.3 Tubular neighborhood embeddings

We define a tubular neighborhood embedding for the pair $(M,N)$ to be an embedding

(5)$$\begin{eqnarray}\unicode[STIX]{x1D713}:\unicode[STIX]{x1D708}(M,N)\rightarrow M,\end{eqnarray}$$

taking the zero section of the normal bundle to $N\subseteq M$, and such that the induced map $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D713})$ is the canonical identification $\unicode[STIX]{x1D708}(V,N)\cong V$ for the vector bundle $V=\unicode[STIX]{x1D708}(M,N)$. A vector field $X\in \mathfrak{X}(M)$ is called Euler-like for $(M,N)$ if it is complete, and

(6)$$\begin{eqnarray}X|_{N}=0,\quad \unicode[STIX]{x1D708}(X)={\mathcal{E}}.\end{eqnarray}$$

Conditions (6) are equivalent to the requirement that if $f\in C^{\infty }(M)$ vanishes along $N$, then ${\mathcal{L}}_{X}f-f$ vanishes to second order along $N$. (The linearized version of this condition says that ${\mathcal{L}}_{\unicode[STIX]{x1D708}(X)}$ acts as the identity on linear functions on the normal bundle, which identifies $\unicode[STIX]{x1D708}(X)$ with the Euler vector field.) Note that an incomplete vector field with the properties (6) can be made complete, by multiplying with a function supported on a suitable open neighborhood of $N$, and equal to $1$ near $N$ (see [Reference Bursztyn, Lima and MeinrenkenBLM19, Rem. 2.9]). A tubular neighborhood embedding (5) determines an Euler-like vector field on its image, by pushing forward the Euler vector field of the normal bundle. In the other direction, we have the following theorem [Reference Bursztyn, Lima and MeinrenkenBLM19, § 2].

The proof in [Reference Bursztyn, Lima and MeinrenkenBLM19] is rather short, but its geometry was further clarified in [Reference Haj Saeedi Sadegh and HigsonHH18], using the deformation space. We will review the viewpoint from [Reference Haj Saeedi Sadegh and HigsonHH18], since it provides an appropriate framework for the applications to more complicated settings, and since we would like to elaborate on some aspects.

See Appendix A for a coordinate description of $W$. We will denote the (local) flow of $W$ by $\unicode[STIX]{x1D711}_{s}=\unicode[STIX]{x1D711}_{s}^{W}$.

Lemma 2.3. For all $Y\in \mathfrak{X}(M)$, the vector field $Y+[X,Y]$ is tangent to $N$, and

(7)$$\begin{eqnarray}[W,\widehat{Y}]={\mathcal{D}}(Y+[X,Y]).\end{eqnarray}$$

If $Y$ is tangent to $N$, then

(8)$$\begin{eqnarray}[W,{\mathcal{D}}(Y)]=\frac{1}{t}{\mathcal{D}}([X,Y]).\end{eqnarray}$$

Proof. We will show that $\unicode[STIX]{x1D711}_{s}(x)$ is defined for all $s>0$ (the proof for $s<0$ is similar). Since $W{\sim}_{\unicode[STIX]{x1D70B}}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$, the map $\unicode[STIX]{x1D70B}$ intertwines the flow of $W$ with the translation flow $t\mapsto t-s$ on $\mathbb{R}$. Hence $\unicode[STIX]{x1D711}_{s}(x)\in M\times \mathbb{R}_{{<}0}$ for $s>0$ sufficiently small. It therefore suffices to show that the solution curves of points $(m,t)\in M\times \mathbb{R}_{{<}0}\subseteq {\mathcal{D}}(M,N)$ are defined for all $s>0$. But on $M\times \mathbb{R}^{\times }$ the vector field $W$ is given by $(1/t)X+\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$, hence we can describe its (local) flow in terms of the flow $\unicode[STIX]{x1D711}_{s}^{X}$ of $X$:

(9)$$\begin{eqnarray}\unicode[STIX]{x1D711}_{s}(m,t)=(\unicode[STIX]{x1D711}_{-\text{log}(1-s/t)}^{X}(m),t-s)\end{eqnarray}$$

for $-\infty <s/t<1$. In particular, if $t<0$ this is defined for all $s>0$.◻

Proof of Theorem 2.1.

For all $s\in \mathbb{R}$, let ${\mathcal{D}}(M,N)_{s}\subseteq {\mathcal{D}}(M,N)$ be the domain of $\unicode[STIX]{x1D711}_{s}$. By the lemma, this is an open neighborhood of $\unicode[STIX]{x1D708}(M,N)$ for all $s$. Hence we obtain a well-defined embedding, for any given $s\neq 0$, by the composition

(10)$$\begin{eqnarray}\unicode[STIX]{x1D708}(M,N)\stackrel{j}{\longrightarrow }{\mathcal{D}}(M,N)_{s}\stackrel{\unicode[STIX]{x1D711}_{s}}{\longrightarrow }{\mathcal{D}}(M,N)\stackrel{\unicode[STIX]{x1D705}}{\longrightarrow }M.\end{eqnarray}$$

Equivalently, this map is the restriction of $\unicode[STIX]{x1D711}_{s}$ to a map from $\unicode[STIX]{x1D70B}^{-1}(0)=\unicode[STIX]{x1D708}(M,N)$ to $\unicode[STIX]{x1D70B}^{-1}(-s)=M$. Since $[W,{\mathcal{D}}(X)]=0$, the map (10) intertwines the restrictions of ${\mathcal{D}}(X)$ to $\unicode[STIX]{x1D708}(M,N)$ and to $M$; it therefore takes ${\mathcal{E}}$ to $X$.

Since $W$ restricts to the vector field $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ along $N\times \mathbb{R}\subseteq {\mathcal{D}}(M,N)$, the map (10) restricts to the identity map on $N$. To find the induced map on normal bundles, consider a section of $\unicode[STIX]{x1D708}(M,N)\cong \unicode[STIX]{x1D708}(\unicode[STIX]{x1D708}(M,N),N)$ represented as $[Y|_{N}]$ for some vector field $Y\in \mathfrak{X}(M)$. Recall that the vector field $\widehat{Y}\in \mathfrak{X}({\mathcal{D}}(M,N))$ is given by $tY$ on $M\times \mathbb{R}^{\times }$, and by $[Y|_{N}]$ (as a fiberwise constant vector field) on $\unicode[STIX]{x1D70B}^{-1}(0)=\unicode[STIX]{x1D708}(M,N)$. Equation (7) shows that the flow $\unicode[STIX]{x1D711}_{s}$ of $W$ preserves $\widehat{Y}$ modulo vector fields that are tangent to $N\times \mathbb{R}$. Hence, the map on normal bundles induced by (10) is $v\mapsto -sv$, and so is the identity map for $s=-1$. We conclude that the map (10) with $s=-1$,

(11)$$\begin{eqnarray}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D705}\circ \unicode[STIX]{x1D711}_{-1}\circ j,\end{eqnarray}$$

is the desired tubular neighborhood embedding.◻

Remark 2.5. Note that the image $U=\unicode[STIX]{x1D713}(\unicode[STIX]{x1D708}(M,N))$ of the tubular neighborhood embedding $\unicode[STIX]{x1D713}$ is simply the attracting set of $N$:

$$\begin{eqnarray}U=\Big\{m\in M|\lim _{s\rightarrow \infty }\unicode[STIX]{x1D711}_{s}^{X}(m)\in N\Big\}.\end{eqnarray}$$

The map ${\mathcal{D}}(\unicode[STIX]{x1D713}):\unicode[STIX]{x1D708}(M,N)\times \mathbb{R}\equiv {\mathcal{D}}(\unicode[STIX]{x1D708}(M,N),N)\rightarrow {\mathcal{D}}(U,N)$ takes $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ to $W$; in particular, the restriction of $W$ to ${\mathcal{D}}(U,N)\subseteq {\mathcal{D}}(M,N)$ is complete.

We also note the following property of $\unicode[STIX]{x1D713}$. Let $m_{t}$ denote scalar multiplication by $t$ on $\unicode[STIX]{x1D708}(M,N)$.

Lemma 2.7. For all $t\in \mathbb{R}$,

(12)$$\begin{eqnarray}\unicode[STIX]{x1D713}\circ m_{t}=\unicode[STIX]{x1D705}\circ \unicode[STIX]{x1D711}_{-t}\circ j.\end{eqnarray}$$

2.4 The splitting theorem for singular foliations

We now illustrate these techniques with an application to the theory of singular foliations. Rather than following the Stefan–Sussmann approach [Reference StefanSte80, Reference SussmannSus73], we will take the viewpoint of Androulidakis and Skandalis [Reference Androulidakis and SkandalisAS09] (see also Hermann [Reference HermannHer62]), and consider a singular foliation on $M$ to be a $C^{\infty }(M)$-submodule ${\mathcal{J}}\subseteq \mathfrak{X}(M)$ such that the following statements hold:

1. (i) ${\mathcal{J}}$ is local. If a vector field $X\in \mathfrak{X}(M)$ has the property that, for all $m\in M$, there exists an element of ${\mathcal{J}}$ equal to $X$ near $m$, then $X\in {\mathcal{J}}$;

2. (ii) ${\mathcal{J}}$ is locally finitely generated. On some neighborhood of any given $m\in M$, it is spanned by finitely many vector fields in ${\mathcal{J}}$;

3. (iii) ${\mathcal{J}}$ is involutive. $[{\mathcal{J}},{\mathcal{J}}]\subseteq {\mathcal{J}}$.

Note that [Reference Androulidakis and SkandalisAS09] considers $C^{\infty }(M)$-modules of compactly supported vector fields on $M$; the equivalence to the definition given above is explained in [Reference Androulidakis and ZambonAZ16] and [Reference WangWan17, ch. 2].

A smooth map $\unicode[STIX]{x1D711}:N\rightarrow M$ is transverse to ${\mathcal{J}}$ if, for all $n\in N$, the range of $T_{n}\unicode[STIX]{x1D711}$ together with $\{Y_{\unicode[STIX]{x1D711}(n)},~Y\in {\mathcal{J}}\}$ spans the entire tangent space $T_{\unicode[STIX]{x1D711}(n)}M$. In this case, a singular foliation $\unicode[STIX]{x1D711}^{!}{\mathcal{J}}\subseteq \mathfrak{X}(N)$ is defined as the set of all $Y\in \mathfrak{X}(N)$ with the property that $T\unicode[STIX]{x1D711}(Y)$ (as a section of the pullback bundle $\unicode[STIX]{x1D711}^{\ast }TM$) can be written as a locally finite sum $\sum _{i}h_{i}X_{i}$, with $h_{i}\in C^{\infty }(N)$ and $X_{i}\in {\mathcal{J}}$. See [Reference Androulidakis and SkandalisAS09, Definition 1.9].

If $N$ is a submanifold of $M$ transverse to ${\mathcal{J}}$, with $i:N\rightarrow M$ the inclusion, then $i^{!}{\mathcal{J}}$ is simply the set of all $X|_{N}$ such that $X\in {\mathcal{J}}$ is tangent to $N$. Letting $p:\unicode[STIX]{x1D708}(M,N)\rightarrow N$ be the projection as in (2), we may regard the foliation

$$\begin{eqnarray}\unicode[STIX]{x1D708}({\mathcal{J}})=p^{!}i^{!}{\mathcal{J}}\end{eqnarray}$$

of $\unicode[STIX]{x1D708}(M,N)$ as the linear approximation for the foliation ${\mathcal{J}}$ around $N$.

Proof. We first show that the $C^{\infty }(M)$-submodule ${\mathcal{J}}$ contains an Euler-like vector field $X$. Let $r=\dim M-\dim N$. Transversality implies that, for all $m\in N$, there is an open neighborhood $V\subseteq M$ and vector fields $Y_{1},\ldots ,Y_{r}\in {\mathcal{J}}|_{V}$, spanning an $r$-dimensional subbundle $K\subseteq TM|_{V}$, such that $TM|_{N\cap V}=K|_{N\cap V}\oplus TN|_{N\cap V}$. By [Reference Bursztyn, Lima and MeinrenkenBLM19, Lemma 3.9], there exists a section of $K$ which vanishes along $N$ and is Euler-like. Since such a section is a $C^{\infty }(V)$-linear combination of the $Y_{j}$, it lies in ${\mathcal{J}}|_{V}$. This shows how to construct Euler-like vector fields $X_{V}\in {\mathcal{J}}|_{V}$ locally; using a partition of unity, we may patch the local definitions to a global Euler-like vector field, $X\in {\mathcal{J}}$. Given a $G$-action preserving all the data, one can produce a $G$-invariant $X\in {\mathcal{J}}$, by averaging over slices for the action.

The vector fields $\unicode[STIX]{x1D6E9}$ and ${\mathcal{D}}(X)$ on the deformation space ${\mathcal{D}}(M,N)$ lie in $\unicode[STIX]{x1D705}^{!}{\mathcal{J}}$, since $\unicode[STIX]{x1D6E9}{\sim}_{\unicode[STIX]{x1D705}}0$ and ${\mathcal{D}}(X){\sim}_{\unicode[STIX]{x1D705}}X$. Hence, the vector field $W=t^{-1}(\unicode[STIX]{x1D6E9}+{\mathcal{D}}(X))$ also lies in $\unicode[STIX]{x1D705}^{!}{\mathcal{J}}$, which implies that its flow $\unicode[STIX]{x1D711}_{s}$ on ${\mathcal{D}}(U,N)\subseteq {\mathcal{D}}(M,N)$ (where $W$ is complete) preserves the foliation [Reference Androulidakis and SkandalisAS09, Proposition 1.6] (see also [Reference Garmendia and YudilevichGY19]). Each of the maps

$$\begin{eqnarray}\unicode[STIX]{x1D708}(M,N)\stackrel{j}{\longrightarrow }{\mathcal{D}}(U,N)\stackrel{\unicode[STIX]{x1D711}_{-1}}{\longrightarrow }{\mathcal{D}}(U,N)\stackrel{\unicode[STIX]{x1D705}}{\longrightarrow }U\end{eqnarray}$$

is foliation preserving; hence, so is their composition $\unicode[STIX]{x1D713}$.◻

4.1 Definitions

The shortest definition of an $L_{\infty }$-algebroid is in super-geometric terms, as split $NQ$-manifolds, that is, split non-negatively graded manifolds with a homological vector field of degree 1 [Reference S̆everaS̆ev05, Reference Sheng and ZhuSZ17, Reference VoronovVor10]. In more classical terms, this is the definition in terms of the associated Chevalley–Eilenberg complex. Consider a non-positively graded vector bundle

$$\begin{eqnarray}E=\bigoplus _{r\leqslant 0}E_{r}\rightarrow M,\end{eqnarray}$$

where each $E_{r}$ has finite rank (but $E$ itself might be of infinite rank). Let $E[1]$ be the graded bundle with the shifted grading $E[1]_{r}=E_{r+1}$, and let $E[1]^{\ast }$ be its (graded) dual, with grading

$$\begin{eqnarray}E[1]^{\ast }=\bigoplus _{r>0}(E[1]^{\ast })_{r}\rightarrow M,\end{eqnarray}$$

where

$$\begin{eqnarray}(E[1]^{\ast })_{r}=(E[1]_{-r})^{\ast }=(E_{1-r})^{\ast }.\end{eqnarray}$$

We denote by $\mathsf{C}(E)\rightarrow M$ the symmetric coalgebra bundle over $E[1]$ (cf.  [Reference Cattaneo and FelderCF07, Appendix A]); its (graded) dual is the symmetric algebra bundle

$$\begin{eqnarray}\mathsf{C}(E)^{\ast }=\operatorname{Sym}(E[1]^{\ast })\rightarrow M.\end{eqnarray}$$

Thus, the fibers of $\mathsf{C}(E)^{\ast }$ are generated by homogeneous elements in fibers of $E[1]^{\ast }$, subject to relations $xx^{\prime }=(-1)^{rr^{\prime }}x^{\prime }x$ whenever $x\in (E[1]^{\ast })_{r},~x^{\prime }\in (E[1]^{\ast })_{r^{\prime }}$. The grading of $\mathsf{C}(E)^{\ast }$ is such that the inclusion of $E[1]^{\ast }$ is degree preserving. The algebra bundle $\mathsf{C}(E)^{\ast }$ also carries another grading coming from the symmetric power degree; we will adopt the terminology of [Reference Pym and SafronovPS18] and call this the weight. The subspace of elements of weight $n$ is a graded subspace $C^{n}(E)^{\ast }=\bigoplus _{r\geqslant 0}C^{n}(E)_{r}^{\ast }$; note that

(21)$$\begin{eqnarray}C^{n}(E)_{r}^{\ast }=0\quad \text{for }r<n.\end{eqnarray}$$

Clearly, $C^{0}(E)^{\ast }=M\times \mathbb{R}$ and $C^{1}(E)^{\ast }=E[1]^{\ast }$. Every section $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(E[1]_{r})=\unicode[STIX]{x1D6E4}(E_{r+1})$ defines a degree $r$ derivation $\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})$ of the algebra $\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })$ lowering the weight by $1$; on $\unicode[STIX]{x1D6E4}(E[1]^{\ast })$ it is the obvious pairing.

Definition 4.1. (i) An $L_{\infty }$-algebroid structure on $E\rightarrow M$ is a degree-$1$ derivation $\text{d}$ of the algebra $\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })$ with $\text{d}\circ \text{d}=0$. If $E_{1-r}=0$ for $r>k$, then $E$ is also called a $\operatorname{Lie}_{k}$- (or $L_{k}$-)algebroid.

(ii) Suppose $E\rightarrow M$ and $F\rightarrow N$ are $L_{\infty }$-algebroids. A morphism of $L_{\infty }$-algebroids

is given by a morphism of graded vector bundles

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}:\mathsf{C}(F)\rightarrow \mathsf{C}(E)\end{eqnarray}$$

such that the pullback map on sections of the dual bundle $\unicode[STIX]{x1D6F7}^{\ast }:\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}(F)^{\ast })$ is a morphism of differential graded algebras (cf. [Reference Bonavolontà and PoncinBP13, Theorem 3]).

Unpacking the data of an $L_{\infty }$-structure on $E=\bigoplus _{r\leqslant 0}E_{r}$ reveals a rich geometric structure [Reference Bonavolontà and PoncinBP13, Reference Sheng and ZhuSZ17]. Let $\text{d}_{n}$ be the component of $\text{d}$ raising the weight by $n$:

$$\begin{eqnarray}\text{d}_{n}:\unicode[STIX]{x1D6E4}(E[1]^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(C^{1+n}(E)^{\ast }).\end{eqnarray}$$

By (21), and since every element of degree $r$ is a sum of products of generators of degree at most $r$, the operator $\text{d}_{n}$ vanishes on elements $x$ of degree $r<n-1$. Hence, on any element of given degree only finitely many terms of

$$\begin{eqnarray}\text{d}=\mathop{\sum }_{n\geqslant 0}\text{d}_{n}\end{eqnarray}$$

act non-trivially. Similarly, we see that all $\text{d}_{n}$ with $n\neq 1$ must vanish on $C^{\infty }(M)$. Hence, by the derivation property, the $\text{d}_{n}$ are $C^{\infty }(M)$-linear maps except for $n=1$. One defines the anchor map

(22)$$\begin{eqnarray}\mathsf{a}:E\rightarrow TM\end{eqnarray}$$

in terms of its action on sections as

$$\begin{eqnarray}{\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D70E})}f=\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})\text{d}f;\end{eqnarray}$$

by the above it is non-vanishing only on $E_{0}$. The restriction of $\text{d}_{0}$ to sections of weight $1$ is dual to a degree-$1$ map $\unicode[STIX]{x1D6FF}:E\rightarrow E$, defining a complex of vector bundles

(23)$$\begin{eqnarray}\cdots \stackrel{\unicode[STIX]{x1D6FF}}{\longrightarrow }E_{-2}\stackrel{\unicode[STIX]{x1D6FF}}{\longrightarrow }E_{-1}\stackrel{\unicode[STIX]{x1D6FF}}{\longrightarrow }E_{0}\longrightarrow 0;\end{eqnarray}$$

one finds that $\mathsf{a}\circ \unicode[STIX]{x1D6FF}=0$. Finally, one has the $n$-brackets for $n\geqslant 2$,

(24)$$\begin{eqnarray}[\cdot ,\cdot ,\ldots ,\cdot ]_{n}:\unicode[STIX]{x1D6E4}(E)\times \cdots \times \unicode[STIX]{x1D6E4}(E)\rightarrow \unicode[STIX]{x1D6E4}(E),\end{eqnarray}$$

given by Voronov’s derived bracket construction [Reference VoronovVor05]:

$$\begin{eqnarray}\unicode[STIX]{x1D704}([\unicode[STIX]{x1D70E}_{1},\ldots ,\unicode[STIX]{x1D70E}_{n}]_{n})=[[\ldots [\text{d}_{n-1},\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E}_{1})],\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E}_{2})],\ldots ,\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E}_{n})];\end{eqnarray}$$

alternatively these may be regarded as the components of the coderivation $Q:\unicode[STIX]{x1D6E4}(\mathsf{C}(E))\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}(E))$ dual to $\text{d}$. The binary bracket $[\cdot ,\cdot ]=[\cdot ,\cdot ]_{2}$ on $\unicode[STIX]{x1D6E4}(E)$ satisfies the Leibniz rule

$$\begin{eqnarray}[\unicode[STIX]{x1D70E},f\unicode[STIX]{x1D70F}]=f[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]+({\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D70E})}f)\unicode[STIX]{x1D70F}\end{eqnarray}$$

while the higher brackets $[\cdot ,\ldots ,\cdot ]_{n}$ for $n\geqslant 3$ are all $C^{\infty }(M)$-linear. These brackets satisfy a sequence of ‘higher Jacobi identities’, and compatibilities with $\unicode[STIX]{x1D6FF}$ (see [Reference Bonavolontà and PoncinBP13] for a detailed discussion). The anchor vanishes on $n$-brackets for $n\geqslant 3$, and satisfies $\mathsf{a}([\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}])=[\mathsf{a}(\unicode[STIX]{x1D70E}),\mathsf{a}(\unicode[STIX]{x1D70F})]$; in particular, an $L_{\infty }$-algebroid $E$ determines a singular foliation $\mathsf{a}(\unicode[STIX]{x1D6E4}(E_{0}))$ on $M$.

Suppose that is an $L_{\infty }$-morphism, given by a bundle map $\unicode[STIX]{x1D6F7}:\mathsf{C}(F)\rightarrow \mathsf{C}(E)$, with base map $\unicode[STIX]{x1D711}:N\rightarrow M$. Since $\unicode[STIX]{x1D6F7}$ is a fiberwise coalgebra morphism, it may be described in terms of its components $\unicode[STIX]{x1D6F7}^{n}:C^{1+n}(F[1])\rightarrow E[1]$, $n=0,1,\ldots .$ The bundle map $\unicode[STIX]{x1D6F7}^{0}:F[1]\rightarrow E[1]$ is a morphism of anchored vector bundles and also intertwines the differentials $\unicode[STIX]{x1D6FF}$. However, this map does not intertwine brackets, in general: one usually obtains a ‘morphism’ only by taking the higher $\unicode[STIX]{x1D6F7}^{n}$ into account. In the special case that $\unicode[STIX]{x1D6F7}^{n}=0$ for $n\neq 0$, we will say that $\unicode[STIX]{x1D6F7}$has weight zero.

The pullback map on sections of the dual bundle $\unicode[STIX]{x1D6F7}^{\ast }:\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}(F)^{\ast })$ is similarly determined by a sequence of maps

$$\begin{eqnarray}(\unicode[STIX]{x1D6F7}^{n})^{\ast }:\unicode[STIX]{x1D6E4}(E[1]^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(C^{1+n}(F)^{\ast })\end{eqnarray}$$

satisfying $(\unicode[STIX]{x1D6F7}^{n})^{\ast }(f\unicode[STIX]{x1D6FC})=(\unicode[STIX]{x1D711}^{\ast }f)((\unicode[STIX]{x1D6F7}^{n})^{\ast }\unicode[STIX]{x1D6FC})$.

4.2 Basic constructions

The usual constructions with Lie algebroids generalize to the $L_{\infty }$-setting.

4.2.2 Products

For any two $L_{\infty }$-algebroids $E\rightarrow M,~E^{\prime }\rightarrow M^{\prime }$, the product $E\times E^{\prime }\rightarrow M\times M^{\prime }$ acquires the structure of an $L_{\infty }$-algebroid. The differential on $\unicode[STIX]{x1D6E4}(\mathsf{C}(E\times E^{\prime })^{\ast })$ is such that $\text{d}(\unicode[STIX]{x1D6FC}\,\otimes \,\unicode[STIX]{x1D6FC}^{\prime })=\text{d}\unicode[STIX]{x1D6FC}\,\otimes \,\unicode[STIX]{x1D6FC}^{\prime }+(-1)^{r}\unicode[STIX]{x1D6FC}\,\otimes \,\text{d}\unicode[STIX]{x1D6FC}^{\prime }$ when $\unicode[STIX]{x1D6FC}\in \mathsf{C}(E),~\unicode[STIX]{x1D6FC}^{\prime }\in \mathsf{C}(E^{\prime })$ have degrees $r,r^{\prime }$, respectively. Projections to the two factors define $L_{\infty }$-morphisms of weight zero,

4.2.3 Restriction to submanifolds

Let $E\rightarrow M$ be an $L_{\infty }$-algebroid, and $N\subseteq M$ a submanifold such that $i^{!}E=\mathsf{a}^{-1}(TN)$ is smooth. Note that $(i^{!}E)_{0}=i^{!}(E_{0})$ (the pullback as an anchored vector bundle), while the summands $(i^{!}E)_{r}$ for $r<0$ are simply the restrictions $E_{r}|_{N}$ as vector bundles.

Lemma 4.3 (Pullback to submanifolds).

If $i:N{\hookrightarrow}M$ is a smooth submanifold such that $i^{!}E=\mathsf{a}^{-1}(TN)$ is smooth, then the subbundle $i^{!}E=\mathsf{a}^{-1}(TN)$ inherits a unique $L_{\infty }$-algebroid structure for which it is an $L_{\infty }$-subalgebroid.

Proof. We have to show that the kernel of the pullback map $\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}(i^{!}E)^{\ast })$ is closed under the differential $\text{d}$. This kernel is the ideal in $\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })$ generated by functions $f\in C^{\infty }(M)$ such that $f|_{N}=0$, and sections $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E4}(E[1]^{\ast })$ such that $\unicode[STIX]{x1D6FC}|_{N}$ takes values in the annihilator of $i^{!}E$. Hence it suffices that the differentials of these generators are again in the ideal.

If $f\in C^{\infty }(M)$ with $f|_{N}=0$, and $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(E)$ with $\mathsf{a}(\unicode[STIX]{x1D70E})$ tangent to $N$, then $\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})(\text{d}f)|_{N}=({\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D70E})}f)|_{N}=0$. This shows that $(\text{d}f)|_{N}$ is a section of the annihilator.

If $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E4}(E[1]^{\ast })$ restricts to a section of the annihilator, and $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(E)$ are two homogeneous sections whose image under the anchor is tangent to $N$, then

$$\begin{eqnarray}\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70F})\text{d}\unicode[STIX]{x1D6FC}=-\unicode[STIX]{x1D704}([\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}])\unicode[STIX]{x1D6FC}\pm \unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FC}\pm \unicode[STIX]{x1D704}(\unicode[STIX]{x1D70F})\text{d}\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})\unicode[STIX]{x1D6FC},\end{eqnarray}$$

where all three terms vanish along $N$. (Here, we are using that $\mathsf{a}([\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}])=[\mathsf{a}(\unicode[STIX]{x1D70E}),\mathsf{a}(\unicode[STIX]{x1D70F})]$ is again tangent to $N$.) This shows that $\text{d}\unicode[STIX]{x1D6FC}$ is in the ideal.◻

Remark 4.4. On the dual side, the degree-$1$ differential $\unicode[STIX]{x1D6FF}$ on the complex $E$ restricts to the differential on $i^{!}E$, and similarly for the anchor. The 2-bracket on $\unicode[STIX]{x1D6E4}(E)$ induces the 2-bracket on $\unicode[STIX]{x1D6E4}(i^{!}E)$, in such a way that $[\unicode[STIX]{x1D70E}|_{N},\unicode[STIX]{x1D70F}|_{N}]=[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]|_{N}$ whenever $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}$ restrict to sections of $i^{!}E$. Likewise, all the higher brackets $[\cdot ,\ldots ,\cdot ]_{n}$ of $E$, with $n\geqslant 3$, ‘restrict’ to the brackets for $i^{!}E$.

4.2.4 Pullbacks

We define the pullback of an $L_{\infty }$-algebroid $E\rightarrow M$, under a smooth map $\unicode[STIX]{x1D711}:N\rightarrow M$ that is transverse to the anchor, as the fiber product

$$\begin{eqnarray}\unicode[STIX]{x1D711}^{!}E=E\times _{TM}TN.\end{eqnarray}$$

In the case of an embedding, the $L_{\infty }$-structure is defined by Lemma 4.3; in general, it is defined by the identification of $\unicode[STIX]{x1D711}^{!}E$ with $i^{!}(E\times TN)$, where $i:N\cong \operatorname{Gr}(\unicode[STIX]{x1D711})\rightarrow M\times N$ is the inclusion as the graph. We have that $(\unicode[STIX]{x1D711}^{!}E)_{0}=\unicode[STIX]{x1D711}^{!}(E_{0})$ (the pullback as an anchored vector bundle), while $(\unicode[STIX]{x1D711}^{!}E)_{r}$ for $r\neq 0$ is just the usual pullback of $E_{r}$ as a vector bundle. The pullback $L_{\infty }$-algebroid comes with a natural $L_{\infty }$-morphism of weight zero,

(25)

given by the inclusion $\unicode[STIX]{x1D711}^{!}E{\hookrightarrow}E\times TN$ as a sub-$L_{\infty }$-algebroid, followed by the projection to $E$.

If $N=M\times Q$, with $\unicode[STIX]{x1D711}$ projection to the first factor, we have that $\unicode[STIX]{x1D711}^{!}E\cong E\times TQ$; this also gives a local description of pullbacks under submersions.

4.2.5 Automorphisms, derivations

For a $\mathbb{Z}$-graded vector bundle $V=\bigoplus _{r}V_{r}\rightarrow M$ with finite-rank graded pieces, we define the degree-zero automorphisms $\widetilde{\unicode[STIX]{x1D719}}\in \operatorname{Aut}_{{\mathcal{V}}{\mathcal{B}}}(V)$ as a collection of vector bundle automorphisms $\widetilde{\unicode[STIX]{x1D719}}_{r}\in \operatorname{Aut}_{{\mathcal{V}}{\mathcal{B}}}(V_{r})$, all with the same base map $\unicode[STIX]{x1D719}:M\rightarrow M$. Similarly, we define a degree-zero infinitesimal vector bundle automorphism $\widetilde{X}\in \mathfrak{aut}_{{\mathcal{V}}{\mathcal{B}}}(V)$ as a collection of vector fields $\widetilde{X}_{r}\in \mathfrak{aut}_{{\mathcal{V}}{\mathcal{B}}}(V_{r})$ on the graded pieces, all with the same base vector field $X\in \mathfrak{X}(M)$. As remarked in § 2, if $X$ is complete then it is automatic that all the $\widetilde{X}_{r}$ are complete; in this case $\widetilde{X}$ determines a one-parameter family of automorphisms $\unicode[STIX]{x1D719}_{s}^{\widetilde{X}}\in \operatorname{Aut}_{{\mathcal{V}}{\mathcal{B}}}(V)$. Any infinitesimal automorphism $\widetilde{X}$ defines a sequence of linear maps $D:\unicode[STIX]{x1D6E4}(V_{r})\rightarrow \unicode[STIX]{x1D6E4}(V_{r})$, satisfying the Leibniz rule (1) for the vector field $X$. Alternatively, we can describe it as a degree-zero linear map on sections of the graded dual $V^{\ast }=\bigoplus _{r}(V_{r})^{\ast }$,

$$\begin{eqnarray}D:\unicode[STIX]{x1D6E4}(V^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(V^{\ast }),\end{eqnarray}$$

again satisfying the Leibniz rule with respect to $X$. The two $D$ are related by

$$\begin{eqnarray}X\langle \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}\rangle =\langle D\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F}\rangle +\langle \unicode[STIX]{x1D6FC},D\unicode[STIX]{x1D70F}\rangle\end{eqnarray}$$

for all $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E4}(V^{\ast })$ and $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(V)$.

Let $E\rightarrow M$ be an $L_{\infty }$-algebroid. An infinitesimal $L_{\infty }$-automorphism of $E$ is given by an infinitesimal automorphism $\widetilde{X}$ of the graded vector bundle $\mathsf{C}(E)$, with base vector field $X$, such that the corresponding (local) flow is by (local) $L_{\infty }$-automorphisms. Using the discussion above, we obtain the following equivalent description in terms of operators on the complex $\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })$.

Lemma 4.5. An infinitesimal $L_{\infty }$-automorphism of an $L_{\infty }$-algebroid $E$ is equivalent to a degree-zero derivation $D:\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })$ which commutes with $\text{d}$.

Note that the restriction of the derivation $D$ to the weight-zero summand $\unicode[STIX]{x1D6E4}(\mathsf{C}^{0}(E)^{\ast })=C^{\infty }(M)$ is the base vector field $X$ of the resulting infinitesimal automorphism $\widetilde{X}$ of $\mathsf{C}(E)$. If $X$ is complete then $\widetilde{X}$ integrates to a flow $\unicode[STIX]{x1D711}_{s}^{\widetilde{X}}:\mathsf{C}(E)\rightarrow \mathsf{C}(E)$, defining a one-parameter family of $L_{\infty }$-algebroid automorphisms, denoted by the same letter .We will mainly be interested in the following situation. Let $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(E_{0})=\unicode[STIX]{x1D6E4}(E[1]_{-1})$. Then the interior multiplication operator $\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})$ is a graded derivation of degree $-1$ of $\unicode[STIX]{x1D6E4}(\mathsf{C}(E)^{\ast })$, and its graded commutator with $\text{d}$ is a degree-zero derivation

$$\begin{eqnarray}{\mathcal{L}}(\unicode[STIX]{x1D70E})=[\text{d},\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})].\end{eqnarray}$$

This satisfies

$$\begin{eqnarray}[\text{d},\,{\mathcal{L}}(\unicode[STIX]{x1D70E})]=[\text{d},[\text{d},\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})]]={\textstyle \frac{1}{2}}[[\text{d},\text{d}],\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})]=0,\end{eqnarray}$$

since $[\text{d},\text{d}]=2\text{d}\circ \text{d}=0$. On functions,

$$\begin{eqnarray}{\mathcal{L}}(\unicode[STIX]{x1D70E})f=\unicode[STIX]{x1D704}(\unicode[STIX]{x1D70E})\text{d}f={\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D70E})}f.\end{eqnarray}$$

It follows that ${\mathcal{L}}(\unicode[STIX]{x1D70E})$ defines an infinitesimal $L_{\infty }$-automorphism $\widetilde{X}$ of $E$, with base vector field $X=\mathsf{a}(\unicode[STIX]{x1D70E})$. If $X$ is complete, then this derivation integrates to a one-parameter family of $L_{\infty }$-automorphisms.

4.3 The splitting theorem

Let $E\rightarrow M$ be an $L_{\infty }$-algebroid, and $N\subseteq M$ a submanifold transverse to the anchor of $E$. Let $i:N\rightarrow M$ be the inclusion, and $p:\unicode[STIX]{x1D708}(M,N)\rightarrow N$ the normal bundle; we take the $L_{\infty }$-algebroid

$$\begin{eqnarray}\unicode[STIX]{x1D708}(E)=p^{!}i^{!}E\end{eqnarray}$$

to be the linear approximation of $E$ along $N$. The following result says that on a tubular neighborhood of $N$, the $L_{\infty }$-algebroid $E$ is $L_{\infty }$-isomorphic to its linear approximation.

Theorem 4.6 (Splitting theorem for $L_{\infty }$-algebroids).

Let $E\rightarrow M$ be an $L_{\infty }$-algebroid, and $N\subseteq M$ a submanifold transverse to the anchor of $E$. Then there exists an Euler-like section $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(E_{0})$ with respect to $N$, and any choice of such a section determines an $L_{\infty }$-isomorphism

with base map the tubular neighborhood embedding $\unicode[STIX]{x1D713}:\unicode[STIX]{x1D708}(M,N)\rightarrow U\subseteq M$ defined by $\mathsf{a}(\unicode[STIX]{x1D70E})$.

Proof. The existence of Euler-like sections follows from Lemma 3.3. Since $\mathsf{a}:E\rightarrow TM$ is transverse to $N$, it is transverse to the map $\unicode[STIX]{x1D705}:{\mathcal{D}}(M,N)\rightarrow M$. We hence obtain an $L_{\infty }$-algebroid

$$\begin{eqnarray}{\mathcal{D}}(E)=\unicode[STIX]{x1D705}^{!}E\rightarrow {\mathcal{D}}(M,N)\end{eqnarray}$$

over the deformation space. By our discussion for anchored vector bundles in § 3, we have the section

$$\begin{eqnarray}w=\frac{1}{t}({\mathcal{D}}(\unicode[STIX]{x1D70E})+\unicode[STIX]{x1D703})\in \unicode[STIX]{x1D6E4}(E_{0})\subseteq \unicode[STIX]{x1D6E4}(E),\end{eqnarray}$$

with $\mathsf{a}(w)=W$. As discussed in § 4.2.5, this section defines a degree-zero derivation

$$\begin{eqnarray}{\mathcal{L}}(w):\mathsf{C}({\mathcal{D}}(E)^{\ast })\rightarrow \mathsf{C}({\mathcal{D}}(E)^{\ast }),\end{eqnarray}$$

commuting with $\text{d}$, and a corresponding infinitesimal $L_{\infty }$-automorphism of ${\mathcal{D}}(E)$. The vector field $W$ is complete over ${\mathcal{D}}(U,N)\subseteq {\mathcal{D}}(M,N)$, and the lifted flow

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D711}}_{s}:\mathsf{C}({\mathcal{D}}(E)|_{{\mathcal{D}}(U,N)})\rightarrow \mathsf{C}({\mathcal{D}}(E)|_{{\mathcal{D}}(U,N)})\end{eqnarray}$$

of the vector field $\widetilde{W}$ on the coalgebra bundle defines a one-parameter family of $L_{\infty }$-automorphisms

with base map the flow $\unicode[STIX]{x1D711}_{s}$ of $W$. The map $\unicode[STIX]{x1D711}_{-1}$ takes $\unicode[STIX]{x1D70B}^{-1}(0)$ to $\unicode[STIX]{x1D70B}^{-1}(1)$, hence it induces an $L_{\infty }$-isomorphism

such that $\widetilde{\unicode[STIX]{x1D711}}_{-1}\circ (j_{0})_{!}=(j_{1})_{!}\circ \widetilde{\unicode[STIX]{x1D713}}$ (cf. (25)). Using the identifications $j_{0}^{!}{\mathcal{D}}(E)|_{{\mathcal{D}}(U,N)}\cong \unicode[STIX]{x1D708}(E)$ and $j_{1}^{!}{\mathcal{D}}(E)|_{{\mathcal{D}}(U,N)}\cong E|_{U}$, this is the desired $L_{\infty }$-isomorphism .◻

Since $\unicode[STIX]{x1D705}\circ j_{1}=\operatorname{id}$, and writing $j=j_{0}$, we may alternatively describe $\widetilde{\unicode[STIX]{x1D713}}$ as a composition of $L_{\infty }$-morphisms

(26)

As another direct application, we obtain a local description of transitive $L_{\infty }$-algebroids.

Corollary 4.9 (Local trivializations of transitive $L_{\infty }$-algebroids).

Let $E\rightarrow M$ be a transitive $L_{\infty }$-algebroid (i.e. such that the anchor map is surjective). Given $m\in M$ and a tubular neighborhood embedding $\unicode[STIX]{x1D713}:T_{m}M\rightarrow M$, with image $U$, there is an $L_{\infty }$-isomorphism

where $\mathfrak{g}$ is an $L_{\infty }$-algebra.

5.1 Basic definitions and constructions

Throughout, we will take metric to be synonymous with ‘non-degenerate symmetric bilinear form’. A Courant algebroid is a vector bundle $\mathbb{A}$ over a manifold $M$, equipped with a metric $\langle \cdot ,\cdot \rangle$ on its fibers, a bundle map $\mathsf{a}:\mathbb{A}\rightarrow TM$ over the identity map on $M$ (called the anchor map), and an $\mathbb{R}$-bilinear bracket $[\![\cdot ,\cdot ]\!]$ on its space of sections, such that the following properties hold, for all sections $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$:

1. (i) $\mathsf{a}(\unicode[STIX]{x1D70E})\langle \unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}\rangle =\langle [\![\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}_{1}]\!],\unicode[STIX]{x1D70F}_{2}\rangle +\langle \unicode[STIX]{x1D70F}_{1},[\![\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}_{2}]\!]\rangle$;

2. (ii) $[\![\unicode[STIX]{x1D70E},[\![\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}]\!]]\!]=[\![[\![\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}_{1}]\!],\unicode[STIX]{x1D70F}_{2}]\!]+[\![\unicode[STIX]{x1D70F}_{1},[\![\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}_{2}]\!]]\!]$;

3. (iii) $[\![\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}]\!]+[\![\unicode[STIX]{x1D70F}_{2},\unicode[STIX]{x1D70F}_{1}]\!]=\mathsf{a}^{\ast }(\text{d}~\langle \unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}\rangle )$.

In the last line, the metric is used to identify $\mathbb{A}^{\ast }\cong \mathbb{A}$, hence $\mathsf{a}^{\ast }:T^{\ast }M\rightarrow \mathbb{A}$. One refers to $[\![\cdot ,\cdot ]\!]$ as the Courant bracket (or sometimes the Courant–Dorfman bracket). From the axioms, one can derive additional properties, such as the derivation property

(27)$$\begin{eqnarray}[\![\unicode[STIX]{x1D70E},~f\unicode[STIX]{x1D70F}]\!]=f[\![\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]\!]+(\mathsf{a}(\unicode[STIX]{x1D70E})f)\unicode[STIX]{x1D70F},\end{eqnarray}$$

for all $f\in C^{\infty }(M)$ and $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$, and the property $\mathsf{a}\circ \mathsf{a}^{\ast }=0$ (see [Reference UchinoUch02]). The anchor map satisfies $\mathsf{a}([\![\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]\!])=[\mathsf{a}(\unicode[STIX]{x1D70E}),\mathsf{a}(\unicode[STIX]{x1D70F})]$, which implies that $(\mathbb{A},\mathsf{a})$ is an involutive anchored vector bundle (in the sense of § 3). As a result, one obtains the following proposition.

A Courant algebroid is called transitive if the anchor map is surjective. Clearly, the restriction $\mathbb{A}|_{{\mathcal{O}}}$ of a Courant algebroid $\mathbb{A}$ to any leaf ${\mathcal{O}}$ of its singular foliation is a transitive Courant algebroid, and $\mathbb{A}$ is the union over these restrictions.

A first example is the standard Courant algebroid $\mathbb{T}M=TM\oplus T^{\ast }M$, with anchor given by the projection to the first summand, the bilinear form $\langle \cdot ,\cdot \rangle$ extending the pairing between $TM$ and $T^{\ast }M$ (thus, $TM$ and $T^{\ast }M$ are Lagrangian), and with the Courant bracket on sections given by

(28)$$\begin{eqnarray}[\![X_{1}+\unicode[STIX]{x1D6FC}_{1},X_{2}+\unicode[STIX]{x1D6FC}_{2}]\!]=[X_{1},X_{2}]+{\mathcal{L}}_{X_{1}}\unicode[STIX]{x1D6FC}_{2}-\unicode[STIX]{x1D704}_{X_{2}}\text{d}\unicode[STIX]{x1D6FC}_{1},\end{eqnarray}$$

for vector fields $X_{1},X_{2}$ and $1$-forms $\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}$. More generally, given a closed $3$-form $\unicode[STIX]{x1D702}\in \unicode[STIX]{x1D6FA}^{3}(M)$, one has the $\unicode[STIX]{x1D702}$-twisted Courant algebroid $\mathbb{T}M^{\unicode[STIX]{x1D702}}$, with the same pairing and anchor as before, and Courant bracket obtained by adding a term $\unicode[STIX]{x1D704}_{X_{1}}\unicode[STIX]{x1D704}_{X_{2}}\unicode[STIX]{x1D702}$ to the right-hand side of (28).

A Courant algebroid over a point is the same as a metrized Lie algebra $\mathfrak{d}$ (a Lie algebra with an $\operatorname{ad}$-invariant metric). More generally, given a Lie algebra action of $\mathfrak{d}$ on a manifold $M$, with coisotropic stabilizers, the trivial bundle $\mathbb{A}=M\times \mathfrak{d}$ has the structure of an action Courant algebroid [Reference Li-Bland and MeinrenkenLM09]. Here the anchor $\mathsf{a}$ is given by the action, and the metric and Courant bracket on sections of $\mathbb{A}$ extend the metric and Lie bracket on $\mathfrak{d}$, regarded as constant sections.

A subbundle $E\subseteq \mathbb{A}$ of a Courant algebroid is called a Dirac structure if it is Lagrangian (i.e. $E=E^{\bot }$) and its space of sections is closed under the Courant bracket. In this case, $E$ becomes a Lie algebroid, with the Lie bracket and anchor obtained by restriction of the Courant bracket and anchor on $\mathbb{A}$. Note that the anchor of a Dirac structure defines another singular foliation of $M$, whose leaves are contained in those of $\mathbb{A}$. Graphs of Poisson structures and graphs of closed $2$-forms define Dirac structures inside the standard Courant algebroid $\mathbb{T}M$. Given a metrized Lie algebra $\mathfrak{d}$ with an action on $M$ with coisotropic stabilizers, any Lagrangian Lie subalgebra defines a Dirac structure $M\times \mathfrak{g}$ inside the action Courant algebroid $M\times \mathfrak{d}$.

5.2 Courant algebroid automorphisms

We denote by $\mathfrak{aut}_{{\mathcal{C}}{\mathcal{A}}}(\mathbb{A})$ the infinitesimal Courant algebroid automorphisms, given by vector fields $\widetilde{X}\in \mathfrak{X}(\mathbb{A})$ on the total space $\mathbb{A}$, with base vector field $X\in \mathfrak{X}(M)$, whose local flows are by Courant algebroid automorphisms (i.e. preserving the vector bundle structure, anchor, bracket, and metric). This is expressed as compatibility of the corresponding linear operator $D$ on $\unicode[STIX]{x1D6E4}(\mathbb{A})$ (cf. § 2.1) with the anchor, metric and bracket:

$$\begin{eqnarray}\displaystyle \mathsf{a}(D\unicode[STIX]{x1D70F}) & = & \displaystyle [X,\mathsf{a}(\unicode[STIX]{x1D70F})],\nonumber\\ \displaystyle {\mathcal{L}}_{X}\langle \unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}\rangle & = & \displaystyle \langle D\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}\rangle +\langle \unicode[STIX]{x1D70F}_{1},D\unicode[STIX]{x1D70F}_{2}\rangle ,\nonumber\\ \displaystyle D[\![\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}]\!] & = & \displaystyle [\![D\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2}]\!]+[\![\unicode[STIX]{x1D70F}_{1},D\unicode[STIX]{x1D70F}_{2}]\!].\nonumber\end{eqnarray}$$

In particular, for every section $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$ the operator $D=[\![\unicode[STIX]{x1D70E},\cdot ]\!]$ defines an ‘inner’ infinitesimal Courant automorphism

(29)$$\begin{eqnarray}\widetilde{\mathsf{a}}(\unicode[STIX]{x1D70E})\in \mathfrak{aut}_{{\mathcal{C}}{\mathcal{A}}}(\mathbb{A})\end{eqnarray}$$

lifting the vector field $\mathsf{a}(\unicode[STIX]{x1D70E})$. Given a closed $2$-form $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{2}(M)$, with associated bundle map $\unicode[STIX]{x1D714}^{\flat }:TM\rightarrow T^{\ast }M$, one finds that

$$\begin{eqnarray}D_{\unicode[STIX]{x1D714}}=\mathsf{a}^{\ast }\circ \unicode[STIX]{x1D714}^{\flat }\circ \mathsf{a}:\unicode[STIX]{x1D6E4}(\mathbb{A})\rightarrow \unicode[STIX]{x1D6E4}(\mathbb{A}^{\ast })\cong \unicode[STIX]{x1D6E4}(\mathbb{A})\end{eqnarray}$$

defines an infinitesimal Courant automorphism, with $X=0$. If $\unicode[STIX]{x1D714}$ is exact, then this infinitesimal Courant automorphism is inner.

Lemma 5.2. For any $1$-form $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}^{1}(M)$,

(30)$$\begin{eqnarray}D_{d\unicode[STIX]{x1D6FC}}=-[\![\mathsf{a}^{\ast }\unicode[STIX]{x1D6FC},\cdot ]\!].\end{eqnarray}$$

In particular, if $\unicode[STIX]{x1D6FC}$ is closed then $[\![\mathsf{a}^{\ast }\unicode[STIX]{x1D6FC},\cdot ]\!]=0$.

Proof. We calculate, for any $\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D701}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$:

$$\begin{eqnarray}\displaystyle \langle [\![\mathsf{a}^{\ast }\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}]\!],\unicode[STIX]{x1D701}\rangle & = & \displaystyle \langle \mathsf{a}^{\ast }\text{d}\langle \mathsf{a}^{\ast }\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}\rangle -[\![\unicode[STIX]{x1D6FE},\mathsf{a}^{\ast }\unicode[STIX]{x1D6FC}]\!],\unicode[STIX]{x1D701}\rangle \nonumber\\ \displaystyle & = & \displaystyle \langle \text{d}\langle \unicode[STIX]{x1D6FC},\mathsf{a}(\unicode[STIX]{x1D6FE})\rangle ,\mathsf{a}(\unicode[STIX]{x1D701})\rangle -{\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D6FE})}\langle \mathsf{a}^{\ast }\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D701}\rangle +\langle \mathsf{a}^{\ast }\unicode[STIX]{x1D6FC},[\![\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D701}]\!]\rangle \nonumber\\ \displaystyle & = & \displaystyle {\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D701})}\langle \unicode[STIX]{x1D6FC},\mathsf{a}(\unicode[STIX]{x1D6FE})\rangle -{\mathcal{L}}_{\mathsf{a}(\unicode[STIX]{x1D6FE})}\langle \unicode[STIX]{x1D6FC},\mathsf{a}(\unicode[STIX]{x1D701})\rangle +\langle \unicode[STIX]{x1D6FC},[\mathsf{a}(\unicode[STIX]{x1D6FE}),\mathsf{a}(\unicode[STIX]{x1D701})]\rangle \nonumber\\ \displaystyle & = & \displaystyle -(\text{d}\unicode[STIX]{x1D6FC})(\mathsf{a}(\unicode[STIX]{x1D6FE}),\mathsf{a}(\unicode[STIX]{x1D701}))\nonumber\\ \displaystyle & = & \displaystyle -\langle D_{d\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6FE}),\unicode[STIX]{x1D701}\rangle .\Box \nonumber\end{eqnarray}$$

We will use the notation

$$\begin{eqnarray}R_{\unicode[STIX]{x1D714}}=\exp (D_{\unicode[STIX]{x1D714}})=\operatorname{id}+D_{\unicode[STIX]{x1D714}}\end{eqnarray}$$

for the global Courant automorphism defined by a closed $2$-form $\unicode[STIX]{x1D714}$, and refer to it as a gauge transformation.

Proposition 5.3. Let $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$, with $\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E}=\mathsf{a}^{\ast }\unicode[STIX]{x1D6FD}$ for some $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}^{1}(M)$, and let $X=\mathsf{a}(\unicode[STIX]{x1D70F})=\mathsf{a}(\unicode[STIX]{x1D70E})$. Then the (local) flows $\widetilde{\unicode[STIX]{x1D713}}_{s},\widetilde{\unicode[STIX]{x1D719}}_{s}$ on $\mathbb{A}$ defined by $\widetilde{\mathsf{a}}(\unicode[STIX]{x1D70F}),\widetilde{\mathsf{a}}(\unicode[STIX]{x1D70E})$ are related by a family of gauge transformations:

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D713}}_{s}=R_{\unicode[STIX]{x1D71B}_{s}}\circ \widetilde{\unicode[STIX]{x1D719}}_{s},\quad \unicode[STIX]{x1D71B}_{s}=-\text{d}\int _{0}^{s}(\unicode[STIX]{x1D719}_{u})_{\ast }\unicode[STIX]{x1D6FD}\,\text{d}u,\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{s}$ is the flow of $X$.

Proof. Recalling the action of automorphisms of $\mathbb{A}$ on $\unicode[STIX]{x1D6E4}(\mathbb{A})$ (see § 2.1), which we denote by a dot, we will show that

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D713}}_{s}\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}.\unicode[STIX]{x1D6FE}=R_{\unicode[STIX]{x1D71B}_{s}}(\unicode[STIX]{x1D6FE})\end{eqnarray}$$

for all sections $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$. Since the two sides coincide for $s=0$, it suffices to compare their $s$-derivatives:

$$\begin{eqnarray}\displaystyle \frac{d}{ds}(\widetilde{\unicode[STIX]{x1D713}}_{s}\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}).\unicode[STIX]{x1D6FE} & = & \displaystyle \widetilde{\unicode[STIX]{x1D713}}_{s}\circ [\![\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70E},\cdot ]\!]\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}.\unicode[STIX]{x1D6FE}\nonumber\\ \displaystyle & = & \displaystyle \widetilde{\unicode[STIX]{x1D713}}_{s}\circ [\![\mathsf{a}^{\ast }\unicode[STIX]{x1D6FD},\cdot ]\!]\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}.\unicode[STIX]{x1D6FE}\nonumber\\ \displaystyle & = & \displaystyle \widetilde{\unicode[STIX]{x1D713}}_{s}\circ D_{-d\unicode[STIX]{x1D6FD}}\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}.\unicode[STIX]{x1D6FE}\nonumber\\ \displaystyle & = & \displaystyle D_{-d(\unicode[STIX]{x1D719}_{s})_{\ast }\unicode[STIX]{x1D6FD}}\circ \widetilde{\unicode[STIX]{x1D713}}_{s}\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}.\unicode[STIX]{x1D6FE}\nonumber\\ \displaystyle & = & \displaystyle D_{-d(\unicode[STIX]{x1D719}_{s})_{\ast }\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6FE})\nonumber\\ \displaystyle & = & \displaystyle \frac{d}{ds}R_{\unicode[STIX]{x1D71B}_{s}}.\unicode[STIX]{x1D6FE},\nonumber\end{eqnarray}$$

where the second-to-last last step used the fact that $\widetilde{\unicode[STIX]{x1D719}_{s}}$ and $\widetilde{\unicode[STIX]{x1D713}}_{s}$ have the same base map $\unicode[STIX]{x1D719}_{s}$, and hence $\mathsf{a}(\widetilde{\unicode[STIX]{x1D713}}_{s}\circ \widetilde{\unicode[STIX]{x1D719}}_{s}^{-1}.\unicode[STIX]{x1D6FE})=\mathsf{a}(\unicode[STIX]{x1D6FE})$.◻

Note that the formula can also be written as

(31)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D713}}_{s}=\widetilde{\unicode[STIX]{x1D719}}_{s}\circ R_{\unicode[STIX]{x1D714}_{s}},\quad \unicode[STIX]{x1D714}_{s}=-\text{d}\int _{0}^{s}(\unicode[STIX]{x1D719}_{u})^{\ast }\unicode[STIX]{x1D6FD}\,\text{d}u.\end{eqnarray}$$

This follows from $R_{\unicode[STIX]{x1D71B}_{s}}\circ \widetilde{\unicode[STIX]{x1D719}}_{s}=\widetilde{\unicode[STIX]{x1D719}}_{s}\circ R_{\unicode[STIX]{x1D719}_{s}^{\ast }\unicode[STIX]{x1D71B}_{s}}$, since $\unicode[STIX]{x1D719}_{s}^{\ast }\unicode[STIX]{x1D71B}_{s}=\unicode[STIX]{x1D714}_{s}$.

5.3 Pullbacks of Courant algebroids

Given a Courant algebroid $\mathbb{A}$ over $M$ and a map $\unicode[STIX]{x1D711}:N\rightarrow M$ transverse to the anchor of $\mathsf{a}:\mathbb{A}\rightarrow TM$, we define the pullback Courant algebroid $\unicode[STIX]{x1D711}^{!}\mathbb{A}$ over $N$ as follows (see [Reference Li-Bland and MeinrenkenLM09, § 2.2]). Let

(32)$$\begin{eqnarray}C\subseteq \mathbb{A}\times \mathbb{T}N\end{eqnarray}$$

be the fiber product with respect to the anchor $\mathsf{a}:\mathbb{A}\rightarrow TM$ and the map $T\unicode[STIX]{x1D711}\circ \mathsf{a}_{\mathbb{T}N}:\mathbb{T}N\rightarrow TM$. Then $C$ is a coisotropic subbundle along $\operatorname{Gr}(\unicode[STIX]{x1D711})\subseteq M\times N$, and $C^{\bot }$ lies in the kernel of the anchor. It follows that the quotient

$$\begin{eqnarray}\unicode[STIX]{x1D711}^{!}\mathbb{A}=C/C^{\bot }\end{eqnarray}$$

inherits a Courant algebroid structure over $N\cong \operatorname{Gr}(\unicode[STIX]{x1D711})$, with the Courant bracket induced from that on sections of $\mathbb{A}\times \mathbb{T}N$. (Note that the pullback $\unicode[STIX]{x1D711}^{!}\mathbb{A}$ as a Courant algebroid is different from the pullback as an anchored vector bundle.)

The diagonal inclusion of $C$ in $\mathbb{A}\times \overline{\unicode[STIX]{x1D711}^{!}\mathbb{A}}$ is a Lagrangian subbundle along the graph of $\unicode[STIX]{x1D711}$, and defines a Courant morphism (in the sense of [Reference Bursztyn, Ponte and SeveraBPS09, § 2.2]),

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{!}:\unicode[STIX]{x1D711}^{!}\mathbb{A}\rightarrow \mathbb{A};\end{eqnarray}$$

see [Reference Li-Bland and MeinrenkenLM09, Proposition 2.10]. Here $\overline{\unicode[STIX]{x1D711}^{!}\mathbb{A}}$ denotes the Courant algebroid with the same bracket and anchor as $\unicode[STIX]{x1D711}^{!}\mathbb{A}$ but with minus the metric $-\langle \cdot ,\cdot \rangle$. We have that $(\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D711}^{\prime })^{!}\mathbb{A}=(\unicode[STIX]{x1D711}^{\prime })^{!}\,\unicode[STIX]{x1D711}^{!}\,\mathbb{A}$ under composition of maps, provided that the appropriate transversality conditions are satisfied.

Given a Dirac structure $E\subseteq \mathbb{A}$ such that $\unicode[STIX]{x1D711}$ is transverse to the anchor of $E$, one obtains a pullback Dirac structure $\unicode[STIX]{x1D711}^{!}E\subseteq \unicode[STIX]{x1D711}^{!}\mathbb{A}$ as the image of $C\cap (E\times TN)$ under the quotient map $C\rightarrow C/C^{\bot }=\unicode[STIX]{x1D711}^{!}E$. (Its Lie algebroid structure coincides with that of the pullback Lie algebroid.)

We give three special cases of the pullback of Courant algebroids.

1. (i) $\unicode[STIX]{x1D711}^{!}\mathbb{T}M=\mathbb{T}N$. (See [Reference Li-Bland and MeinrenkenLM09, Proposition 2.9].)

2. (ii) If $\unicode[STIX]{x1D711}:M\times Q\rightarrow M$ is the projection onto the first factor, then $\unicode[STIX]{x1D711}^{!}\mathbb{A}=\mathbb{A}\times \mathbb{T}Q$.

3. (iii) If $\unicode[STIX]{x1D711}$ is a diffeomorphism, then $\unicode[STIX]{x1D711}^{!}\mathbb{A}$ is the usual pullback $\unicode[STIX]{x1D711}^{\ast }\mathbb{A}$ as a vector bundle.

Given a section $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$ and a vector field $Z\in \mathfrak{X}(N)$ such that $Z{\sim}_{\unicode[STIX]{x1D711}}\mathsf{a}(\unicode[STIX]{x1D70F})$, we obtain a section of $\unicode[STIX]{x1D711}^{!}\mathbb{A}$ by restricting $\unicode[STIX]{x1D70F}\times Z\in \unicode[STIX]{x1D6E4}(\mathbb{A}\times \mathbb{T}N)$ to a section of $C\rightarrow \operatorname{Gr}(\unicode[STIX]{x1D711})\cong N$, followed by the quotient map to $C/C^{\bot }$. We will denote this section by

(33)$$\begin{eqnarray}\unicode[STIX]{x1D70F}\times _{\unicode[STIX]{x1D711}}Z\in \unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}^{!}\mathbb{A});\end{eqnarray}$$

its image under the anchor is $Z$. Given another map $\unicode[STIX]{x1D711}^{\prime }:N^{\prime }\rightarrow N$, transverse to the anchor of $\unicode[STIX]{x1D711}^{!}\mathbb{A}$, and a vector field $Z^{\prime }\in \mathfrak{X}(N^{\prime })$ such that $Z^{\prime }{\sim}_{\unicode[STIX]{x1D711}^{\prime }}Z$, we have that

$$\begin{eqnarray}\unicode[STIX]{x1D70F}\times _{\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D711}^{\prime }}Z^{\prime }=(\unicode[STIX]{x1D70F}\times _{\unicode[STIX]{x1D711}}Z)\times _{\unicode[STIX]{x1D711}^{\prime }}Z^{\prime }.\end{eqnarray}$$

5.4 The splitting theorem

Suppose that $\mathbb{A}\rightarrow M$ is a Courant algebroid whose anchor $\mathsf{a}$ is transverse to a given submanifold $i:N{\hookrightarrow}M$. The description of the pullback Courant algebroid simplifies [Reference Li-Bland and MeinrenkenLM09, Proposition 2.8]:

(34)$$\begin{eqnarray}i^{!}\mathbb{A}=\mathsf{a}^{-1}(TN)/\mathsf{a}^{-1}(TN)^{\bot }.\end{eqnarray}$$

Any section $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$ such that $\mathsf{a}(\unicode[STIX]{x1D70F})$ is tangent to $N$ defines a section

(35)$$\begin{eqnarray}i^{!}\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(i^{!}\mathbb{A}),\end{eqnarray}$$

by taking the image of $\unicode[STIX]{x1D70F}|_{N}\in \unicode[STIX]{x1D6E4}(C)$ under the quotient map; in terms of (33) we have that $i^{!}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}\times _{i}\mathsf{a}(\unicode[STIX]{x1D70F})|_{N}$. As usual, we denote by $p:\unicode[STIX]{x1D708}(M,N)\rightarrow N$ the projection; we take

(36)$$\begin{eqnarray}\unicode[STIX]{x1D708}(\mathbb{A})=p^{!}i^{!}\mathbb{A}\rightarrow \unicode[STIX]{x1D708}(M,N)\end{eqnarray}$$

to be the linear approximation for $\mathbb{A}$ around $N$. This bundle has a distinguished Euler section

(37)$$\begin{eqnarray}\unicode[STIX]{x1D716}=0\times _{p}{\mathcal{E}}\in \unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D708}(\mathbb{A})).\end{eqnarray}$$

By construction, $\langle \unicode[STIX]{x1D716},\unicode[STIX]{x1D716}\rangle =0$, $\unicode[STIX]{x1D716}|_{N}=0$, and $\mathsf{a}(\unicode[STIX]{x1D716})={\mathcal{E}}$. For $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6E4}(\mathbb{A})$ such that $\mathsf{a}(\unicode[STIX]{x1D70F})$ is tangent to $N$, we have the section $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70F})=i^{!}\unicode[STIX]{x1D70F}\times _{p}\unicode[STIX]{x1D708}(\mathsf{a}(\unicode[STIX]{x1D70F}))$. If $\unicode[STIX]{x1D70E}$ is an Euler-like section, then $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D716}$. Sometimes, it is useful to take $\unicode[STIX]{x1D70E}$ to be isotropic.

The previous proof used the following fact.

Proof. This is well known in case where the metric on $V$ is split, and $C$ is Lagrangian. In this case, one takes $F$ to be the midpoint between $F_{1}$ and $F_{1}^{\bot }$ in the affine space of complementary subspaces (see, for example, [Reference MeinrenkenMei13, Proposition 1.4]). We may reduce to this case by considering the subbundle $V^{\prime }=C^{\bot }\oplus F_{1}$. Since

$$\begin{eqnarray}V^{\prime }\cap (V^{\prime })^{\bot }=(C^{\bot }+F_{1})\cap C\cap F_{1}^{\bot }=C^{\bot }\cap F_{1}^{\bot }=(C+F_{1})^{\bot }=V^{\bot }=0,\end{eqnarray}$$

we see that $V^{\prime }$ is a metrized subbundle, containing $C^{\prime }=C^{\bot }$ as a Lagrangian subbundle and $F_{1}$ as a complementary subbundle.◻

Given a Dirac structure $E\subseteq \mathbb{A}$ such that its anchor $\mathsf{a}|_{E}$ is transverse to $N$, we obtain a Dirac structure

(38)$$\begin{eqnarray}\unicode[STIX]{x1D708}(E)=p^{!}i^{!}E\subseteq \unicode[STIX]{x1D708}(\mathbb{A}).\end{eqnarray}$$

The proof will be given in § 5.5.

Note that if the normal bundle to the submanifold $N$ is trivial, then the choice of a trivialization $\unicode[STIX]{x1D708}(M,N)=N\times \mathsf{P}$ simplifies the model for the Courant algebroid $\mathbb{A}$ in a neighborhood of $N$ to

(39)$$\begin{eqnarray}\unicode[STIX]{x1D708}(\mathbb{A})=i^{!}\mathbb{A}\times \mathbb{T}\mathsf{P}.\end{eqnarray}$$

In the case of a Dirac structure $E\subseteq \mathbb{A}$ with $\mathsf{a}_{E}$ transverse to $N$, the model for $E$ becomes

(40)$$\begin{eqnarray}\unicode[STIX]{x1D708}(E)=i^{!}E\times T\mathsf{P},\end{eqnarray}$$

as a Dirac structure in $\unicode[STIX]{x1D708}(\mathbb{A})$. In particular, taking $N$ to be a ‘small’ transversal to a leaf, one obtains a local normal form for Courant algebroids, as a product of the standard Courant algebroid over the leaf and a Courant algebroid over the transversal. We will discuss this local splitting theorem in § 6.1, but let us already point out the following consequence.

Corollary 5.8 (Local trivializations of transitive Courant algebroids).

Let $\mathbb{A}\rightarrow M$ be a transitive Courant algebroid. Given a point $m\in M$ and a tubular neighborhood embedding $\unicode[STIX]{x1D713}:T_{m}M\rightarrow M$, with image $U$, there is an isomorphism with the product Courant algebroid

$$\begin{eqnarray}\mathbb{A}|_{U}\cong \mathbb{T}U\times \mathfrak{q},\end{eqnarray}$$

where $\mathfrak{q}$ is a metrized Lie algebra. Given a Dirac structure $E\subseteq \mathbb{A}$ which is transitive in the sense that $\mathsf{a}|_{E}$ is surjective, we may choose this isomorphism in such a way that it identifies

$$\begin{eqnarray}E|_{U}\cong TU\times \mathfrak{g},\end{eqnarray}$$

where $\mathfrak{g}\subseteq \mathfrak{q}$ is a Lagrangian Lie subalgebra.

5.5 Proof of the splitting theorem

The fact that the inclusion $i:N\rightarrow M$ is transverse to $\mathsf{a}$ implies that $\unicode[STIX]{x1D705}:{\mathcal{D}}(M,N)\rightarrow M$ is transverse to $\mathsf{a}$. So we can define the Courant algebroid

$$\begin{eqnarray}{\mathcal{D}}(\mathbb{A})=\unicode[STIX]{x1D705}^{!}\mathbb{A}\rightarrow {\mathcal{D}}(M,N).\end{eqnarray}$$

This satisfies

$$\begin{eqnarray}{\mathcal{D}}(\mathbb{A})|_{M\times \mathbb{R}^{\times }}=\mathbb{A}\times \mathbb{T}\mathbb{R}^{\times },\quad j^{!}{\mathcal{D}}(\mathbb{A})=\unicode[STIX]{x1D708}(\mathbb{A}),\end{eqnarray}$$

since over $M\times \mathbb{R}^{\times }$, $\unicode[STIX]{x1D705}$ is just projection to the first factor, and since $\unicode[STIX]{x1D705}\circ j=i\circ p$. Making use of the construction (33), we define various sections of ${\mathcal{D}}(\mathbb{A})$ as follows.