Lagrangian Submanifolds of Standard Multisymplectic Manifolds

Abstract

We give a detailed, self-contained proof of Geoffrey Martin’s normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein’s famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential topology, i.e., a foliated tubular neighborhood theorem and a foliated relative Poincaré lemma.

Appendix: Two Results in Foliated Differential Topology

In this appendix we give proofs for two “folkloristic” but subtle (and useful) extensions of well-known results in differential topology. Both are used in [8] but ask for a detailed proof. A brief sketch of a proof of the first result is given on the pages 88–89 in [2].

We begin with the tubular neighborhood theorem, in the presence of a foliation:

Theorem 2

(Foliated tubular neighborhood theorem) Let M be a manifold, $W\subset TM$ an integrable distribution, and N a submanifold complementary to W in the sense that $W|_N\oplus TN = TM|_N$. Then there exist an open neighborhood U of N in M, and a diffeomorphism $\phi$ from U onto an open subset of $W|_N$ containing the zero section, such that $\phi |_N=id_N$, the differential of $\phi$ at any point of N is the identity, and $\phi$ maps for all p in N the leaf of the foliation defined by W passing through it to the fiber $\phi (U)\cap (W|_p)$ of $W\vert _N \rightarrow N$, intersected with $\phi (U)$.

Proof

Let g be a fixed (auxiliary) Riemannian metric on the manifold M.

Given $q\in M$, the leaf $\mathcal {W}_q$ of the foliation $\mathcal {W}$ defined by the distribution W and containing q is given as an injectively immersed submanifold $j_q:F_q\rightarrow M$ (with image $j_q(F_q)=\mathcal {W}_q$). The induced Riemannian metric $j_q^{*}(g)$ defines an exponential map $exp^{\mathcal {W}}$, notably one has $exp^{\mathcal {W}}_q:T_q(F_q)\rightarrow F_q$, defined on an open neighborhood of 0. Since $T_q(F_q)$ is canonically identified with $W_q=T_q(\mathcal {W}_q)$ via the differential of $j_q$, and $j_q$ is smooth, $exp^{\mathcal {W}}_q$ is a smooth map from an open neighborhood of 0 in $W_q$ to M, having values in $\mathcal {W}_q$. Restricting q to be an element of N we obtain a map $exp^{\mathcal {W},N}$ from a subset of $W|_N$ containing N to M.

Let us now show that $exp^{\mathcal {W},N}$ is, in fact, smooth on an open neighborhood of N in $W|_N$. Fix q in N and a coordinate chart $M\supset U\xrightarrow {\varphi } V_1\times V_2\subset \mathbb {R}^{m-d}\times \mathbb {R}^d$, such that the fibers of $\pi :V_1\times V_2\rightarrow V_1$ are the leaves of the foliation $\mathcal {W}$ (d is the rank of this foliation). Furthermore, we can assume that $\varphi (q)=0$ and denote the elements of $\mathbb {R}^{m-d}$ resp. $\mathbb {R}^d$ by x resp. z.

We denote $\varphi (U\cap N)$ by N and $T\varphi (W)$ by W if no ambiguities are possible. By the assumption $TM|_N=TN\oplus W|_N$ we have $\forall q \in N \subset V_1\times V_2$ that $\mathbb {R}^m=T_q(V_1\times V_2)= T_qN\oplus W_q=T_qN\oplus \mathbb {R}^d$ and thus the natural projection $\pi _q: T_qN\rightarrow \mathbb {R}^{m-d}$ is a linear isomorphism. Thus $\pi \vert _N: N\rightarrow V_1$ has everywhere maximal rank equal to the dimension of $V_1$. Shrinking $V_1$ and $V_2$ if necessary, we can assume that $\pi |_N:N\rightarrow V_1$ is a diffeomorphism whose inverse is described by $(id_{V_1},f):V_1\rightarrow V_1\times V_2$, where $f:V_1\rightarrow V_2$ is smooth and $N=\Gamma _f$, the graph of f. The map $\psi$ given by $\psi (x,z)=(x,z-f(x))=:(x,y)$ is a diffeomorphism of $V_1\times V_2$ to an open subset of $\mathbb {R}^m$. Restricting $\psi$ to an appropriate open neighborhood of 0, the image of $\psi$ is a product of open subsets of $\mathbb {R}^{m-d}$ and $\mathbb {R}^d$. Furthermore, $\psi (0)=0$, $\psi$ preserves the leaves of $\mathcal {W}$, and maps $N=\Gamma _f$ to $\{y=0\}$.

Post-composing $\varphi$ with $\psi$ yields a chart of M near q compatible with the foliation $\mathcal {W}$ and “adapted” to N. Obviously, we can construct a locally finite covering of N by open subsets of M that are domains of such charts, again denoted by $\varphi :U\rightarrow V_1\times V_2$ for simplicity.

In these coordinates $exp^{\mathcal {W},N}_{(x,0)}$ is given as the time-one value of the solution of the following differential equation:

$$\begin{aligned} \frac{d^2 y^k}{dt^2} + \sum _{i,j}\Gamma _{i,j}^k(x,y)\frac{d y^i}{dt} \frac{d y^j}{dt}=0 \,\,\,\, \hbox {for}\, \,\, \, 1\le k \le d, \end{aligned}$$

subject to the initial condition that $x\in V_1$, $y(0)=0$ and $\frac{dy}{dt}(0)\in W_{(x,0)}$. Standard results on smooth ordinary differential equations depending smoothly on parameters imply that there exists an open subset $\widetilde{O}\subset W|_N$ containing N, where $exp^{\mathcal {W},N}$ is uniquely defined and smooth.

Upon identifying, for $q\in N$, $T_q\widetilde{O}=T_qN\oplus W_q=T_qM$, we obtain that $D(exp^{\mathcal {W},N}_q)=id_{T_qM}$. By the below cited Proposition 3, it follows that there exists an open neighborhood O of N in $\widetilde{O}\subset W|_N$ such that $exp^{\mathcal {W},N}|_O$ is a diffeomorphism onto its image U, an open neighborhood of N in M. Calling its inverse $\phi$, this latter map fulfills the conditions stated in Theorem 2.    $\square$

The last argument relies on a standard result in differential topology (cf., e.g., Proposition 7.3 in [6]):

Proposition 3

Let Y and $Y'$ be two manifolds, and $X\subset Y$, $X'\subset Y'$ two regular submanifolds. Let $f:Y\rightarrow Y'$ be a smooth map satisfying:

• $f|_X:X\rightarrow X'$ is a diffeomorphism

• $T_xf:T_xY\rightarrow T_{f(x)}Y'$ is an isomorphism for all $x\in X$

Then there exists an open neighborhood V of X in Y such that f(V) is open in $Y'$, and $f|_V$ is a diffeomorphism.

Now we show the relative Poincaré lemma, again in the presence of a foliation:

Theorem 3

(Foliated relative Poincaré lemma) Let M be a smooth manifold and $N{\subset } M$ a submanifold. Let $\omega$ be a closed $(k{+}1)$-form on M which vanishes when pulled back to N. Then there exists a neighborhood U of N in M, and a k-form $\mu$ defined on U, such that $d\mu =\omega |_U$ and $\mu |_N =0$. Moreover, if there exists an integrable distribution $W\subset TM$ complementary to N, and such that $\iota _{u\wedge v}\omega =0$ whenever x is in M and u and v are in $W_x \subset T_xM$, we may choose $\mu$ such that $\iota _{X}\mu =0$, for all vector fields X taking value in W and defined on an open subset of U.

Proof

By the (standard) tubular neighborhood theorem, there exist U and V neighborhoods of N in M respectively E (where $E\rightarrow N$ can be chosen to be any vector bundle such that $E\oplus TN=TM|_N$), and a diffeomorphism $\phi : U\rightarrow V$ fixing N pointwise. Thus in what follows, we can and will assume to be in a open neighborhood U of N in M, which is also a vector bundle $\pi : E=U\rightarrow N$ over N. Let us consider the map:

$$\begin{aligned} H:[0,1]\times U\rightarrow U\, , \, (t,x)\mapsto t\cdot x =tx. \end{aligned}$$

If we denote $H_t(x):=H(t,x)$ then $H_0=\iota \circ \pi$ (where $\iota :N\rightarrow U=E$ is the inclusion of N as the zero-section of E), and $H_1=id_E=id_U$. Let $Y_t(x):=\frac{d}{dt} H_t(x)$. The smooth map $Y_t$ is not a vector field since H is not a flow, but the following formula still holds:

$$\begin{aligned} \frac{d}{dt}(H_t^*\omega )=d(H_t^*\iota _{Y_t}\omega )+H_t^*\iota _{Y_t}d\omega , \end{aligned}$$

(*)

where, for $\alpha$ a $(k{+}1)$-form, $H_t^*\iota _{Y_t}\alpha$ is the following (well-defined!) k-form:

$$\begin{aligned} (H_t^*\iota _{Y_t}\alpha )_x(v_1,...,v_k)=\alpha _{tx}(Y_t(x),T_xH_t(v_1),...,T_xH_t(v_k)), \end{aligned}$$

for $x\in U$ and $v_j \in T_xU$. For a proof of $(*)$ see, e.g., [7].

Since $\iota ^*\omega =0$ and $\omega$ is closed we obtain:

$$\begin{aligned} \omega |_U&=H_1^*\omega -H_0^*\omega \\&=\int _{[0,1]}\left( \frac{d}{dt}H_t^*\omega \right) dt \\&=\int _{[0,1]}(d(H_t^*\iota _{Y_t}\omega ))dt \\&=d\int _{[0,1]}(H_t^*\iota _{Y_t}\omega )dt \\&=d\mu , \end{aligned}$$

where we set $\mu :=\int _{[0,1]}(H_t^*\iota _{Y_t}\omega )dt$. Moreover $\mu |_N=0$ because $Y_t|_N=0$.

To prove the last part of the theorem, we apply Theorem 2 in order to choose a foliated tubular neighborhood U of N with respect to W. We can thus assume that the fibers of $U\rightarrow N$ are the fibers of $W|_N\rightarrow N$. Then for $x\in U$, $Y_t(x)\in W_{tx}$, implying for X a vector field tangent to W:

$$\begin{aligned} (\iota _XH_t^*\iota _{Y_t}\omega )_x(v_1,...,v_{k{-}1})&=(H_t^*\iota _{Y_t}\omega )_x(X(x),v_1,...,v_{k{-}1}) \\&=\omega _{tx}(Y_t(x),T_xH_t(X(x)),T_xH_t(v_1),...,T_xH_t(v_{k-1})) \\&=0 \end{aligned}$$

since $Y_t(x)$ and $T_xH_t(X(x))$ are both in $W_{tx}$.    $\square$