Lagrangian Gauss map - Imported Knowlpedia Content

Definition

Let $f:L^n\to(M^{2n},\omega)$ be a Lagrangian immersion into a symplectic manifold. Write $\operatorname{Lag}(TM)\to M$ for the bundle whose fiber at $p$ is the Lagrangian Grassmannian of $T_pM$ . The Lagrangian Gauss map is

\gamma_f:L\longrightarrow\operatorname{Lag}(TM),\qquad \gamma_f(x)=df_x(T_xL)\subset T_{f(x)}M,

and it covers $f$ . Equivalently, it is a section of the pullback bundle $f^*\operatorname{Lag}(TM)$ . If $M=V$ is a symplectic vector space, translation canonically identifies all tangent spaces with $V$ , so $\gamma_f$ is an ordinary map $L\to\Lambda(V)$ . For a general $M$ , such a fixed-target map requires a symplectic trivialization of $f^*TM$ .

Why the target is Lagrangian

Because $f^*\omega=0$ , the subspace $df_x(T_xL)$ is isotropic. Its dimension is $n$ , half the dimension of $T_{f(x)}M$ , so it is Lagrangian. Thus the Lagrangian condition is exactly what makes the ordinary tangent-plane Gauss map factor through the Lagrangian-Grassmannian subbundle rather than the full Grassmann bundle.

Trivialized form

Given a symplectic trivialization

\tau:f^*TM\overset{\sim}{\longrightarrow}L\times(V,\omega_0),

the Gauss map becomes $\gamma_{f,\tau}(x)=\tau_x(df_x(T_xL))\in\Lambda(V)$ . Replacing $\tau$ by a varying symplectic transformation acts pointwise on $\Lambda(V)$ . Therefore constructions that pull back a class from one fixed Lagrangian Grassmannian must record the trivialization or an equivalent Maslov datum when the ambient tangent bundle is not canonically trivialized.

Examples

For the zero section $Q\hookrightarrow T^*Q$ , the Gauss map selects the horizontal Lagrangian tangent planes along the zero section. If $L\subset\mathbb R^{2n}$ is an affine Lagrangian plane, $\gamma_L$ is constant. By contrast, a half-dimensional immersed submanifold on which $\omega$ does not vanish has tangent planes outside $\operatorname{Lag}(TM)$ , so its tangent-plane map is not a Lagrangian Gauss map.

Relation to Maslov data

In a symplectic vector space, pulling the universal Maslov class on $\Lambda(V)$ back along $\gamma_f$ gives the Maslov class of the immersion. For a general symplectic manifold, the intrinsic section into $f^*\operatorname{Lag}(TM)$ is still canonical, while an integral phase or grading can require additional ambient Maslov-covering data Seidel, §2.

References

V. I. Arnol'd, “On a characteristic class entering into conditions of quantization,” Functional Analysis and Its Applications 1 (1967), 1–14. DOI record. Relevant: the Gauss map of a Lagrangian submanifold and the Maslov class.
Paul Seidel, “Graded Lagrangian submanifolds,” Bulletin de la Société Mathématique de France 128 (2000), 103–149. arXiv record. Relevant: §2, Maslov coverings and gradings.