Isotropic submanifold - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold. A smooth immersion $\iota:L\to M$ is an isotropic submanifold if

\iota^*\omega=0.

Equivalently, for every $p\in L$ , the image $d\iota_p(T_pL)$ is an isotropic subspace of the symplectic vector space $T_{\iota(p)}M$ . When $L\subseteq M$ is embedded, $\iota$ is the inclusion and the condition says $\omega_p(u,v)=0$ for all $u,v\in T_pL$ . The definition concerns the pullback to $L$ ; self-intersections of an isotropic immersion do not alter it.

Dimension bound and the Lagrangian case

If $\dim M=2n$ , isotropic linear algebra gives

\dim L\leq n.

An isotropic submanifold of dimension $n$ is Lagrangian. Thus isotropic submanifolds range from curves, which are automatically isotropic, up to the maximal half-dimensional case. The zero pullback condition is much stronger than the automatic identity $\omega(v,v)=0$ for an individual tangent vector.

Examples and non-examples

Every immersed curve in a symplectic manifold is isotropic because a two-form vanishes on a one-dimensional tangent space. In standard $\mathbb R^{2n}$ with coordinates $(q_i,p_i)$ , the coordinate submanifold

\{p_1=\cdots=p_n=0,\ q_{k+1}=\cdots=q_n=0\}

is isotropic of dimension $k\leq n$ .

By contrast, an open subset of a positive-dimensional symplectic manifold is not isotropic: its pulled-back form remains nondegenerate rather than vanishing. A surface in a symplectic four-manifold is isotropic exactly when it is Lagrangian.

Immersed versus embedded terminology

Some authors say “isotropic immersion” for the map $\iota$ and reserve “isotropic submanifold” for an embedded image. The defining equation is the same. When an immersed image has self-intersections, tangent planes coming from different preimages need not be mutually symplectically orthogonal; only each individual tangent plane must be isotropic.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Oxford DOI record. Relevant: §3.3, isotropic and Lagrangian submanifolds.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Chapter DOI record. Relevant: “Lagrangian Submanifolds,” pp. 17–23.