Conormal bundle is Lagrangian

Statement

Let $S\subseteq Q$ be an embedded submanifold of a smooth $n$ -manifold. Its conormal bundle

N^*S=\{\alpha_q\in T^*Q:q\in S,\ \alpha_q(v)=0\text{ for every }v\in T_qS\}

is an embedded $n$ -dimensional submanifold of the cotangent bundle $T^*Q$ . With the canonical symplectic form on $T^*Q$ , the submanifold $N^*S$ is Lagrangian. This includes the zero section as the case $S=Q$ . The statement uses the full conormal bundle over $S$ , including its zero covectors; deleting the zero section gives another Lagrangian submanifold, which is nonclosed when S is nonempty of positive codimension.

Proof mechanism

Let $\lambda$ denote the tautological one-form on $T^*Q$ , so that at $\alpha_q$ ,

\lambda_{\alpha_q}(\xi)=\alpha_q(d\pi(\xi)).

If $\xi\in T_{\alpha_q}N^*S$ , then $d\pi(\xi)\in T_qS$ , while $\alpha_q$ annihilates $T_qS$ . Hence $\lambda|_{N^*S}=0$ , and therefore the canonical symplectic form, whether defined as $d\lambda$ or $-d\lambda$ , restricts to zero on $N^*S$ . The conormal bundle has rank $\operatorname{codim}_Q S$ over $S$ , so

\dim N^*S=\dim S+\operatorname{codim}_Q S=n=\tfrac12\dim T^*Q.

Isotropy plus this dimension count proves the theorem.

Examples and local coordinates

If local coordinates $(x^1,\ldots,x^k,y^1,\ldots,y^{n-k})$ make $S$ the locus $y=0$ , then

N^*S=\{y=0,\ p_x=0\}

in the induced cotangent coordinates $(x,y,p_x,p_y)$ . The free coordinates are $x$ and $p_y$ , visibly giving dimension $n$ , and the canonical form restricts to zero.

For a point $q\in Q$ , the conormal bundle is the entire cotangent fiber $T_q^*Q$ , which is Lagrangian. For an open submanifold $S\subseteq Q$ , it is the zero section over $S$ . The normal bundle of a Riemannian embedding is not the same object until a metric identifies tangent and cotangent vectors.

Geometric significance

Conormal bundles are the basic Lagrangians attached functorially to submanifolds. Their punctured versions are conic Lagrangians and provide the phase-space geometry used to record singular directions of distributions and kernels in microlocal analysis. The theorem depends only on the embedding $S\hookrightarrow Q$ , whereas identifying $N^*S$ with a normal bundle requires auxiliary metric data.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer, 1990. DOI record. Relevant: §21.2, conormal bundles as canonical Lagrangian submanifolds.
Victor Guillemin and Shlomo Sternberg, Geometric Asymptotics, revised ed., Mathematical Surveys and Monographs 14, American Mathematical Society, 1977. DOI record. Relevant: Chapter IV, cotangent symplectic geometry and conormal examples.