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Normal addition is a local diffeomorphism along the zero section

Statement

Let $S \subseteq R^{m}$ be an embedded smooth submanifold, and let $E : N^{⊥} S \to R^{m}$ be its normal addition map. For every $p \in S$ the differential $d E_{(p, 0)} : T_{(p, 0)} (N^{⊥} S) \to T_{p} R^{m} ≅ R^{m}$ is an isomorphism. Consequently, $E$ is a local diffeomorphism at every point of the zero section.

Facts & Assumptions

Given: An embedded smooth submanifold $S \subseteq R^{m}$ and its normal addition map $E$ .

[F1]

The map is $E (p, v) = p + v$ (The normal addition map for a Euclidean submanifold).

[L1]

A smooth map with invertible differential at a point is a local diffeomorphism there (The smooth inverse function theorem on manifolds).

Proof

technique · direct

1.1F1givenalgebra

At a zero vector $(p, 0)$ , the tangent space of the normal bundle splits as $T_{(p, 0)} (N^{⊥} S) ≅ T_{p} S \oplus N_{p}^{⊥} S .$ With the formula in [F1], the differential sends $(u, w)$ to $u + w \in R^{m}$ .

2.1step 1.1algebra

Because $T_{p} S$ and $N_{p}^{⊥} S$ are orthogonal complementary subspaces of $R^{m}$ , the map $(u, w) \mapsto u + w$ is a linear isomorphism. Hence $d E_{(p, 0)}$ is invertible.

3.1L1step 2.1∎

Apply [L1] at each $(p, 0)$ . The map $E$ is a local diffeomorphism along the zero section.

Depends on

Dependency tree · 2 levels

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 11, Theorem 3.54