Immersion and submersion

A $𝐶 𝑘$ differentiable function $𝑓 : 𝑋 \to 𝑌$ between $𝐶 𝑘$ differentiable manifolds $𝑋, 𝑌$ of dimensions $𝑛, 𝑚$ respectively is (im/sub)mersive at $𝑥 \in 𝑋$ iff the Tangent map $𝑇 𝑥 𝑓 : 𝑇 𝑥 𝑋 \to 𝑇 𝑓 (𝑥) 𝑌$ at $𝑥$ is a linear (mono/epi)morphism. Such function is said to be an (im/sub)mersion iff it is an (im/sub)mersion everywhere. #m/def/geo/diff

An immersion may be thought of as a map which locally resembles the canonical immersion defined for $𝑛 \leq 𝑚$ as

𝑖 : ℝ 𝑛 \to ℝ 𝑛 \times ℝ 𝑚 - 𝑛 𝑥 \mapsto (𝑥; 0)

whereas submersion may be thought of as a map which locally resembles the canonical submersion defined for $𝑛 \geq 𝑚$ as

𝑗 : ℝ 𝑚 \times ℝ 𝑛 - 𝑚 \to ℝ 𝑚 (𝑥; 𝑦) \mapsto 𝑥

This point of view is justified by the Local (im/sub)mersion theorem.

Local (im/sub)mersion theorem

Let $𝑓 : 𝑋 \to 𝑌$ be a $𝐶 \infty$ differentiable map between $𝐶 \infty$ differentiable manifolds $𝑋, 𝑌$ of dimension $𝑛, 𝑚$ respectively, and let $𝑓 : 𝑥 \mapsto 𝑦$ . Then $𝑓$ is an (im/sub)mersion at $𝑥$ iff there exist $𝐶 \infty$ charts $𝜑 : 𝑈 \to 𝑉$ on $𝑋$ about $𝑥$ and $˜ 𝜑 : ˜ 𝑈 \to ˜ 𝑉$ on $𝑌$ about $𝑦$ with $𝑓 (𝑈) \subseteq ˜ 𝑈$ such that $𝜓 𝑓 𝜑 - 1$ is a restriction of the canonical (im/sub)mersion, #m/thm/geo/diff i.e. in the immersion case

˜ 𝜑 𝑓 𝜑 - 1 (𝑣) = 𝑖 (𝑣) = (𝑣; ⃗ 𝟎)

and in the submersion case

˜ 𝜑 𝑓 𝜑 - 1 (𝑣; 𝑤) = 𝑗 (𝑣; 𝑤) = 𝑣

Proof

Without loss of generality, we can consider $𝑋$ and $𝑌$ to be open subsets of $ℝ 𝑛$ and $ℝ 𝑚$ respectively, since we are only interested in local properties and locally these are diffeomorphic. We also assume $𝑛 \neq 𝑚$ , since otherwise this is a special case of the Inverse function theorem.

Assume $𝑓$ is an immersion at $𝑥$ . Let $𝑊 = 𝐷 𝑓 (𝑥) (ℝ 𝑛) \leq ℝ 𝑚$ where $d i m 𝑊 = 𝑛$ . Choose some complement subspace $𝑊 𝑐$ where $d i m 𝑊 𝑐 = 𝑚 - 𝑛$ . We can then define

𝐹 : 𝑋 \times 𝑊 𝑐 \to ℝ 𝑚 (𝜉; 𝑤) \mapsto 𝑓 (𝜉) + 𝑤

which has the total derivative

𝐷 𝐹 (𝑥, 0) : ℝ 𝑛 \times 𝑊 𝑐 \to ℝ 𝑚 (⃗ 𝐚, ⃗ 𝐛) \mapsto 𝐷 𝑓 (𝑥 0) (⃗ 𝐚) + ⃗ 𝐛

which is a Linear isomorphism, so by the Inverse function theorem $𝐹$ is locally a diffeomorphism. Thus taking the canonical immersion $𝑖 : 𝑋 \to 𝑋 \times 𝑊 𝑐 = ℝ 𝑚$ , we have $𝐹 𝑖 = 𝑓$ , as required.

Now assume $𝑓$ is a submersion at $𝑥$ . Let $𝐾 = k e r 𝐷 𝑓 (𝑥)$ where $d i m 𝐾 = 𝑛 - 𝑚$ by the Rank-nullity theorem and let $𝑃 : ℝ 𝑛 ↠ 𝐾$ be a projection operator onto $𝐾$ (where we make the natural identification of $ℝ 𝑛$ with $𝑇 𝑥 𝑋$ ). We can then define

𝐹 : 𝑋 \to ℝ 𝑚 \times 𝐾 𝜉 \mapsto (𝑓 (𝜉), 𝑃 (𝜉))

which has the total derivative

𝐷 𝐹 (𝑥) : ℝ 𝑛 \to ℝ 𝑚 \times 𝐾 ⃗ 𝐚 \mapsto (𝐷 𝑓 (𝑥) ⃗ 𝐚, 𝑃 ⃗ 𝐚)

which is a Linear isomorphism, so by the Inverse function theorem $𝐹$ is locally a diffeomorphism. Thus taking the canonical submersion $𝑗 : ℝ 𝑚 \times 𝐾 \to ℝ 𝑚$ , we have $𝑗 𝐹 = 𝑓$ , as required.

For the converses note, that the composition of (im/sub)mersions is an (im/sub)mersion, and coördinate charts are diffeomorphisms and hence both immersions and submersions.

Properties

Iff $𝑓$ is immersive at $𝑥$ , then $r a n k (𝑇 𝑥 𝑓) = 𝑛 \leq 𝑚$
Iff $𝑓$ is submersive at $𝑥$ , then $r a n k (𝑇 𝑥 𝑓) = 𝑚 \leq 𝑛$

#state/tidy | #lang/en | #SemBr