Differential geometry MOC

Immersion and submersion

A differentiable function between differentiable manifolds of dimensions respectively is (im/sub)mersive at iff the Tangent map at 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 as

whereas submersion may be thought of as a map which locally resembles the canonical submersion defined for as

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

Local (im/sub)mersion theorem

Let be a differentiable map between differentiable manifolds of dimension respectively, and let . Then is an (im/sub)mersion at iff there exist charts on about and on about with such that is a restriction of the canonical (im/sub)mersion, #m/thm/geo/diff i.e. in the immersion case

and in the submersion case

Proof

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

Assume is an immersion at . Let where . Choose some complement subspace where . We can then define

which has the total derivative

which is a Linear isomorphism, so by the Inverse function theorem is locally a diffeomorphism. Thus taking the canonical immersion , we have , as required.

Now assume is a submersion at . Let where by the Rank-nullity theorem and let be a projection operator onto (where we make the natural identification of with ). We can then define

which has the total derivative

which is a Linear isomorphism, so by the Inverse function theorem is locally a diffeomorphism. Thus taking the canonical submersion , we have , 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

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