Differential geometry MOC

Preïmage theorem

Let be a differentiable map between differentiable manifolds of dimensions respectively. Let be a regular value of . Then the fibre is a submanifold of of dimension . diff

Proof

Since is a regular value of , is submersive at every , so by the local submersion theorem we may define charts such that the following diagram commutes in

with and . Now

Therewithal

so which is diffeomorphic to an open subset of . Thus is an -dimensional differentiable manifold.

Direct proof

Since Lyle Noakes has an irrational distaste for the local submersion theorem, we present a direct proof here. Note that this is essentially the same as the above proof, just with the content of the proof of the local submersion theorem absorbed.

Let . Since is a regular value, the tangent map is a linear epimorphism (i.e. has full rank). We define where by the Rank-nullity theorem, and let be a projection operator onto (note ). We can then define

which has the tangent map

which is clearly a Linear isomorphism, so by the inverse function theorem is locally a diffeomorphism at , i.e. maps some open neighbourhood of diffeomorphically onto a neighbourhood of , Thus maps diffeomorphically onto which is diffeomorphic to an open subset of . Thus is an dimensional differentiable manifold.

Further properties

---

tidy | en | sembr