Differential geometry MOC

Preïmage theorem

Let 𝑓 :𝑋 →𝑌 be a 𝐶∞ differentiable map between 𝐶∞ differentiable manifolds 𝑋,𝑌 of dimensions 𝑛,𝑚 respectively. Let 𝑦 ∈𝑌 be a regular value of 𝑓. Then the fibre 𝑆 =𝑓−1{𝑦} is a 𝐶∞ submanifold of 𝑋 of dimension 𝑛 −𝑚. diff

Proof

Since 𝑦 is a regular value of 𝑓, 𝑓 is submersive at every 𝑥 ∈𝑆 =𝑓−1{𝑦}, so by the local submersion theorem we may define charts such that the following diagram commutes in 𝖬𝖺𝗇∞

with 𝜑 :𝑥 ↦𝑣 and ˜𝜑 :𝑦 ↦˜𝑣. Now

𝑉∩𝑗−1{˜𝑣}=𝑉∩({˜𝑣}×ℝ𝑛−𝑚)

Therewithal

𝑓−1{𝑦}=𝜑−1𝑗−1˜𝜑{𝑦}=𝜑−1𝑗−1{˜𝑣}

so 𝜑(𝑈 ∩𝑆) =𝑉 ∩({˜𝑣} ×ℝ𝑛−𝑚) which is diffeomorphic to an open subset of ℝ𝑛−𝑚. Thus 𝑓−1{𝑦} is an (𝑛 −𝑚)-dimensional 𝐶∞ 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 𝑥 ∈𝑓−1{𝑦}. Since 𝑦 is a regular value, the tangent map 𝑇𝑥𝑓 :𝑇𝑥𝑋 ↠𝑇𝑦𝑌 is a linear epimorphism (i.e. has full rank). We define 𝐾 =ker⁡𝑇𝑥𝑓 where dim⁡𝐾 =𝑛 −𝑚 by the Rank-nullity theorem, and let 𝑃 :ℝ𝑁 ↠𝐾 be a projection operator onto 𝐾 (note 𝐾 ≤𝑇𝑥𝑋 ≤ℝ𝑁). We can then define

𝐹:𝑋→𝑌×𝐾𝜉↦(𝑓(𝜉),𝑃(𝜉))

which has the tangent map

𝑇𝑥𝐹:𝑇𝑥𝑋→𝑇𝑥𝑌×𝐾⃗𝐚↦(𝑇𝑥𝑓⃗𝐚,𝑃⃗𝐚)

which is clearly a Linear isomorphism, so by the inverse function theorem 𝐹 is locally a diffeomorphism at 𝑥, i.e. maps some open neighbourhood 𝑈 of 𝑥 diffeomorphically onto a neighbourhood ˜𝑈 of (𝑦,𝑃(𝑥)), Thus 𝐹 maps 𝑓−1{𝑦} ∩𝑈 diffeomorphically onto ({𝑦} ×𝐾) ∩˜𝑈 which is diffeomorphic to an open subset of ℝ𝑛−𝑚. Thus 𝑓−1{𝑦} is an (𝑛 −𝑚) dimensional 𝐶∞ differentiable manifold.

Further properties

• 𝑇𝑥𝑆 =ker⁡𝑇𝑥𝑓

---

tidy | en | SemBr