Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 1

A submersion is a smooth map between manifolds $F:X^n\to Y^m$ for whichis surjective at every $p\in X$.

The local submersion theorem says that around every $p\in X$ there are coordinates $(x^1,\dots ,x^n)$ centered at $p$ and $(y^1,\dots ,y^m)$ centered at $F(p)$ in whichTo prove it, surjectivity lets us choose $m$ source coordinates such that $d{F}^1,\dots ,d{F}^m$ are independent. Complete them by source coordinates $x^{m+1},\dots ,x^n$ and defineThe derivative of $G$ is invertible at $p$, so the inverse function theorem makes $G$ a local coordinate system; in these coordinates $F$ is the displayed projection.

For $q\in Y$, the fiber is locally given byThese are slice coordinates, so $F^{-1}(q)$ is an embedded submanifold of dimension $n-m$.

Choose a Riemannian metric on $X$, using a partition of unity if necessary. For each $p$, letSince $F$ is a submersion, $D_{p}F{|}_{H_p}:H_p\to T_{F(p)}Y$ is an isomorphism. Its inverse depends smoothly on $p$, sodefines a smooth vector field on $X$ satisfying $D_{p}F(v(p))=w(F(p))$. The formula also gives $v(p)=0$ whenever $w(F(p))=0$.

Choose a coordinate ball $W$ around $q_1$ and a smaller convex coordinate ball $U$ whose closure lies in $W$. For $q_2\in U$, let $a$ be the constant coordinate vector from $q_1$ to $q_2$. Choose a smooth bump function $\rho$ supported in $W$ and equal to one on $U$, and define in the chartextending it by zero outside $W$. This is a compactly supported smooth vector field. The segment $q_1+ta$ stays in $U$, where $w=a$, so uniqueness for ordinary differential equations givesIn particular, ${\Psi }^1(q_1)=q_2$.

Fix $q_1$ and choose $U$ as in part c. For $q_2\in U$, choose the compactly supported $w$ moving $q_1$ to $q_2$, and lift it by part b to $v$. The support of $v$ is contained inwhich is compact because $F$ is a proper map. Hence $v$ is compactly supported and complete. If ${\Phi }^t$ and ${\Psi }^t$ are the flows of $v$ and $w$, thenUniqueness of integral curves givesTherefore the diffeomorphism ${\Phi }^1$ maps $F^{-1}(q_1)$ onto $F^{-1}(q_2)$. Every equivalence class is open. Its complement, being a union of the other open classes, is also open; thus each class is clopen. If $Y$ is connected, there is only one class. This proves the fiber-diffeomorphism conclusion of the Ehresmann fibration theorem.

Properness is essential. The projectionis a submersion but is not proper. Its fiber over $x\ne 0$ is diffeomorphic to $\mathbb{R}$, whereas its fiber over zero is $\mathbb{R}\setminus \{0\}$ and has two connected components.

Merely requiring every fiber to be a submanifold is also insufficient. The surjective maphas every fiber finite and therefore a zero-dimensional embedded submanifold. Some regular values have three preimages and others have one, so the fibers are not all diffeomorphic. The map fails to be a submersion at $x=\pm 1/\sqrt{3}$.