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

Solution

Fix

q_{1}

and choose

U

as in part c. For

q_{2} \in U

, choose the compactly supported

w

moving

q_{1}

q_{2}

, and lift it by part b to

v

. The support of

v

is contained in

F^{- 1} (supp w),

which is compact because

F

v

is compactly supported and complete. If

Φ^{t}

and

Ψ^{t}

are the flows of

v

and

w

, then

\frac{d}{d t} F (Φ^{t} (p)) = D_{Φ^{t} (p)} F (v) = w (F (Φ^{t} (p))) .

F \circ Φ^{t} = Ψ^{t} \circ F .

Φ^{1}

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.

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