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The Ehresmann fibration theorem says that a surjective submersion $p:M\to B$ between smooth manifolds without boundary is a smooth fiber bundle if $p$ is a proper map. In particular, the fibers over a connected base are mutually diffeomorphic.

Locally lift coordinate vector fields from $B$ to $M$. Properness confines each lifted integral curve over a compact base path to a compact inverse image, so its flow exists for the required time. Composing these flows yields a local trivialization; inverse flows give its inverse. This explains why merely being a submersion does not suffice.

If $M$ is a manifold with boundary and $B$ has no boundary, require both $p$ and its restriction to the boundary to be submersions. The lifts can then be chosen tangent to the boundary and the same argument applies.^[ref]

At an isolated zero $p$ of a smooth vector field $X$ on an oriented $n$-manifold, its index is the mapping degree of the normalized field on a sufficiently small surrounding coordinate sphere:The sphere contains no other zeros. Shrinking it without crossing another zero gives a homotopy of normalized maps, so the degree stays unchanged.

For a nondegenerate zero, the derivative $D{X}_p$ is invertible and the index is $\mathrm{sign}\det D{X}_p$. In the plane, sources and sinks therefore have index $+1$, while a saddle has index $-1$. Degenerate zeros can have other values: $X(x,y)=(x^2-y^2,2xy)$ winds twice around a small circle and has index two.

The Poincaré-Hopf theorem sums these local indices to the Euler characteristic on a closed manifold, with the corresponding boundary condition needed when a boundary is present. Thus the local winding detects a global obstruction to a nowhere-zero field.^[ref]

Peter Petersen authored an exposition of smooth manifolds, transversality and de Rham cohomology. Among its structural results is Ehresmann fibration theorem, which turns a proper submersion into a locally trivial fiber bundle. This connects a differential condition on a map with a uniform local description of all its fibers.^[ref]

For a closed, oriented smooth manifold $M$ and a smooth vector field $X$ with isolated zeros, the Poincaré-Hopf theorem statesThe index of a vector field at an isolated zero is the mapping degree of $X/\Vert X\Vert$ on a small surrounding sphere, computed in an oriented coordinate chart. Thus local winding at the zeros adds up to the global Euler characteristic.^[ref]

For the gradient of height on a two-dimensional sphere, the north and south zeros each have index one; their sum is $\chi (S^2)=2$. For a compact manifold with boundary, the same sum requires an outward-pointing boundary condition. A constant nonzero vector field on a closed disk shows why unrestricted boundary behavior cannot be allowed.