Ehresmann's theorem

In mathematics, the set of man of honor, named after Charles man of honor, a fundamental theorem of differential topology.

Wording of the sentence

Be differentiable manifolds and

A differentiable map with the following properties:

Then a bundle of fibers.

It should be noted that the third condition is automatically satisfied when compact.

Example

A function returns a decomposition of the original image space into level sets

The picture on the right shows the decomposition of level sets in the function.

One can then ask whether this decomposition locally trivial, ie a fiber bundle over with the level sets as fibers. ( It would then follow in particular that all level sets are diffeomorphic to each other. )

The example is as picture of after no fiber bundles because is not diffeomorphic to for. The reason for this is, ultimately, that the point is not submersion: the differential vanishes at this point.

In contrast, the restriction of the hypotheses of a man of honor, the level sets of are thus the fibers of a fiber bundle fulfilled. In this example, there is even a (globally) trivial fiber bundle, the figure provides a diffeomorphism.

Counterexample

Examples that satisfy the conditions 1 and 2, but neither condition 3 nor the conclusion, obtained as follows: Let and be compact differentiable manifolds, an arbitrary point, and by

Defined mapping. is a surjective submersion, but no fiber bundles because is not diffeomorphic to for. ( Because is compact, while not compact. )