Ehresmann's theorem
Encyclopedia
In mathematics
, Ehresmann's fibration theorem states that a smooth mapping
where M and N are smooth manifolds, such that
is a locally trivial fibration
. This is a foundational result in differential topology
, and exists in many further variants. It is due to Charles Ehresmann
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Ehresmann's fibration theorem states that a smooth mapping
- f:M → N
where M and N are smooth manifolds, such that
- f is a surjective submersionSubmersion (mathematics)In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
, and - f is a proper mapProper mapIn mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...
, (in particular if M is compact)
is a locally trivial fibration
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
. This is a foundational result in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, and exists in many further variants. It is due to Charles Ehresmann
Charles Ehresmann
Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...
.