In
mathematics, or specifically, in
differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a
smooth mapping , where
and
are
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, is
# a surjective
submersion, and
# a
proper map
In mathematics, a 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
There are several competing definit ...
(in particular, this condition is always satisfied if ''M'' is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
),
then it is a
locally trivial fibration. This is a foundational result in
differential topology due to
Charles Ehresmann, and has many variants.
See also
*
Thom's first isotopy lemma In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f : M \to N between smooth manifolds and S \subset M a closed Whitney stratified subset, if f, _S is proper and f, _A is a submersion for eac ...
References
*
* {{cite book, last1=Kolář, first1=Ivan, last2=Michor, first2=Peter W., last3=Slovák, first3=Jan, title=Natural operations in differential geometry, publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, year=1993, isbn=3-540-56235-4, mr=1202431, zbl=0782.53013, url=https://www.emis.de///monographs/KSM/
Theorems in differential topology