In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, or specifically, in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, Ehresmann's lemma or Ehresmann's fibration theorem states that if a
smooth mapping
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
, where
and
are
smooth manifolds, is
# a surjective
submersion, and
# a
proper map (in particular, this condition is always satisfied if ''M'' is
compact),
then it is a
locally trivial fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
. This is a foundational result in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
due to
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differential ...
, and has many variants.
See also
*
Thom's first isotopy lemma
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, 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