Editing Differential topology (section)

== Description ==
Differential topology considers the properties and structures that require only a [[smooth structure]] on a manifold to be defined.  Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and [[Deformation theory|deformations]] that exist in differential topology. For instance, volume and [[Riemannian curvature]] are [[Invariant (mathematics)|invariants]] that can distinguish different geometric structures on the same smooth manifold&mdash;that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.
{{Citation needed|date=August 2020}}

On the other hand, smooth manifolds are more rigid than the [[topological manifold]]s. [[John Milnor]] discovered that some spheres have more than one smooth structure—see [[Exotic sphere]] and [[Donaldson's theorem]].  [[Michel Kervaire]] exhibited topological manifolds with no smooth structure at all.<ref name="Kervaire">{{harvnb|Kervaire|1960}}</ref> Some constructions of smooth manifold theory, such as the existence of [[tangent bundle]]s,<ref name="lashof">{{harvnb|Lashof|1972}}</ref> can be done in the topological setting with much more work, and others cannot.

One of the main topics in differential topology is the study of special kinds of smooth mappings between manifolds, namely [[immersion (mathematics)|immersions]] and [[submersion (mathematics)|submersions]], and the intersections of submanifolds via [[transversality (mathematics)|transversality]]. More generally one is interested in properties and invariants of smooth manifolds that are carried over by [[diffeomorphisms]], another special kind of smooth mapping. [[Morse theory]] is another branch of differential topology, in which topological information about a manifold is deduced from changes in the [[rank (differential topology)|rank]] of the [[Jacobian matrix and determinant|Jacobian]] of a function.

For a list of differential topology topics, see the following reference: [[List of differential geometry topics]].