Differential topology

Differential topology is the field dealing with infinitely-differentiable functions on infinitely-differentiable manifolds. It arises naturally from the study of the theory of differential equations. At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of R^n. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first deriviative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability.

A vector field is a function from a manifold to the disjoint union of its tangent spaces, such that at each point, the value is a member of the tangent space at that point. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time independant differential equations. A function from the reals to the manifold defines a function from the reals to the tangent spaces -- the functional of a function will be the deriviative of the composition at that point. A function from the reals will be said to satisfy the vector field if at every point, the function to the tangent is a member of the vector field.

n-dimensional linear forms are functions from the n'th tensor of the tangent space to the reals which are anti-symmetric. They will be called differentiable if whenever operated on n differentiable fields, the result is differentiable. A space form is a linear form with the dimensionality of the manifold.

Subareas include Symplectic topology, the study of manifolds with non-degenerate bi-linear forms.