Differential geometry

Merging from differential topology

Differential topology is the field of mathematics dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations.

A differential manifold is a topological space with a collection of homeomorphisms from open sets to the open unit ball in Rⁿ such that the open sets cover the space, and if f, g are homeomorpisms then the function f^-1 o g from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every homemorphism results in an infinitely differentiable function from the open unit ball to R.

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 Rⁿ. 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 derivative 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-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point.

An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V^* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.

Subareas include Symplectic topology, the study of symplectic manifolds. A symplectic manifold is a differentiable manifold equipped with a closed non-degenerate bi-linear alternating form (a closed 2-form for short).

In mathematics, differential geometry is basically the study of geometry using calculus.

It has many applications in physics, especially in the theory of relativity.

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, p-forms, integrals of p-forms, exterior derivatives, wedge products, Lie derivatives, and Stokes' theorem. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature.

Intrinsic vs. Extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces and other objects were considered as lying in a space of higher dimension (for example a surface in an ambient space of three dimensions). Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. This is more powerful, and for example necessary in relativity where space-time cannot naturally be taken as extrinsic. (In order to define curvature, some structure such as a connection is necessary, so there is a price to pay.) The Nash embedding theorem shows that the points of view can be reconciled for Riemannian geometry even for global properties.

Riemannian geometry

A special case of differential geometry is Riemannian manifolds (see also Riemannian geometry): geometrical objects such as surfaces which locally look like Euclidean space and therefore allow the definition of analytical concepts such as tangent vectors and tangent space, differentiability, and vector and tensor fields.

The manifolds are equipped with a metric, which introduces geometry because it allows to measure distances and angles locally and define concepts such as geodesics, curvature and torsion.