Differential geometry

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.

See also the list of differential geometry topics.