Exponential map

The exponential map is a mathematical object in the field of differential geometry, in particular from the two branches of Riemannian geometry and the theory of Lie groups.

In the field of Riemannian geometry can be any tangent vector to a Riemannian manifold in the point exactly one geodesic be associated with and. This follows from the differential equation for geodesics and applies to locally. The exponential map of the point, written as then called, the point. With this figure, an environment of a point of the manifold with a neighborhood of the zero are identified at this point in the tangent space. This leads to the Riemannian normal coordinates.

Riemannian geometry

Definition

Let be a Riemannian manifold. The tangent space is described in point. Be a sufficiently small neighborhood of zero in. The exponential map at the point

Assigns to each tangent vector to the point, where is the unique geodesic with starting point and (directed ) is speed.

This definition can be extended to the tangent bundle. Be

The set of all vectors of the geodesic is defined on the whole interval. Then for the exponential

Properties

• In importance is exponential in that it maps a neighborhood of the origin in the tangent space at p is diffeomorphic to a neighborhood of the point p in the manifold. It forms straight line through the zero point p of the tangent to geodesic from isometric. In directions perpendicular to the geodesic is not shown isometrically in general.

• The images in the environment around p under this figure are based on the geodesic normal coordinates. This property is also based the designation that a neighborhood of a point a (simple ) is convex environment if every pair of points can be connected in this environment by a single geodesic of the manifold, lying entirely in this environment. 1932 was shown by Whitehead that any semi- Riemannian manifold contains such convex environments for each point and hence normal coordinates exist in the neighborhood of the point. This environment is then also called convex normal environment.

• Another special feature applies to these environments in the Lorentzian geometry. So all the points p are in this neighborhood U (q ) to q, which can be reached from q by time-like curves in Us, points of the form p = EXPQ (v ) for some v in TqM with g (v, v) < 0, where g ( ·, · ) denotes the metric of the manifold. Intuitively, this means that in these environments, all points that can be achieved by a time-like curve can be achieved by a time-like geodesic.