Exponential mapping

The exponential mapping is a mathematical object from the field of differential geometry , in particular from the two sub-areas of Riemannian geometry and the theory of Lie groups .

In the area of Riemannian geometry, exactly one geodesic with and can be assigned to each tangential vector to a Riemannian manifold at the point . This follows from the differential equation for geodesics and applies locally to . The exponential map of in the point , written as , then denotes the point . With this mapping, a neighborhood of a point of the manifold with a neighborhood of the zero in the tangent space at that point can be identified. This leads to the Riemannian normal coordinates . ${\ displaystyle v}$ ${\ displaystyle M}$ ${\ displaystyle p \ in M}$ ${\ displaystyle \ gamma _ {v}}$ ${\ displaystyle \ gamma _ {v} (0) = p}$ ${\ displaystyle \ gamma _ {v} ^ {\ prime} (0) = v}$ ${\ displaystyle p}$ ${\ displaystyle v}$ ${\ displaystyle p}$ ${\ displaystyle \ exp _ {p} (v)}$ ${\ displaystyle \ gamma _ {v} (1)}$

Riemannian geometry

definition

Let be a Riemannian manifold . With which is tangent to the point described. Let the zero in be a sufficiently small neighborhood . The exponential mapping in the point ${\ displaystyle M}$ ${\ displaystyle T_ {p} M}$ ${\ displaystyle M}$ ${\ displaystyle p \ in M}$ ${\ displaystyle B _ {\ epsilon} (0)}$ ${\ displaystyle T_ {p} M}$ ${\ displaystyle p}$

{\ displaystyle \ exp _ {p} \ colon B _ {\ epsilon} (0) \ to M}

assigns the point to each tangential vector , which is the uniquely determined geodesic with the starting point and (directed) speed . ${\ displaystyle v \ in B _ {\ epsilon} (0)}$ ${\ displaystyle \ gamma _ {v} (1)}$ ${\ displaystyle \ gamma _ {v}}$ ${\ displaystyle p}$ ${\ displaystyle v}$

This definition can be extended to the tangential bundle . Be ${\ displaystyle TM}$

{\ displaystyle {\ mathcal {E}}: = \ {v \ in TM | \ gamma _ {v} \ colon [0,1] \ to M \} \ subset TM}

the set of all vectors for which the geodesic is defined over the whole interval . The following then applies to the exponential mapping ${\ displaystyle \ gamma _ {v}}$ ${\ displaystyle [0,1]}$ ${\ displaystyle \ exp \ colon {\ mathcal {E}} \ to M}$

{\ displaystyle \ exp (V): = \ gamma _ {V} (1) \ ,.}

properties

Exponential mapping is important because it maps a neighborhood of the origin in the tangential space at p to a neighborhood of the point p in the manifold diffeomorphically . It maps straight lines through the zero point p of the tangential space on geodetic isometric . In general, the mapping is not isometric in directions perpendicular to the geodetic.

The images in the area around p below this figure are the basis of the geodetic normal coordinates . The designation that a neighborhood around a point is a ( simple ) convex neighborhood is based on this property , if every pair of points in this neighborhood can be connected by a single geodesic of the manifold that lies completely in this neighborhood. In 1932 it was shown by Whitehead that every semi-Riemannian manifold contains such convex neighborhoods for every point and consequently normal coordinates exist in the neighborhood of the point. This environment is then also called the convex normal environment.

Another special property applies to these environments in the Lorentz geometry . Thus all points p in this neighborhood U ( q ) around q , which are reached from q by time-like curves within U s, are points of the form p = exp _q ( v ), for a v in T _q M with g (v , v) <0, where g (·, ·) denotes the metric of the manifold. In clear terms, this means that in this environment all points that can be reached by a time-like curve can also be reached by a time-like geodesic.

Individual evidence

↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , p. 72.
^ Whitehead JHC: Convex regions in the geometry of paths , Quart. J. Math. Oxford Ser. 3, 33-42 (1932)
^ Hawking SW, Ellis GFR: The Large Scale Structure of Spacetime , Cambridge University Press, Cambridge. pp.103-105 (1976)

literature

Beem, JK, Ehrlich, PE, Easley, KL: Global Lorentzian Geometry, Pure and Applied Mathematics 202, 2nd Edition. New York: Marcel Dekker, Inc. 1996

[1] John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , p. 72.

[2] Whitehead JHC: Convex regions in the geometry of paths , Quart. J. Math. Oxford Ser. 3, 33-42 (1932)

[3] Hawking SW, Ellis GFR: The Large Scale Structure of Spacetime , Cambridge University Press, Cambridge. pp.103-105 (1976)