Let be a differentiable manifold and a point of. An affine connection on allows one to define the notion of a straight line through the point. Let be a tangent vector to the manifold at. Then there is a uniquegeodesic satisfying with initial tangent vector. The corresponding exponential map is defined by. In general, the exponential map is only locally defined, that is, it only takes a small neighborhood of the origin at, to a neighborhood of in the manifold. This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.
Properties
Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and goes in that direction, for a unit time. Since v corresponds to the velocity vector of the geodesic, the actual distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define expp = β where β is the unit-speed geodesic going in the direction of v. As we vary the tangent vector v we will get, when applying expp, different points on M which are within some distance from the base point p—this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of "linearization" of the manifold. The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a metric space. In particular, compact manifolds are geodesically complete. However even if expp is defined on the whole tangent space, it will in general not be a global diffeomorphism. However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of TpM on which the exponential map is an embedding. The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically via expp is called the injectivity radius of M at p. The cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum. An important property of the exponential map is the following lemma of Gauss : given any tangent vector v in the domain of definition of expp, and another vector w based at the tip of v and orthogonal to v, w remains orthogonal to v when pushed forward via the exponential map. This means, in particular, that the boundary sphere of a small ball about the origin in TpM is orthogonal to the geodesics in M determined by those vectors. This motivates the definition of geodesic normal coordinates on a Riemannian manifold. The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface through the point p in consideration. Via the exponential map, it now can be precisely defined as the Gaussian curvature of a surface through p determined by the image under expp of a 2-dimensional subspace of TpM.
In the case of Lie groups with a bi-invariant metric—a pseudo-Riemannian metric invariant under both left and right translation—the exponential maps of the pseudo-Riemannian structure are the same as the exponential maps of the Lie group. In general, Lie groups do not have a bi-invariant metric, though all connected semi-simple Lie groups do. The existence of a bi-invariant Riemannian metric is stronger than that of a pseudo-Riemannian metric, and implies that the Lie algebra is the Lie algebra of a compact Lie group; conversely, any compact Lie group has such a Riemannian metric. Take the example that gives the "honest" exponential map. Consider the positive real numbersR+, a Lie group under the usual multiplication. Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product .. Consider the point 1 ∈ R+, and x ∈ R an element of the tangent space at 1. The usual straight line emanating from 1, namely y = 1 + xt covers the same path as a geodesic, of course, except we have to reparametrize so as to get a curve with constant speed. To do this we reparametrize by arc length : and after inverting the function to obtain t as a function of s, we substitute and get Now using the unit speed definition, we have giving the expected ex. The Riemannian distance defined by this is simply a metric which should be familiar to anyone who has drawn graphs on log paper.
