Exponential map

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself.

Definition

For v ∈ TpM, there is a unique geodesic γv satisfying γv(0) = p such that the tangent vector γ′v(0) = v. Then the corresponding exponential map is defined by expp(v) = γv(1). In general, the exponential map really is only locally defined, that is, it only takes a small neighborhood of the origin at TpM, to a neighborhood of p in the manifold (this is simply due to the fact that it relies on the theorem on existence and uniqueness of ODEs which is local in nature).

Properties

Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and going in that direction, for a unit time. Since v corresponds to the velocity vector of the geodesic, the actual (Riemannian) distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define expp(v) = β(|v|) where β is the unit-speed geodesic (geodesic parameterized by arc length) 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 (which justifies the usual term geodesically complete for a manifold having an exponential map with this property). 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 (Riemannian manifolds would be pretty boring if that were the case). 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 (i.e. the exponential map is a local diffeomorphism). 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.

An important property of the exponential map, is the following lemma of Gauss (yet another Gauss's lemma): given any tangent vector v in the domain of definition of expp, and another vector w based at the tip of v (hence w is actually in the double-tangent space Tv(TpM)) and orthogonal to v, 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 (i.e. the geodesics are radial). 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 (i.e. a slicing of the manifold by a 2-dimensional submanifold) 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.

Why "exponential"?

The name comes from the fact that it coincides with the matrix exponential in the case of bi-invariant metrics on Lie groups, when one is using a matrix representation of the group, and its Lie algebra as tangent space at the identity.

As a special case of the above, take the example that gives the "honest" exponential map. Consider the positive real numbers R+, 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

<u,v>y = uv/y2

(multiplying them as usual real numbers but scaling by y2). (This is what makes the metric left-invariant, for left multiplication by a factor will just pull out of the inner product, twice-canceling the square in the denominator).

Consider the point 1 ∈ R+, and xR an element of the tangent space at 1. The usual straight line emanating from 1, namely y(t) = 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 ("constant speed", remember, is not going to be the ordinary constant speed, because we're using this funny metric). To do this we reparametrize by arc length (the integral of the length of the tangent vector in the norm |.|y induced by the modified metric):

$\displaystyle s(t) = \int_0^t |x|_{y(\tau)} d\tau = \int_0^t \frac{|x|}{1 + \tau x} d\tau = |x| \int_0^t \frac{d\tau}{1 + \tau x} = \frac{|x|}{x} \ln|1 + tx|$

and after inverting the function to obtain t as a function of s, we substitute and get

y(s) = esx/|x|.

Now using the unit speed definition, we have

exp1(x) = y(|x|1) = y(|x|),

giving the expected ex.

The Riemannian distance defined by this is simply

dist(a,b) = |ln(b/a)|,

a metric which should be familiar to anyone who has drawn graphs on log paper.