Encyclopedia
There are two different notions of an
exponential map in differential geometry, both of which generalize the ordinary
exponential function of mathematical analysis.
Lie theory
The
exponential map is a fundamental construction in the theory of Lie groups. It is a map from the
Lie algebra of a Lie group to the group which allows one to completely recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras.
The ordinary
exponential function of mathematical analysis may be viewed as a special case of the exponential map when
G is the multiplicative group of positive real numbers . The Lie-theoretic exponential map satisfies many properties analogous to those the ordinary exponential function, however, it also differs in many important respects.
Definition
Let be a Lie group and be its
Lie algebra . The
exponential map is a map
given by where
is the unique one-parameter subgroup of whose tangent vector at the identity is equal to . It follows easily from the chain rule that
. The map may be constructed as the
integral curve of either the right- or left-invariant
vector field associated with . That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
If is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:
Properties
- For all , the map is the unique one-parameter subgroup of whose tangent vector at the identity is . It follows that:
- The exponential map is a smooth map. Its derivative at the identity, , is the identity map . The exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in to a neighborhood of 1 in .
- The image of the exponential map always lies in the identity component of . When is compact, the exponential map is surjective onto the identity component.
- The map is the integral curve through the identity of both the right- and left-invariant vector fields associated to .
- The integral curve through of the left-invariant vector field associated to is given by . Likewise, the integral curve through of the right-invariant vector field is given by . It follows that the flows generated by the vector fields are given by:Since these flows are globally defined, every left- and right-invariant vector field on is complete.
- Let be a Lie group homomorphism and let be its derivative at the identity. Then the following diagram commutes:
- In particular, when applied to the adjoint action of a group we have
Riemannian geometry
In Riemannian geometry, an
exponential map is a map from a subset of a tangent space T
pM of a Riemannian manifold
M to
M itself.
Definition
For
v ∈ T
pM, there is a unique geodesic γ
v satisfying γ
v =
p such that the tangent vector γ′
v =
v. Then the corresponding
exponential map is defined by exp
p = γ
v. In general, the exponential map really is only
locally defined, that is, it only takes a small neighborhood of the origin at T
pM, to a neighborhood of
p in the manifold .
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 distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define exp
p = 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 exp
p, 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 exp
p 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 T
pM on which the exponential map is an embedding . The radius of the largest ball about the origin in T
pM that can be mapped diffeomorphically via exp
p is called the
injectivity radius of
M at
p.
An important property of the exponential map is the following lemma of Gauss : given any tangent vector
v in the domain of definition of exp
p, and another vector
w based at the tip of
v 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 T
pM 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 exp
p of a 2-dimensional subspace of T
pM.
Relationships
The two notions of the exponential map coincide in the case of Lie groups equipped with bi-invariant metrics . In this case the geodesics through the identity are precisely the one-parameter subgroups of
G.
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
. .
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
- y = esx/|x|.
Now using the unit speed definition, we have
- exp1 = y = y,
giving the expected
ex.
The Riemannian distance defined by this is simply
- dist = |ln|,
a metric which should be familiar to anyone who has drawn graphs on
log paper.
See also
References
- Manfredo P. do Carmo, Riemannian Geometry, Birkhäuser . ISBN 0-8176-3490-8. See Chapter 3.
- Jeff Cheeger and David G. Ebin, Comparison Theorems in Riemannian Geometry, Elsevier . See Chapter 1, Sections 2 and 3.
