Exponential map

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This article is about the exponential map in differential geometry. For discrete dynamical systems, see Exponential map (discrete dynamical systems).
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography.

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection. Two important special cases of this are the exponential map for a manifold with a Riemannian metric, and the exponential map from a Lie algebra to a Lie group.

Definition[edit]

Let M be a differentiable manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p.[1]

Let v ∈ TpM be a tangent vector to the manifold at p. Then there is a unique geodesic γv satisfying γv(0) = p with initial tangent vector γ′v(0) = v. The corresponding exponential map is defined by expp(v) = γv(1). In general, the exponential map 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 because it relies on the theorem on 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.

Lie theory[edit]

In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group which allows one to 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 is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.

Definitions[edit]

Let G be a Lie group and \mathfrak g be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map

\exp\colon \mathfrak g \to G

which can be defined in several different ways as follows:

  • It is the exponential map of a canonical left-invariant affine connection on G, such that parallel transport is given by left translation.
  • It is the exponential map of a canonical right-invariant affine connection on G. This is usually different from the canonical left-invariant connection, but both connections have the same geodesics (orbits of 1-parameter subgroups acting by left or right multiplication) so give the same exponential map.
  • It is given by \exp(X) = \gamma(1) where
\gamma\colon \mathbb R \to G
is the unique one-parameter subgroup of G whose tangent vector at the identity is equal to X. It follows easily from the chain rule that \exp(tX) = \gamma(t). The map \gamma may be constructed as the integral curve of either the right- or left-invariant vector field associated with X. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
\exp (X) = \sum_{k=0}^\infty\frac{X^k}{k!} = I + X + \frac{1}{2}X^2 + \frac{1}{6}X^3 + \cdots
(here I is the identity matrix).
  • If G is compact, it has a Riemannian metric invariant under left and right translations, and the exponential map is the exponential map of this Riemannian metric.

Examples[edit]

  • The unit circle centered at 0 in the complex plane is a Lie group (called the circle group) whose tangent space at 1 can be identified with the imaginary line in the complex plane, \{it:t\in\mathbb R\}. The exponential map for this Lie group is given by
    it \mapsto \exp(it) = e^{it} = \cos(t) + i\sin(t),\,
that is, the same formula as the ordinary complex exponential.
  • In the split-complex number plane z = x + y \jmath , \quad \jmath^2 = +1, the imaginary line \lbrace \jmath t : t \in \mathbb R \rbrace forms the Lie algebra of the unit hyperbola group \lbrace \cosh t + \jmath \ \sinh t : t \in \mathbb R \rbrace since the exponential map is given by
    \jmath t \mapsto \exp(\jmath t) = \cosh t + \jmath \ \sinh t.
  • The unit 3-sphere S^3 centered at 0 in the quaternions H is a Lie group (isomorphic to the special unitary group SU(2)) whose tangent space at 1 can be identified with the space of purely imaginary quaternions, \{it+ju + kv :t, u, v\in\mathbb R\}. The exponential map for this Lie group is given by
    \bold{w} = (it+ju+kv) \mapsto \exp(it+ju+kv) = \cos(|\bold{w}|) + \sin(|\bold{w}|)\frac{\bold{w}}{|\bold{w}|}.\,
This map takes the 2-sphere of radius R inside the purely imaginary quaternions to \{s\in S^3 \subset \bold{H}: \operatorname{Re}(s) = \cos(R)\} = a 2-sphere of radius \sin(R) when R\not\equiv 0\pmod{2\pi}. Compare this to the first example above.

Properties[edit]

  • For all X\in\mathfrak g, the map \gamma(t) = \exp(tX) is the unique one-parameter subgroup of G whose tangent vector at the identity is X. It follows that:
    • \exp(t+s)X = (\exp tX)(\exp sX)\,
    • \exp(-X) = (\exp X)^{-1}.\,
  • The exponential map \exp\colon \mathfrak g \to G is a smooth map. Its derivative at the identity, \exp_{*}\colon \mathfrak g \to \mathfrak g, is the identity map (with the usual identifications). The exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in \mathfrak g to a neighborhood of 1 in G.
  • The exponential map is not, however, a covering map in general – it is not a local diffeomorphism at all points. For example, so(3) to SO(3) is not a covering map; see also cut locus on this failure.
  • The image of the exponential map always lies in the identity component of G. When G is compact, the exponential map is surjective onto the identity component.
  • The image of the exponential map of the connected but non-compact group SL2(R) is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with module 1, and of non-diagonalizable trigonalizable matrices with eigenvalue 1.
  • The map \gamma(t) = \exp(tX) is the integral curve through the identity of both the right- and left-invariant vector fields associated to X.
  • The integral curve through g\in G of the left-invariant vector field X^L associated to X is given by g \exp(t X). Likewise, the integral curve through g of the right-invariant vector field X^R is given by \exp(t X) g. It follows that the flows \xi^{L,R} generated by the vector fields X^{L,R} are given by:
    • \xi^L_t = R_{\exp tX}
    • \xi^R_t = L_{\exp tX}.
    Since these flows are globally defined, every left- and right-invariant vector field on G is complete.
  • Let \phi\colon G \to H be a Lie group homomorphism and let \phi_{*} be its derivative at the identity. Then the following diagram commutes:
ExponentialMap-01.png
  • In particular, when applied to the adjoint action of a group G we have
    • g(\exp X)g^{-1} = \exp(\mathrm{Ad}_gX)\,
    • \mathrm{Ad}_{\exp X} = \exp(\mathrm{ad}_X).\,

Riemannian geometry[edit]

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.

Properties[edit]

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. 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. 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 (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.

Relationships[edit]

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 (or reductive) 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 (or abelian) Lie group has such a Riemannian metric.

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

\langle u,v\rangle_y = \frac{uv}{y^2}

(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 |\cdot|_y induced by the modified metric):

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)=e^{sx/|x|}

Now using the unit speed definition, we have

\exp_1(x)=y(|x|_1)=y(|x|),

giving the expected ex.

The Riemannian distance defined by this is simply

\operatorname{dist}(a,b) = |\ln(b/a)|,

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

See also[edit]

Notes[edit]

  1. ^ A source for this section is Kobayashi & Nomizu (1975, §III.6), which uses the term "linear connection" where we use "affine connection" instead.

References[edit]