There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... The exponential function is one of the most important functions in mathematics. ...

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. This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

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 (whose Lie algebra is the additive group of all 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. The exponential function is one of the most important functions in mathematics. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Definition

Let G be a Lie group and be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

given by exp(X) = γ(1) where

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) = γ(t). The map γ 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. In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

If G is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion: In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ...

Properties

• For all , the map γ(t) = exp(tX) is the unique one-parameter subgroup of G whose tangent vector at the identity is X. It follows that:
  • 
  • 
• The exponential map is a smooth map. Its derivative at the identity, , is the identity map (with the usual identifications). The exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in to a neighborhood of 1 in G.
• 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 map γ(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 of the left-invariant vector field X^L associated to X is given by gexp(tX). Likewise, the integral curve through g of the right-invariant vector field X^R is given by exp(tX)g. It follows that the flows ξ^L,R generated by the vector fields X^L,R are given by:
  • 
  • 
  Since these flows are globally defined, every left- and right-invariant vector field on G is complete.
• Let be a Lie group homomorphism and let φ _* be its derivative at the identity. Then the following diagram commutes:

• In particular, when applied to the adjoint action of a group G we have
  • 
  • 

In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, flow refers to the group action of a one-parameter group on a set. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Image File history File links ExponentialMap-01. ... In mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its own Lie algebra. ...

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. In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

Definition

For v ∈ T_pM, there is a unique geodesic γ_v satisfying γ_v(0) = p such that the tangent vector γ′_v(0) = v. Then the corresponding exponential map is defined by exp_p(v) = γ_v(1). 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 (this is simply due to the fact that it relies on the theorem on existence and uniqueness of ODEs which is local in nature). In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an differential equation that contains functions of only one independent variable, and derivatives in that variable. ...

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 exp_p(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 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 (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 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 (i.e. the exponential map is a local diffeomorphism). 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. If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...

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 exp_p, and another vector w based at the tip of v (hence w is actually in the double-tangent space T_v(T_pM)) 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 (i.e. the geodesics are radial). This motivates the definition of geodesic normal coordinates on a Riemannian manifold. In mathematics, more than one statement is known as Gausss lemma; all of them are named after Carl Friedrich Gauss. ...

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 exp_p of a 2-dimensional subspace of T_pM. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds. ... Curvature is the amount by which a geometric object deviates from being flat. ...

Relationships

The two notions of the exponential map coincide in the case of Lie groups equipped with bi-invariant metrics (i.e. Riemannian metrics invariant under left and right translation). 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

(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 x ∈ R 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):

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

exp₁(x) = y(|x|₁) = y(|x|),

giving the expected e^x.

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. Graph paper is paper that is printed with fine lines making up a grid. ...

See also

• Exponential function
• Matrix exponential
• List of exponential topics

The exponential function is one of the most important functions in mathematics. ... In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ... This is a list of exponential topics, by Wikipedia page. ...

References

• Manfredo P. do Carmo, Riemannian Geometry, Birkhäuser (1992). ISBN 0-8176-3490-8. See Chapter 3.
• Jeff Cheeger and David G. Ebin, Comparison Theorems in Riemannian Geometry, Elsevier (1975). See Chapter 1, Sections 2 and 3.