Exponential map - Citizendia

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. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In the mathematical field of Differential geometry, an affine connection is a geometrical object on a Smooth manifold which connects nearby Tangent 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. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

1 Definition
2 Lie theory
3 Riemannian geometry
- 3.1 Properties
4 Relationships
5 See also
6 References

Definition

An affine connection on a manifold M allows one to define the notion of a geodesic. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces

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. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces Then the corresponding exponential map is defined by exp_p(v) = γ_v(1). In general, the exponential map 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 because it relies on the theorem on existence and uniqueness of ODEs which is local in nature). In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its

Lie theory

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. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie 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 non-zero real numbers (whose Lie algebra is the additive group of all real numbers). The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the real numbers may be described informally in several different ways 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

Let $G$ be a Lie group and $\mathfrak g$ be its Lie algebra (thought of as the tangent space to the identity element of $G$ ). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since 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 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 $exp(X) = γ(1)$ 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

. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R It follows easily from the chain rule that

exp(t X) = γ(t)

. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. The map

γ

may be constructed as the integral curve of either the right- or left-invariant vector field associated with

X

. In Mathematics, an integral curve for a Vector field defined on a Manifold is a curve in the manifold whose tangent vector (i In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.

If $G$ is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:

$\exp (X) = \sum_{k=0}^\infty\frac{X^k}{k!} = I + X + \frac{1}{2}X^2 + \frac{1}{6}X^3 + \cdots$

(here

I

is the identity matrix). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

If G is compact, it has a Riemannian metric invariant under left and right translations, and the exponential map is the exponenetial map of this Riemannian metric.

Examples

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

$it \mapsto \exp(it) = e^{it},\,$

that is, the same formula as the ordinary complex exponential. In Mathematics, a unit circle is In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

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:
- $\exp(t+s)X = (\exp tX)(\exp sX)\,$
- $\exp(-X) = (\exp X)^{-1}\,$
The exponential map $\exp\colon \mathfrak g \to G$ is a smooth map. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability Its derivative at the identity, $\exp_{*}\colon \mathfrak g \to \mathfrak g$ , is the identity map (with the usual identifications). Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some The exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in $\mathfrak g$ to a neighborhood of 1 in $G$ . In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable
The image of the exponential map always lies in the identity component of $G$ . In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity 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.
The map $γ(t) = exp(t X)$ is the integral curve through the identity of both the right- and left-invariant vector fields associated to $X$ . In Mathematics, an integral curve for a Vector field defined on a Manifold is a curve in the manifold whose tangent vector (i
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:
- $\xi^L_t = R_{\exp tX}$
- $\xi^R_t = L_{\exp tX}.$
Since these flows are globally defined, every left- and right-invariant vector field on G is complete. In Mathematics, a flow formalizes in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in Engineering
Let $\phi\colon G \to H$ be a Lie group homomorphism and let $φ *$ be its derivative at the identity. Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some Then the following diagram commutes:

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

In Riemannian geometry, an exponential map is a map from a subset of a tangent space T_pM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its Elliptic geometry is also sometimes called Riemannian geometry. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseduo) Riemannian manifold is given by the exponential map of this connection.

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 Mathematics, the Hopf–Rinow theorem is a set of statements about the Geodesic completeness of Riemannian manifolds It is named after In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined 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. This article is about the Identity Map software design pattern In Mathematics, the inverse function theorem gives sufficient conditions for a Vector-valued function to be Invertible on an Open region containing 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.

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. In Riemannian geometry, Gauss's lemma asserts that any sufficiently small Sphere centered at a point in a Riemannian manifold is perpendicular to every Gauss's lemma can mean any of several lemmas named after Carl Friedrich Gauss: Gauss's lemma (polynomial Gauss's lemma 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 Riemannian geometry, the normal coordinates at p consist of a chart such that locally the symmetric part of the Christoffel symbols vanish i

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. In Mathematics, specifically Differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures 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.

Relationships

In the case of Lie groups with a pseudo-Riemannian metric invariant under both left and right translation the exponential maps of the pesudo-Riemannian structure are the same as the exponential maps of the Lie group. In general Lie groups do not have pseudo-Riemannian metrics invariant under both right and left translations, though all connected semisimple (or reductive) Lie groups do. The existence of a Riemannian metric invariant under right and left translations 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

<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):

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

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

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. Format and availability of graph paper Graph paper is available either as Loose leaf paper or bound in Notebooks It is becoming

References

Manfredo P. This is a list of exponential topics, by Wikipedia page See also List of logarithm topics. 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.

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Contents

Definition

Lie theory

Definitions

Examples

Properties

Riemannian geometry

Properties

Relationships

See also

References