Exponential Map

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	lie groups
	riemannian geometry

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LIE THEORY The exponential map is a fundamental construction in the theory of Lie Group s. 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 Number s (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. Definition Let $G$ be a Lie Group and $\backslash mathfrak\; g$ be its Lie Algebra (thought of as the Tangent Space to the Identity Element of $G$). The exponential map is a map :$\backslash exp\backslash colon\; \backslash mathfrak\; g\; o\; G$ given by $\backslash exp(X)\; =\; \backslash gamma(1)$ where :$\backslash gamma\backslash colon\; \backslash mathbb\; R\; o\; G$ 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 $\backslash exp(tX)\; =\; \backslash gamma(t)$. The map $\backslash gamma$ 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. If $G$ 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 $X\backslash in\backslash mathfrak\; g$, the map $\backslash gamma(t)\; =\; \backslash exp(tX)$ is the unique one-parameter subgroup of $G$ whose Tangent Vector at the identity is $X$. It follows that: ---$\backslash exp(t+s)X\; =\; (\backslash exp\; tX)(\backslash exp\; sX)\backslash ,$ ---$\backslash exp(-X)\; =\; (\backslash exp\; X)^\{-1\}\backslash ,$ The exponential map $\backslash exp\backslash colon\; \backslash mathfrak\; g\; o\; G$ is a Smooth Map . Its Derivative at the identity, $\backslash exp\_\{---\}\backslash colon\; \backslash mathfrak\; g\; o\; \backslash mathfrak\; g$, is the identity map (with the usual identifications). The exponential map, therefore, restricts to a Diffeomorphism from some neighborhood of 0 in $\backslash mathfrak\; g$ 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 $\backslash gamma(t)\; =\; \backslash exp(tX)$ is the Integral Curve through the identity of both the right- and left-invariant vector fields associated to $X$. The integral curve through $g\backslash in\; G$ of the left-invariant vector field $X^L$ associated to $X$ is given by $g\; \backslash exp(t\; X)$. Likewise, the integral curve through $g$ of the right-invariant vector field $X^R$ is given by $\backslash exp(t\; X)\; g$. It follows that the . Let $\backslash phi\backslash colon\; G\; o\; H$ be a Lie group homomorphism and let $\backslash phi\_\{---\}$ 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 ---$g(\backslash exp\; X)g^\{-1\}\; =\; \backslash exp(\backslash mathrm\{Ad\}\_gX)\backslash ,$ ---$\backslash mathrm\{Ad\}\_\{\backslash exp\; X\}\; =\; \backslash exp(\backslash mathrm\{ad\}\_X)\backslash ,$ RIEMANNIAN GEOMETRY In Riemannian Geometry , an exponential map is a map from a subset of a Tangent Space T''p''''M'' of a Riemannian Manifold ''M'' to ''M'' itself. Definition For ''v'' ∈ T''p''''M'', 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'') = γv(1). In general, the exponential map really is only locally defined, that is, it only takes a small neighborhood of the origin at T''p''''M'', 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
	Consider The Point 1 &isin '''R'''<sup>+</sup>, And ''x'' &isin '''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 <sub>''y''</sub> induced by the modified metric):

  :''y''(''s'') ''e''^''sx''/''x''
  :exp₁(''x'') ''y''(''x''₁) = ''y''(''x''),
  :''dist''(''a'',''b'') ln(''b''/''a''),