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Exponential map - Relationships | | Exponential map - Relationships: Encyclopedia II - Exponential map - 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< ...
See also:Exponential map, Exponential map - Lie theory, Exponential map - Definition, Exponential map - Properties, Exponential map - Riemannian geometry, Exponential map - Definition, Exponential map - Properties, Exponential map - Relationships | | | Exponential map, Exponential map - Definition, Exponential map - Lie theory, Exponential map - Properties, Exponential map - Relationships, Exponential map - Riemannian geometry, Exponential function, Matrix exponential, List of exponential topics | | |
| | | Exponential map: Encyclopedia II - Exponential map - Relationships
Exponential map - 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) = esx/|x|.
Now using the unit speed definition, we have
exp1(x) = y(|x|1) = y(|x|),
giving the expected ex.
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.
Other related archivesExponential function, Gauss's lemma, Gaussian curvature, Hopf-Rinow theorem, Lie algebra, Lie group, Lie groups, List of exponential topics, Matrix exponential, ODEs, Riemannian geometry, Riemannian manifold, abstract definition of curvature, adjoint action, chain rule, commutes, compact, derivative, diffeomorphism, differential geometry, existence and uniqueness, exponential function, flows, geodesic, identity component, identity element, identity map, integral curve, inverse function theorem, log paper, matrix Lie group, matrix exponential, metric space, one-parameter subgroup, real numbers, sectional curvature, smooth map, tangent space, tangent vector, vector field
 Adapted from the Wikipedia article "Relationships", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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