Bounding the Norm of the Derivative of the Lie Exponential Map for Connected Lie Groups

Let G be a real connected Lie group with a left invariant metric d, g its Lie algebra, exp : g G be the Lie exponential map, and ad be the adjoint representation of g.In this paper we use matrix algebra and Jordan normal form to derive a set of upper and lower bounds for |d exp x (y)|, x, y g that generally are exponential type functions of the eigenvalues of ad x .These bounds provide useful information about the exponential map and the way it relates the Euclidean metric of g and the left invariant metric of G .For Lie groups for which the exponential map is a diffeomorphism, we investigate conditions under which the exponential map is a quasi-isometry.This is obviously true if G is isomorphic to R n .We prove that the exponential map is a quasiisometry only when G is isomorphic to R n .