Journal of Lie Theory 35 (2025), No. 2, 345--358
Copyright Heldermann Verlag 2025
Bounding the Norm of the Derivative of the Lie Exponential Map for Connected Lie Groups
Reza Bidar
Faculty of Mathematics, University of Dayton, Dayton, Ohio, U.S.A.
rbidar1@udayton.edu
[Abstract-pdf]
Let $G$ be a real connected Lie group with a left invariant metric $d$, $\mathfrak{g}$ its Lie algebra, $\exp: \mathfrak{g} \rightarrow G$ be the Lie exponential map, and $\mathrm{ad}$ be the adjoint representation of $\mathfrak{g}$. In this paper we use matrix algebra and Jordan normal form to derive a set of upper and lower bounds for $|d\exp_{x}(y)|,\ x,y \in \mathfrak{g}$ that generally are exponential type functions of the eigenvalues of $\mathrm {ad}_x$. These bounds provide useful information about the exponential map and the way it relates the Euclidean metric of $\mathfrak{g}$ 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 $\mathbb{R}^n$. We prove that the exponential map is a quasi-isometry only when $G$ is isomorphic to $\mathbb{R}^n$.
Keywords: Lie group, exponential map, adjoint, quasi-isometry.
MSC: 22E15, 22E60.
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