Journal of Lie Theory

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 $\mathbb{R}^n$.