On the singularity of the exponential map on a Lie group

Abstract:

Let $\mathfrak {G}$ be a connected (real or complex) Lie group with Lie algebra G. Define a conjugate point g of $\mathfrak {G}$ as a point $g = \exp x$ for some $x \in G$ and $d{\exp _x}$ is a noninvertible linear map. We prove that $g \in \mathfrak {G}$ is a conjugate point if and only if $g = \exp {x_\lambda }$ for at least a (complex parameter) family of elements ${x_\lambda }(\lambda \in {\mathbf {C}})$ in G.