On the singularity of the exponential map on a Lie group

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