Geodesics and Maximal Tori

Abstract

An important theorem of Cartan asserts that any two maximal tori in a compact Lie group are conjugate. There are different ways of proving this. We will deduce it from the surjectivity of the exponential map, which we will prove by showing that a geodesic between the origin and an arbitrary point of the group has the form $t\mapsto {\mathrm{e}}^{tX}$ for some X in the Lie algebra.