Riemannian Exponential Maps and Decompositions of Reductive Lie Groups

Abstract

Let X be a complete connected riemannian manifold, Y a closed submanifold, and ℕ _Y , _X →Y the normal bundle of Y in X. Then the exponential map exp _Y , _X : ℕ _Y , _X → X is surjective. When X is a riemannian symmetric space X = G/K, G reductive, this extends a number of decomposition theo-rems of the form G = H•exp _G (s∩τ) • K, and when Y is totally geodesic in X it extends a number of “Euler angle type” formulae of the form G = HAK. The principal new features here are that H can be any reductive subgroup of G and the symmetric space X may have compact and/or euclidean factors. There are also some consequences for pseudo-riemannian manifolds and for open G-orbits on complex flag manifolds G _C/Q. The papers [11] and [12] use the result with compact factors, and [3] uses the pseudo-riemannaian result.