Applications To Splitting Theorems

Abstract

As we know from the semi-Riemannian regular interval theorem (Theorem 6.3.1), the existence of an affine solution f of a nonnull eikonal equation on a semi-Riemannian manifold (M, g) yields a splitting of (M, g) into a semi-Riemannian product manifold, provided that ∇ f is a complete vector field on (M, g). We also know from Proposition 6.2.5 that being affine for a solution f of a semi-Riemannian eikonal equation is related to the Ricci curvature of (M,g) and the Hessian tensor of f. Now, by combining these results, one can obtain splitting theorems for certain semi-Riemannian manifolds which admit solutions to nonnull eikonal equations. In fact, in this chapter we obtain such splitting theorems in a more general context for semi-Riemannian, Riemannian and Lorentzian manifolds. We devote sections 7.1, 7.2 and 7.3 to such splitting theorems for semi-Riemannian, Riemannian and Lorentzian manifolds, respectively. Section 7.3 may be considered of separate interest for applications of semi-Riemannian maps in General Relativity.