Second Fundamental Form of a Map

Abstract

In this chapter, we introduce the second fundamental form of a map between semi-Riemannian manifolds. The second fundamental form of a map is used largely in the analysis of certain properties of maps between manifolds in the literature. However, in this chapter, we study the second fundamental form of a map only as much as we need to carry out our work on semi-Riemannian maps. We devote the first section to the definition and basic properties of the second fundamental form of a map between semi-Riemannian manifolds. In section 2, we define affine and harmonic maps between semi-Riemannian manifolds and derive certain properties of affine maps. Then in section 3, we define the divergence of a vector field along a map between semi-Riemannian manifolds and generalize the semi-Riemannian divergence theorem to such vector fields. Finally in section 4, we derive the Bochner identity of a map between semi-Riemannian manifolds which will be used in showing a Riemannian map to be affine under certain curvature conditions. Throughout this chapter, and in all following chapters, let M ₁ M ₂ and M be manifolds with dimensions n ₁, n ₂ and n, respectively. Also let (M _l, g ₁), (M ₂, g ₂ and (M, g) be semi-Riemannian manifolds of indices v _l, v ₂ and v with Levi-Civita connections $\mathop \nabla \limits^1 ,\mathop \nabla \limits^2$ and ∇, respectively, unless otherwise stated.