Chapter 1 - Basic Geometric Structures on Manifolds

Abstract

In this chapter, we give brief information about geometric structures which will be used in the following chapters. In addition to the basic concepts, some geometric notions such as symmetry conditions, parallelity conditions, new product structures on manifolds, generalized Einstein manifolds, and biharmonic maps introduced very recently will be presented in this chapter. The chapter consists of six sections. In the first section, the main notions and theorems of Riemannian geometry are reviewed, such as the Riemannian manifold, Riemannian metric, Riemannian connection, curvatures (Riemannian, sectional Ricci, Scalar), derivatives (exterior, covariant, inner), operators (Hessian, divergence, Laplacian), Einstein manifolds and their generalizations, symmetry conditions, and very brief notions from integration. In the second section, we look at vector bundles and related notions. In the third section, we recall basic notions from submanifold theory and distributions on manifolds. In the fourth section, we mention Riemannian submersions, O’Neill’s tensor fields, and curvature relations of Riemannian submersions. In this section, we also recall the notion of horizontally weakly conformal maps, which will be a crucial tool for a characterization of harmonic maps. In the fifth section, we study various product structures including warped product, twisted product, oblique warped product, and convolution product on Riemannian manifolds. We also provide connection relations and curvature expressions of each case. In the last section, we give a general setting for a map between manifolds such as a vector field along a map, a connection along a map, a curvature tensor field along a map, a second fundamental form of a map, a tension field of a map. We also recall harmonic maps and biharmonic maps in detail.