Publisher Summary This chapter focuses on semi-Riemannian manifolds, which are ordered pairs (M, g) and are often known as pseudo-Riemannian manifolds. Two different metric tensors on the same manifold constitute different semi-Riemannian manifolds and the identity map of a semi-Riemannian manifold is an isometry. The object preserved in an appropriate sense by all isometries is called an isometric invariant. If M is an arbitrary semi-Riemannian manifold, its metric tensor makes each of its tangent spaces a semi-Euclidean space of the same dimension and index as M itself. The chapter demonstrates the process of defining a new vector field DvW on M whose value at each point p is the vector rate of change of W in the Vp, direction. Thus in semi-Riemannian geometry, one can freely transform a vector field into a one-form, and vice versa.