Semi-Riemannian Manifolds

Abstract

Each tangent space $M_P$ of a $C^{\infty }$-manifold M is an n-dimensional vector space. Thereby for nonnegative integers p and q, one obtains the tensor spaces $(M_P)_q^p$. In particular, one forms the dual space $M_P^*=(M_P)_1^0$. The dual space $M_P^*=(M_P)_1^0$ to the tangent space $M_P$ is called the cotangent space, and its elements are called cotangent vectors or covector or covariant vectors.