Riemannian manifolds

Abstract Let M be a differentiable manifold. We say that M carries a pseudo Riemannian metric if there is a differentiable field g = (gm} , m ∈ M, of non-degenerate symmetric bilinear forms gm on the tangent spaces Mm of M. This makes the tangent space into an inner product space. Let Y, Z be differentiable vector fields defined over an open set U of M. By asserting that g is differentiable we mean that the function g( Y, Z) is differentiable. Often we will write < Y, Z> instead of g( Y, Z). If in addition the forms gm are positive definite we call the metric Riemannian. A differentiable manifold with a (pseudo-)Riemannian metric is called a (pseudo-)Riemannian manifold.