More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on M consists of a metric tensor at each point p of M that varies smoothly with p.Ī metric tensor g is positive-definite if g( v, v) > 0 for every nonzero vector v. In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. For the specific case of spacetime of relativity, see Metric tensor (general relativity). This article is about metric structures on manifolds.
