Let M be a smooth manifold of dimension n ; for instance a surface (in the case n = 2) or hypersurface in the Cartesian space R n + 1 . At each point p ∈ M there is a vector space T p M, called the tangent space, consisting of all tangent vectors to the manifold at the point p. A metric tensor at p is a function g p (X p, Y p) which takes as inputs a pair of tangent vectors X p and Y p at p, and produces as an output a real number (scalar), so that the following conditions are satisfied: