Metric tensor
Definition

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 {\displaystyle \mathbb {R} ^{n+1}}. At each point p ∈ M there is a vector space T 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 (X, Y) which takes as inputs a pair of tangent vectors X and Y at p, and produces as an output a real number (scalar), so that the following conditions are satisfied:

A metric tensor field g on M assigns to each point p of M a metric tensor g in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function g (X, Y) (p) = g p (X p, Y p) {\displaystyle g(X,Y)(p)=g_{p}(X_{p},Y_{p})} is a smooth function of p.
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