A metric tensor g on M assigns to each point p of M a metric gp 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
is a smooth function of p.
Read more about Metric Tensor: Components of The Metric, Intrinsic Definitions of A Metric, Arclength and The Line Element, Canonical Measure and Volume Form