Question

Is it correct to sum over either index of the metric the same way?

Answer 1

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectorsv and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v