Do Metric Tensor Commute?

As general tensors, metric tensors are not commutative in general (try in dimension 2 for example to construct two symmetric matrices that do not commute).

What does a metric tensor do?

In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v 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 ...

Are metric tensors invariant?

It is a basic result of special relativity that the Minkowski metric tensor is invariant under the Lorentz group. ... Any (0 2) tensor on M that is invariant under the Lorentz group is a scalar multiple of the Minkowski metric tensor γ0. (See Note 1 for the extension to a Lorentzian spacetime.)