Why is the metric of a Riemannian manifold required to be symmetric?

Question

• As Lee Mosher pointed out int the comments, asymmetric metrics do naturally occur in some contexts. So it is definitely justified to wonder about the mathematical necessity of symmetry in the formulation of the theory of Riemannian manifolds (i.e. which theorems require symmetry).

• If we drop the assumption that $g$ is symmetric, then $g^{ij}$ is only the transpose of the actual inverse of $g_{ij}$, but this doesn't seem like a huge deal. I assume that there are other reasons for the requirement of symmetry. Is it needed for the existence of orthonormal bases? Is it needed for the uniqueness of the signature? If yes, why is the uniqueness of the signature important? Are there more reasons to require symmetry?

Answer 1

We definitely need symmetry to prove the existence of an orthonormal basis: If an orthonormal basis exists, then the corresponding Gram matrix is symmetric and thus the bilinear form is symmetric.