What is an example of a non-Riemannian manifold? A Riemannian manifold $(M,g)$ is a manifold equipped with a metric tensor ($g$) that is both symmetric and positive definite. Now if the metric tensor is symmetric but not positive definite, then it's a pseudo-Riemannian manifold. But what the case of a manifold equipped with a metric tensor $(M,g)$ such that $g$ is not symmetric(regardless of whether it is or is not positive definite)?
That is, $\forall (x,y)\in T_{p}(M) \ g(x,y)\neq g(y,x)$ at any point $p \in M$.

If you are familiar with relativity ie Minkowski spacetime or curved Minkowski spacetime.  In this case the metric allows distinct points to have zero 'distance' or negative distance.  This is also an example of a pseudo-Riemannian manifold.
