In heard from a someone in verbatim that if you take the taylor series of a certain function, if the Hessian is positive definite, then it is a metric.
This quote is without context and therefore much of my confusion. Can someone please clarify what is the connection between a Hessian and a "metric"?
A metric on a smooth manifold is by definition a positive definite quadratic form on the tangent space of the manifold at a point, defined at every point in such a way that the dependence of the form on the point is smooth. If you like you can generate the quadratic from from the symmetric bilinear form defined by the Hessian of a multivariate function. You would have to clarify in what context this occurs to see how useful such an approach is.