I would like to know how the definiteness/semi-definiteness(positive or negative)of a scalar product (i.e. a symmetric bilinear form) influence the geometric nature of a space on which they are defined??I know that they characterise Reimannian and Semi-Riemannian spaces and so on, but in what way?? Maybe it is a bit vague but I am unable to phrase my question better at this moment, any thoughts would be most appreciated. Thank you.
P.S. It would also help if the explaination could include something about the index. It is defined as the largest possible dimension of a subspace of the given vector space $W \subset V$ so that the bilinear form restricted to that space is negative definite.