It is of full rank: if the manifold is of dimension n, then the rank is n.
This is because a riemannian metric g, when evaluated at a point p of the manifold M, is a positive definite symmetric bilinear form g_p(.,.)on TpM. In particular, its matrix can be diagonalized: there is a basis e_1,...,e_n of T_pM such that g(e_i,e_j) = 0 if i and j are different. Then the rank is equal to the number of nonzero diagonal element. Suppose the ith diagonal element is 0. Then g_p(e_i,e_i) = 0, violating the positive definiteness of g_p.
More generally, a bilinear form has maximal rank iff it is nondegenerate.
the fundamental tensor g is also a symmetric bilinear form defined poitive USUALLY so it will rank n, I find no reason why it is of rank n-1, is a property called strong convexity
For a bilinear form B:V x V --> R, positive definiteness on V-0 or on V is the same thing since positive definiteness is a property that concerns vectors in V-0.
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