Homework Help: Showing that a given metric is Riemannian

1. Feb 12, 2012

demonelite123

So I have the hyperboloid in $\mathbb{R}^3$ given by $x^2 + y^2 - z^2 = -1$. I define the scalar product of 2 tangent vectors (a1, b1, c1) and (a2, b2, c2) at a point as a1a2 + b1b2 - c1c2.

I want to show that this defines a Riemannian metric on the hyperboloid. I know that a Riemannian metric must be linear, symmetric, and positive definite. Linearity and symmetry I have checked easily but I am confused on the positive definite portion. The problem tells me to show that it is a Riemannian metric but it seems like it does not satisfy the positive definite requirement. The matrix associated with $a_1^2 + b_1^2 - c_1^2$ is just the 3x3 identity matrix with the (3, 3) entry being -1 instead of 1. Thus, the eigenvalues of this matrix are 1, 1, and -1 so it seems to me that this is actually indefinite since the eigenvalues are not all the same sign.

Am I correct in saying this or is there something I am missing? thanks!