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Showing that a given metric is Riemannian

  1. Feb 12, 2012 #1
    So I have the hyperboloid in [itex] \mathbb{R}^3 [/itex] given by [itex] x^2 + y^2 - z^2 = -1 [/itex]. 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 [itex] a_1^2 + b_1^2 - c_1^2 [/itex] 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!
     
  2. jcsd
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