So, by accident, while deriving the induced metric for a sphere in 3 dimensions I realized that the transpose of the jacobi matrix multiplied by the jacobi matrix (considering it as 3 row/column vectors)will work out the induced metric. Why is it that i≠j ends up being superfluous. One would have X=a 2-sphere in parameterized coordinates, and then g_ij= <X_;i,X_;j>. Thus one would compute <X_;1,X_;2> and the same for 2,3. Is this because the embedded manifold is an immersion, or is there something else? Thanks for any elucidation and best.