Taylor expansion of the square square of the distance function on a Riemannian manifold

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asked May 25, 2021 at 16:05

Luis Yanka Annalisc

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Taylor expansion of the square square of the distance function on a Riemannian manifold

I have recently read the problem named "Square of the distance function on a Riemannian manifold"(enter link description here) and I am interested in the formula
$ d^2(exp_{x_0}(tv),exp_{x_0}(tw))=|v-w|^2t^2-\frac{1}{3}R(v,w,w,v)t^4+O(t^5) $.
However I can not solve it by myself. Can one give me a complete solution?

riemannian-geometry curvature

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Taylor expansion of the square square of the distance function on a Riemannian manifold

Taylor expansion of the square square of the distance function on a Riemannian manifold