I have recently read the problem named "Square of the distance function on a Riemannian manifold"([enter link description here](https://mathoverflow.net/questions/215573/square-of-the-distance-function-on-a-riemannian-manifold)) 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?



Taylor expansion of the square of the distance function on a Riemannian manifold
The "standard" proof using Jacobi fields can be found in section 1.3 here [https://www2.math.upenn.edu/~wziller/math660/TopogonovTheorem-Myer.pdf](https://www2.math.upenn.edu/%7Ewziller/math660/TopogonovTheorem-Myer.pdf)


