I am watching an online lecture by John Morgan on lektorium, and at about 33:00, he claimed a theorem which I have never heard before, that
There is a formula in terms of Riemann curvature $R$ for $\exp^*(g_{ij}(x^1,\cdots,x^n))$, $\exp^*$ is the pullback mapping by Gaussian map.
But when I searched via the Internet, I cannot find the name of this theorem, or the proof. Hope to find some reference, thanks!
I cannot really tell from this picture what he meant. However, it could be that he was referring to Cartan's theorem, which indeed allows one to reconstruct metric from its curvature in a small normal ball via the inverse of the exponential map. You can find this theorem with the proof for instance in do Carmo's "Riemannian Geometry" book, chapter 8, section 2.