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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. enter image description here

But when I searched via the Internet, I cannot find the name of this theorem, or the proof. Hope to find some reference, thanks!

Golbez
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  • I think, he meant something else, namely, that one can recover 2nd order term in the Taylor expansion of $g_{ij}$ at the origin via normal coordinates (exponential map). This result is quite standard. – Moishe Kohan Jan 29 '14 at 12:40
  • @studiosus, No, I think he meant that we can recover the metric by just Riemann curvature. Maybe I can take the picture of this theorem. – Golbez Jan 29 '14 at 12:48

1 Answers1

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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.

Moishe Kohan
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