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What complete local invariants are there in Riemannian geometry? Specifically, is Riemannian curvature tensor such a complete local invariant?

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what do you mean by complete local invariant? –  user20266 Jan 20 '12 at 19:14
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@Thomas: I mean roughly a way to assign to each Riemannian manifold $M$ a mapping $W_M$ with the domain $M$ such that for each two Riemannian manifolds $M$, $N$ there exists a local isometry from some neighborhood of $p \in M$ to $N$ iff $W_M(p) = W_N(f(p))$. I'm sorry, I couldn't Google anything rigorous about it. –  Alexei Averchenko Jan 20 '12 at 19:29
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Curvature doesn't work that locally, but if you know the 2 curvatures are "the same" in some neighborhood (not just at a point), then the two neighborhoods are isometric. –  Jason DeVito Jan 20 '12 at 21:01

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