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Let $(M,g)$ be a Riemannian manifold. Then, it is usually said that $M$ has local invariants associated to $g$. For example, the curvature of the Levi-Civita connection associated to $g$. My question is: in which sense these curvature-related objects are local invariants?

In addition, it is usually said that a symplectic manifold $(M,\omega)$ has no local invariants associated to the symplectic form $\omega$. I guess this is related to the Darboux theorem, which implies that locally every symplectic form is the standard symplectic form, but, what is the precise form of this statement?

Thanks.

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Local invariants in a broad sense are quantities that can be used to distinguish manifolds endowed with some structure, at least locally. There are global invariants too. For instance, on a Euclidean plane, given two circles of unequal radii we can tell them one from another by a global Euclidean invariant (the perimeter), or by a local invariant (the inverse of the radius, aka the curvature).

To make the concept of local invariant more precise let us recall that usually we identify isomorphic objects in a category. If we deal with a manifold equipped with a geometric structure (usually given as a tensor, or a set of tensors, satisfying, perhaps, some additional conditions), then we may think of a category of local diffeomorphisms, preserving this structure. For a Riemannian structure we speak about isometries, and for the symplectic structure we have symplectomorhisms.

Local invariants are again some tensorial (coordinate-independent) quantities, which do not change under structure-preserving diffeomorphisms. In the Riemannian case we have the Riemann curvature, and all its covariant derivatives, and their contractions, and all possible linear combinations of such quantities. The fact that the Riemannian curvature is a local invariant is sometimes referred to as "the naturality of the Riemannian curvature".

The Darboux theorem for the symplectic geometry states that all symplectic manifolds are locally symplectomorhpic (which is equivalent to the existence of Darboux coordinates). Hence no local invariants.

A more advanced example would be the unformization theorem, which implies that in dimension two there are no local conformal invariants. By the way, there is a famous example of a global conformal invariant of a surface (2-dimensional) in a 3-dimensional manifold, the Willmore energy.

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