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Background:

Let $M$ be a smooth, Riemannian manifold with metric $g$ and dimension $n$. Let $R^a_{bcd}$ be the Riemann tensor with respect to the Levi-Civita connection for $g$.

Question:

Is there any rigorous result that gives a good intuitive sense of the meaning of the scalar curvature $R = R^{ab} R_{ab}$?

Discussion:

What I have in mind is something like the following:

For $n =2$, the volume of a geodesic ball of radius $\epsilon$ in $M$ is $\pi \epsilon^2 [1 - (R / 48) \epsilon ^2 + O(\epsilon^4) ]$. I may have the numerical factors wrong, but the point is this: $R$ tells you the difference between the volume of a geodesic ball and an ordinary Euclidean ball (for small radius). That's the kind of result I'm looking for.

My problem with this result is that it holds in normal geodesic coordinates but not in general coordinates. (Note that a choice of coordinates is necessary to define 'a geodesic ball of radius $\epsilon$'). If you know how to generalize this result to arbitrary coordinates, or you know another result that gives some intuition for $R$, please let me know.

Thanks for any help!

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    Have you looked up comparison theorems, such as Bishop-Gromov or Heintze-Karcher? Also, coordinates are not necessary to define geodesic $\epsilon$-balls; using geodesics to define distance induces a metric space structure on a Riemannian manifold.2011-12-12
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    @Neal: I think that Bishop-Gromov only applies when you have a bound on Ricci curvature, which is a significantly stronger condition than having a bound on scalar curvature. Likewise, Heintze-Karcher needs sectional curvature bounds, as I understand it (but I'm much less familiar with this).2011-12-12
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    I wasn't clear. My idea is that those theorems might help form intuition about the relationship between curvature (but not necessarily scalar) and other geometric quantities, such as the volume of distance balls.2011-12-12

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