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 Jan 22 comment Interpreting the scalar curvature in a semi-Riemannian manifold @emiliocba I'm afraid I don't see it. A hyperbolic space is an example of a Riemannian manifold, isn't it? The scalar curvature may be negative, but the metric is still positive definite. Dec 23 comment Alternate definition of a 'geodesic ball' I will look into the numerical factor. Unfortunately that doesn't account for the discrepancy. I added an appendix to my original post to explain the problematic terms I am finding in my calculation of $Vol(B_{alt})$. Dec 23 comment Alternate definition of a 'geodesic ball' Thanks for your reply. With this correction, do you think $B = B_{alt}$? I'm still not sure, for the following reason. A standard result I have seen is that the volume of a geodesic ball in two dimensions is $\pi \epsilon^2 [1 - (R / 48) \epsilon ^2 + O(\epsilon^4) ]$, where $R$ is the scalar curvature. I calculated the volume using the second definition ($B_{alt}$) in the special case of a diagonal metric and found terms that could not possibly occur in the scalar curvature. Changing $\epsilon$ to $\epsilon^{1/2}$ in my calculation does not make it match the standard result. Dec 19 comment What is the intuitive meaning of the scalar curvature R? Thanks for your reply. I should have mentioned that I ultimately want to do the same thing for a semi-Riemannian manifold. I understand how (in the Riemannian case) one can use geodesics to define a metric on the manifold, and then use the same definition of 'ball' as is used for any metric space. I wonder if you might take a look at the comment I posted below Paul's answer, in which I try to give a definition of 'geodesic ball' that also works in the semi-Riemannian case. I think something is wrong with my definition -- I'd appreciate any help. Dec 19 comment What is the intuitive meaning of the scalar curvature R? I am having trouble proving this in n=2. We want to find the volume of the geodesic ball of radius $\epsilon$ centered at $p$. Every source I've seen has a 'geodesic ball' as the set of all points $\gamma(1) \in M$, where $\gamma$ is any geodesic satisfying $\gamma'(0) = v$ for some $v\in T_pM$ s.t. $\lt \epsilon$. Would it possible to reverse E and 1, considering instead the set of all points $\gamma(\epsilon) \in M$, where $\gamma$ is any geodesic satisfying $\gamma'(0) = v$ for some $v\in T_pM$ for which $<1$? I tried the latter in a special case but it appears not to work. Dec 19 comment What is the intuitive meaning of the scalar curvature R? I have been trying to prove this in the n=2 case, without success. Oops - I posted this comment before I finished typing. More is coming. $\times$