marlow
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 Jul2 awarded Curious Mar3 asked Axiom of choice in proof of Wigner's theorem? Mar3 awarded Scholar Mar3 accepted Fourier series/transform with cutoff Feb11 asked Fourier series/transform with cutoff Dec19 awarded Promoter Dec16 asked Infinite dimensional reps of the rotation group Jan29 awarded Tumbleweed Jan22 revised Geometric algebra approach to Lorentz group representations deleted 5 characters in body Jan22 asked Geometric algebra approach to Lorentz group representations Jan22 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. Jan22 asked Interpreting the scalar curvature in a semi-Riemannian manifold Dec27 revised examples of 'continuous bases of functions,' like the Fourier transform added another property I want the functions to have Dec27 asked examples of 'continuous bases of functions,' like the Fourier transform Dec23 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 $Vol(B_{alt})$. Dec23 awarded Editor Dec23 revised Alternate definition of a 'geodesic ball' added Appendix Dec23 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. Dec22 asked Alternate definition of a 'geodesic ball' Dec19 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.