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So I wrote up this small derivation drawing insights from the answer by Deane Yang and Steve Huntsman, With respect to the Riemann-Christoffel connection on a riemannian manifold the laplacian on th …

Just write $SU(2)$ in some local coordinates (some of the standard systems are the double-polar system (a.k.a Hyperspherical coordinates) or the single angle coordinates (a.k.a Hopf coordinates) and t …

I am not very sure of everything here but I wonder if the example given in Appendix A on Page 21 of this paper by one of my professors meets your criteria.

One is familiar from Quantum Theory that each of the angular momentum generators $L_{x,y,z}$ are Killing Fields for the standard metric on $S^2$ and the sum of the squares of these generators gives th …

A version of the uniformization theorem for the 2-manifolds that I have read about says that any connected 2-manifold is diffeomorphic to either of the 3 constant curvature model spaces modulo an acti …

How does one prove that on $S^n$ (with the standard connection) any geodesic between two fixed points is part of a great circle? For the special case of $S^2$ I tried an naive approach of just writin …

Can anyone give me a reference which explain the derivation of the partial differential operator expression for the laplacian on the euclidean n-dimensional space and on $S^n$ ? One generally writes …

There were errors in the way I framed the question last time. So doing a major revision this time. Consider $SU(2)$ as a homogeneous space $SU(2)\times SU(2)/SU(2)$ and sections of this principle …

This is an expansion which frequently occurs in the papers of Camporesi and Higuchi but I couldn't find a derivation of it in either their review papers or in standard books on homogenous vector bundl …

If $M$ is a $n-1$ dimensional Riemannian submanifold in a $1+n$ dimensional space-time manifold $(V,g)$ of pseudo-Riemannian signature $(1,n)$ and $\nabla$ be the Riemann-Christoffel connection on it. …

Let $(\hat{M},\hat{g})$ be the conformal compactification of a space-time $(M,g)$. Let $I^+$ be the conformal null infinity and $J^{-}(I^+)$ be its causal past. Then the spacetime will be called "asy …

I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry. Jurgen Jost's book does give somewhat of an argument fo …

At least about the definition of "complete future null infinity" I found some answers on the 8th page of these lectures by Klainerman. I would be glad to hear of some explanations about how the two …

I have only recently started exploring this region of homogeneous spaces and its geometry and the question is born from that and given the beginner state of my exploration the questions might sound i …

In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system. Now if on the same space one has two such metrics given as matrices then …