Search Results

Let $(M,g)$ be a Riemannian manifold and let $p$ and $q$ be two points on it and define $d(p,q)$ as the length of the minimizing geodesic between them. Now given two rectifiable paths $\gamma_1$ and $ …

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 …

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 …

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 …

I suppose given a Lie Group ($G$) and its corresponding Lie Algebra ($\mathfrak{g}$) every element in its dual defines a Maurer-Cartan form on the whole Lie Group? Let $\omega \in \mathfrak{g}^*$ be …

I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space. Coming from a background of studying Quantum Field Theory from the books like t …

I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. I guess in the condensed matter p …

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 …

I am using usual physics notations and I guess the physics motivations of this question are obvious. Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, $[S^ …

I am using these lectures by Rodnianksi and Dafermos as the reference for this question. In third point in the list on the top of page 19 they emphasize the importance of completeness of the future n …

This in reference to this fascinating lecture by Nicolai Reshetikhin- http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf Given what is said on page 13 in section 4.1 its not clear to me why …

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 been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is es …

I want to precisely understand in what sense is (if it is!) $SL(2,\mathbb{C})$ the "complexified" version of $SU(2)$? Can I think of it like choosing a natural matrix basis of the real three dimen …

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 …