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Results tagged with dg.differential-geometry
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user 5993
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
1
vote
Lower bound for domain of exponential map on Lorentzian manifolds
You've waited a long time for an answer. And I am surprised that no one has written one.
Let $M=\mathbb{R}\times(0,\infty)$. The exponential map isn't defined at $(x,t)$ for vectors $(u,s)$ with $s<-t …
- 490
2
votes
Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstru...
A conformal structure with Lorentzian representative produces a conformally invariant causal structure. Causal structure + volume = unique metric is claimed in Bombelli and Meyer (page 2) 1976. I'm un …
- 490
2
votes
1
answer
168
views
What are the spacelike boundaries referred to in theorem 4.1 of "Smoothing and extending cos...
I've been struggling with this question for a while.
In theorem 4.1 of "Smoothing and extending cosmic time functions" Seifert proves that a time function defined on a compact subset of a stably causa …
- 1,446
1
vote
Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths
The Raychaudhuri Equation is called the Raychaudhuri Equation because of a physist called Raychaudhuri. In mathematics the same equation occurs, for the reason you have pointed out, but it is called s …
- 490
7
votes
Penrose’s singularity theorem
Wille's answer is technically true, but he doesn't talk about the historical context of the result. I think that is important for understanding why such a "simple" result is deserving of a Nobel prize …
- 1,097
1
vote
Conjugate point to spacelike hypersurface
There are two cases: Jacobi fields defined in terms of a geodesic spray from a point and a geodesic spray from a surface. In both cases the differential equation that defines the Jacobi tensor is the …
- 490
2
votes
1
answer
914
views
Reference for existence and uniqueness of differential equations for low differentiability?
My specific situation is that I have a non-spacelike continuous future directed curve $\gamma:[0,a)\to M$ in a Lorentzian manifold. The curve must necessarily satisfy a local Lipschitz condition and t …
- 8,086
2
votes
Eikonal equation and double null coordinates
Per Willie's answer, locally this is the same thing. The global situation is, predictably, very different. I don't know anything specifically about the eikonal equation, but I do know about global sol …
- 490
4
votes
Accepted
Conditions for existence of Penrose diagrams
There is no definitive answer to your specific question, so I'm going to talk around the topic and hope that it's informative.
As you've noted the classical examples all basically look like special c …
- 490
4
votes
Exponential map and covariant derivative
There are many ways to talk about covariant derivatives and the exponential maps on manifolds. I will discuss the exponential map from the coordinate point of view.
The covariant derivative can be us …
- 490