All Questions
Tagged with riemannian-geometry mp.mathematical-physics
44 questions
1
vote
0
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48
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Question on gamma matrices
Let $(M,g)$ be a pseudo-Riemannian spin manifold and let us denote by $S$ the spinor bundle, i.e. the associated vector bundle with respect to the spin representation. Usually, the "gamma ...
- 1,171
4
votes
1
answer
547
views
Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
1
vote
0
answers
117
views
Question on globally hyperbolic manifolds and coordinates
Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as
$$M=\mathbb{R}...
- 1,171
6
votes
1
answer
334
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Tensor component calculation
First of all, this question may be more suited for the Math stack exchange site. If anyone finds this question irrelevant here, please transfer to the relevant site.
Recall that in terms of Weyl and ...
- 319
0
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0
answers
134
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Positive mass theorem and Seiberg-Witten equations
Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis:
"We also investigate some
basic facts on Spin$^c$ structure on $4$-...
- 954
5
votes
1
answer
368
views
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
- 134
3
votes
0
answers
126
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On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,429
1
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0
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59
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Number of divergence free symmetric two tensor in dimension 4 [duplicate]
In a $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any ...
- 319
3
votes
1
answer
333
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Definitions fundamental forms and their geometric Intuition
Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many ...
- 319
2
votes
0
answers
92
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Linearization stability condition
The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs.
Theorem.
Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
- 319
3
votes
2
answers
222
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$2\mathrm{d}$ area maximizing short embeddings
Think of a beach ball on an pool of water or sand.
Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a ...
- 134
2
votes
1
answer
182
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Signs of curvatures of integrals lines of frames with constant principal values
Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ...
- 134
0
votes
1
answer
683
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Self duality and anti-self duality of Weyl curvature in four dimension
I am trying to compute explicitly in terms of extrinsic curvature the self dual part $W^+$ and the anti-self dual part $W^-$ of the Weyl tensor $W$ associated with a codimension 1 submanifold into ...
- 11
6
votes
0
answers
129
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Deriving (Gaussian) curvature bounds from bounds on the metric
I am trying to understand a bound in Christodoulou's 2008 paper on black hole formation.
The paper considers a spacelike surface $S$ diffeomorphic to a sphere, with two metrics:
the induced metric $\...
- 419
3
votes
2
answers
297
views
Classification of conformal diffeomorphisms of Minkowski space, part 2
This is a continuation of Classification of conformal diffeomorphisms of Minkowski space
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^...
- 21.8k
0
votes
1
answer
155
views
Classification of similarity transformations of Minkowski space
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^n)^2.$$
Is there a classification of diffeomorphisms $F\colon \mathbb{R}^{n+1}\tilde\to ...
- 21.8k
4
votes
1
answer
274
views
Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric
Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric
...
- 5,405
4
votes
2
answers
390
views
Applications of flat submanifolds to other fields of mathematics
Developable surfaces in $\mathbb{R}^{3}$ have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).
I am a curious about potential or actual applications to other ...
- 965
7
votes
1
answer
1k
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Moduli space of flat connections over a Riemann surface
If I understand correctly, in the Refs below:
We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$
Namely,
$$
M_{\...
- 10.5k
36
votes
7
answers
5k
views
Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?
I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free.
The idea is to use ...
- 395
9
votes
2
answers
568
views
Some Mathematical Questions on Gravitational Waves and Numerical Relativity
Due to the recent spate of detections of gravitational waves by LIGO, my amateurish interest in the mathematics of general relativity has been revived.
The wave-forms of the detected gravitational ...
- 1,396
6
votes
1
answer
580
views
A step in the proof on the uniqueness of mass
I am reading the survey paper "The Yamebe Problem" by Lee and Parker. In section 9, Theorem 9.6 in P.78, it was proved that the mass is well defined in the sense that $m(g)$ depends only on the metric ...
- 193
2
votes
0
answers
193
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Chebyshev polynomials on a Riemannian manifold
I have been wondering whether it is possible to extend the notion of Chebyshev polynomials to non-Euclidean domains, if such a generalization is possible, or if one already exists. I am particularly ...
- 91
2
votes
0
answers
271
views
Einstein's field equation on orbifolds
I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).
Here, by an orbifold I mean the "stacky" quotient of, ...
user95283
6
votes
1
answer
435
views
The Yamabe problem and $\phi^4$ scalar field theory?
The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap
In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the ...
- 787
13
votes
4
answers
3k
views
General Relativity and Differential Geometry intuitions of Second Bianchi Identity
In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-
$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$
It is said that ...
- 3,574
4
votes
2
answers
528
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Obtaining Killing fields from the tetrad
I'm reading the following article by Newman
http://scitation.aip.org/content/aip/journal/jmp/4/7/10.1063/1.1704018
about the generalization of the Schwarzschild metric. My question is the following: ...
- 475
19
votes
2
answers
4k
views
Exact Definition of Dirac Operator
Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...
- 2,091
1
vote
1
answer
213
views
Some quantities which definitions are (somehow) similar to the classical Divergence
Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
- 356
2
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0
answers
255
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The Cauchy Problem in General Relativity: Existence of a Hausdorff Development
This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...
- 307
2
votes
1
answer
279
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Is the structure constant additive on connected components?
This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...
8
votes
1
answer
336
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Short examples that are/are not quantum-ergodic
Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if almost ...
- 131
1
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1
answer
370
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On the Geroch's argument
During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below:
Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
- 1,303
5
votes
2
answers
3k
views
Van Vleck-Morette Determinant
There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...
- 9,853
1
vote
1
answer
177
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pre-symplectic and foliation and its trajectories
Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?
user21574
2
votes
1
answer
335
views
Time has dimension $2$ with respect to the Ricci flow scaling
Terence Tao in his lecture notes on Ricci flow has written:
If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the ...
- 1,303
7
votes
0
answers
996
views
On Perelman's paper
In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Perelman has written:
Fix a closed manifold $M$ with a probability measure $m$, and suppose
that our system is ...
- 1,303
4
votes
2
answers
2k
views
Energy functional
During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works $\mathcal{F}(g,f)=\...
- 1,303
2
votes
1
answer
1k
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recognizing Kahler manifolds of complex dimension n
Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...
user21574
4
votes
1
answer
1k
views
Conformal Killing spinors
In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).
If $\epsilon$ and the $\bar{\epsilon}$ ...
- 3,541
6
votes
1
answer
1k
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How the Jacobi metrics may be useful in mechanics with or without constraints?
A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$
If $V{<}...
- 4,306
4
votes
0
answers
242
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Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds
I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...
- 1,431
8
votes
2
answers
630
views
"Noncommutative heat equation" -- a strange generalization of Killing vectors for a flat metric
Let $(M,g)$ be a smooth (pseudo)Riemannian manifold with a flat metric $g$, and $X$, $Y$ be vector fields on $M$ such that
$$
L_X^2 (g)=L_Y(g). \hspace{70mm} \mbox{(1)}
$$
where $L_Z$ is the Lie ...
- 389
12
votes
2
answers
1k
views
Special Holonomy Groups for Lorentzian Manifolds
Let $X$ be a Riemannian manifold. If $X$ is simply connected, irreducible, and not a symmetric space then we know that the possible holonomy groups of the metric on $X$ are:
1) $O(n)$ General ...
- 2,087