# Questions tagged [semi-riemannian-geometry]

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37
questions

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### Lorentzian manifolds of negative spacelike sectional curvature

Suppose $(M,g)$ is a simply connected Lorentzian manifold of signature $(-,+,\ldots,+)$ and such that the sectional curvature of any space-like surface is non-positive. Is it true that there are no ...

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16 views

### Distance function and Hessian in Lorentzian geometries with positive curvature

Suppose $(M,g)$ is a Lorentzian manifold with signature $(-,+,\ldots,+)$ and a positive curvature. Let $p \in M$. Let $U$ be a sufficiently small neighborhood of $p$ in the exterior of the double null ...

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52 views

### Analogous $H^1$-space for pseudo inner products

Perhaps this is a naive question but I could not find anything related to this.
Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...

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**1**answer

60 views

### Lower bound for domain of exponential map on Lorentzian manifolds

Let $M$ denote a manifold admitting a Lorentzian metric $g_{ab}$. Essentially, I would like to know the "minimum domain" on which the exponential map is defined at $p\in M$. To make this concrete, ...

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87 views

### Can the Lie derivative of a Riemannian metric be expressed in terms of the Lie derivative of a Lorentzian metric?

On a Lorentzian manifold with metric (M,g) with a vanishing Euler-Poincare characteristic, there exists a line element vector X which has a collinear vector u (Manifold Theory: An introduction for ...

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297 views

### Maximal symmetry at the speed of light

Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)?
Here is a (...

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**1**answer

64 views

### Existence of transverse null vector bundle in a degenerate Lorentzian hypersurface

This question is cross-posted at https://math.stackexchange.com/questions/3234895/existence-of-transverse-null-vector-bundle
Let $(M,g)$ be a Lorentzian manifold. Let $\dim(M)=d$. Given a null ...

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171 views

### Completeness is a conformal invariant

In this article about completeness of semi-riemannian manifolds the author writes (in section 2.2) that it is unkown if the following statement holds:
A compact indefinite manifold which is conformal ...

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**1**answer

139 views

### $(M,g)$ is complete iff $(\tilde{M},\tilde{g})$ is complete (non-Riemannian version)

I'm not sure if this question is too low level for Math Overflow (so feel free to move this to SE if you think it is).
Inspired by this and this question I'm wondering if the following statement is ...

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124 views

### Compatibility of Kirillov-Kostant-Souriau form and Killing form

Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-...

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73 views

### Compact, incomplete semi-Riemannian manifold of constant curvature

In the Riemannian setting, Hopf-Rinow tells us that any compact Riemannian manifold is complete. The Clifton-Pohl torus gives a counter example for indefinite metrics.
However, in the Lorentz setting,...

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54 views

### What is the meaning and role of global hyperbolicity condition for semi-Riemannian manifolds

What is the heuristic meaning of the global hyperbolicity condition for semi-Riemannian manifolds?
Also what is the role of this condition in the study of geodesic connectedness?

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110 views

### (Semi-)Riemannian geometry for working PDE analysts

What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)?
The closest thing I know to this, are two books by ...

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105 views

### Analytic approach to geodesic connectedness in Semi-Riemannian manifolds

Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?

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87 views

### Surjectivity of Pseudo-Riemannian exponential map on geodesically complete manifolds

Suppose one has a geodesically complete pseudo-Riemannian manifold $M$ i.e. the exponential map is defined for all tangent vectors on the manifold. Can one make a sensible statement about whether (or ...

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252 views

### Lorentzian analogue to Thurston geometries

Is there an analogue to the eight Thurston geometries for Lorentz metrics?
If so, how many "disctinct" geometries are there in the Lorentzian case?
And which closed 3-manifolds admit metrics which ...

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84 views

### Positive and non-negative sectional curvature of semi-riemannian metrics

I am currently studying the techniques related to geometry of positive and non-negative sectional curvature of Riemannian metrics. In particular, I have done some work with Cheeger deformations. I ...

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214 views

### Closed Semi-Riemannian manifolds with non-compact isometry group

Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?
Edit: My goal is to understand a bit better what the intuition ...

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97 views

### Reference for connection of a Hessian metric

Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a ...

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151 views

### Parallel frame for marginally trapped bi-harmonic surfaces in $\Bbb R^4_2$

I'm reading the paper Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its applications to biharmonic surfaces by B. Y. Chen.
Summarizing it quickly: he first ...

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**2**answers

141 views

### Why are they called “screen” distributions?

If $V$ is a vector space and $g$ is a symmetric degenerate bilinear form on $V$, every complementary subspace to the radical ${\rm rad}(V)$ is called a "screen subspace" of $V$: we have an orthogonal ...

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**1**answer

123 views

### Does every submanifold of $\Bbb S^{n+2}_\nu$ contained in a lightlike hyperplane have lightlike Second Fundamental Form?

Let $1 \leq \nu \leq n+1$ and $M^n \subseteq \Bbb S^{n+2}_\nu$ be a non-degenerate submanifold. Assume that $\renewcommand{\vec}[1]{{\bf #1}} \vec{L}_0 \in \Bbb R^{n+3}_\nu$ is lightlike and $M \...

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254 views

### Possible mistake in classification of marginally trapped submanifolds of $\Bbb R^{n+2}_{p+1}$

I am trying to read the paper Marginally Trapped Submanifolds in Space Forms with Arbitrary Signature by Henri Anciaux, but I think that there is a mistake in Lemma $1$, in page $5$:
The second ...

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**1**answer

155 views

### Relation of pseudo-torsion with curvature in degenerate plane

Question: I'd like to know if there is some reference or reasonable way to develop curve theory in a plane with degenerate metric $(\Bbb R^2, {\rm d}s^2 ={\rm d}x^2)$.
Context: In Lorentz-Minkowski ...

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132 views

### Limit of curvature near lightlike points

Let $\alpha \colon I \to \Bbb R^2_1$ be a regular curve and $t_0 \in I$ be such that $\alpha$ is lightlike at $t_0$, and not lightlike at $]t_0-r,t_0[$ for some $r>0$. Then, in that interval the ...

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444 views

### Tensor Field Decomposition in Space time

For vector fields in $R^3$ one knows that there exists a unique decomposition of vector fields in to solenoidal (divergence free) and potential parts. A generalization of this Theorem for symmetric ...

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290 views

### “Correct” definition of signed curvature in Minkowski plane

We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively.
Looking at references such as ...

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**1**answer

252 views

### Induced connection on null hypersurfaces

I will use a local coordinate formalism here, since this is related to research in general relativity, and my supervisor only tolerates local coordinate formalisms. Plus the research papers I base my ...

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326 views

### Gauss' theorem for null boundaries

Prenote: I have asked this question first on math stackexhange, but a user suggested that mathoverflow might be a better place for this question. Upon thinking about it I have agreed with him and copy-...

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**1**answer

227 views

### Eikonal equation and double null coordinates

I"m trying to understand the exact/technical link between the Eikonal equation and a double-null form of the metric (if such a direct link even exists). R. Wald, in his "General Relativity", doesn't ...

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435 views

### Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary

It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in I^-(p)...

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302 views

### The momentum constraints in the ADM formulation of general relativity

Suppose that the space-time has a time function. Let $g_{ij}$ be
the Riemannian metrics of the time slices, and $K_{ij}$ be the second
fundamental forms. It is by Codazzi equation that
$$
D^{i}(K_{ij}...

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**1**answer

145 views

### Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously

My question will rely on a clarification of a proof, which I simply don't understand.
Let us denote by $X$ a pseudo-riemannian symmetric space and define
$$
Z_{\mathrm{Iso}\left(X\right)}G(X) = \{\, ...

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416 views

### Diffusion on a semi-Riemannian manifold?

A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ...

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123 views

### Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$

Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...

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197 views

### Are all orbits semi-Riemannian submanifolds?

Let $M$ be a semi-Riemannian manifold and $G\subset Iso(M)$ a closed connected Lie subgroup which acts properly on $M$. It is known that every orbit of the action is a (closed) submanifold of $M$. My ...

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2k views

### Hodge decomposition in Minkowski space

This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section.
Let us recall Maxwell’s equations: Given a semi-...