# Questions tagged [semi-riemannian-geometry]

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27 questions

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### 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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83 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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48 views

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### Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...

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82 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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36 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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209 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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78 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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172 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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59 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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148 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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117 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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120 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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247 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

151 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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112 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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312 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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235 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

200 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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240 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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188 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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417 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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265 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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135 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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375 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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113 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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188 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-...