# Questions tagged [lorentzian-geometry]

Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of signature $(+,+,+,…,+)$ (which yields Euclidean geometry) but with the pseudo-Riemannian metric of signature $(−,+,+,…,+).$

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### Properties that only Lorentzian manifolds have

I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case. I admit things like: &...
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### In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure?

Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of Hawking, King, and McCarthy (see Thm 5) says that $M$ is determined up to conformal isomorphism by its class of ...
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### Decomposition of a bivector of a Lorentzian manifold [closed]

In the case of the $(1, -1, -1, -1)$ metric, a bivector (or a 2-form) $F$ decomposes uniquely as the sum of two orthogonal simple bivectors if $F^2 \ne 0$. I have the impression that it is very little ...
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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 ...
113 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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### Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
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### A Riemannian manifold with a non-degenerate metric and an inner product $u_{\beta} u^{\beta}=1$

The question is: given a Riemannian manifold with a non-degenerate metric g and an inner product $u_{\beta}u^{\beta}=1$, is $\nabla_{\mu} (u_{\alpha}u_{\beta})=0$ without demanding the trivial ...
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### $(M,g)$ 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 ...
1 vote
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### Are there Lorentzian complex manifolds?

Quick and simple... Is it possible to define complex structures on Lorentzian manifolds? If so, Can you point me to some example(s)?
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### Decomposition of a Jacobi field along a lightlike geodesic

Consider a Lorentzian manifold of dimension $1+n$ (with $n\geq1$) and a lightlike geodesic $\gamma(t)$ on it. One can define a Jacobi field $J(t)$ along $\gamma$ in the usual way without issues. In ...
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### 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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### Is the set of Lorentzian metrics metrizable?

Fix a differentiable non-compact manifold $M$. Denote by $\mathrm{Lor}(M) := \{\text{Lorentzian metrics on$M$}\}.$ One can define a topology on this set via: fix any open covering $\mathcal{A}$ on $M$...
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### 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 ...
336 views

### Canonical Metrics on 3- and 4-Manifolds

From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
1 vote
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### Support of functions on Minkowski space and their Fourier transform

Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x^2<0$ and its Fourier transform $\hat f(p)$ vanishes for $p^2<0$. Are then $f$ ...
### Find the fixed geodesic of an orientation-preserving isometry of the $3D$ hyperboloid model
Let $\mathcal{I}^3\subset\mathbb{R}^4$ be the standard hyperboloid model for hyperbolic $3$-space and consider the usual $\mathrm{SO}(3,1)$ action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathcal{I}^3$. ...
Let $$P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$$ be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...