# Questions tagged [semi-riemannian-geometry]

Manifolds with a non-degenerate symmetric bilinear form in each tangent space varying differentiably but with constant index and signature.

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### How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?

We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...

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### Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that
$$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$
where $\Sigma$...

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### Comparing Honda's construction to mine

Consider a finite family $\Psi_\alpha$ where $\alpha=1,2,3,4$ of mutually transversely intersecting mixed type surfaces of $\mathbf L^3$ (Lorentz-Minkowski $3$-space) inscribed in $[-1,1]^3.$ Let the $...

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### Metric with a constant Chern–Pontryagin scalar

Do there exist 3+1 dimensional spacetimes (i.e. Lorentzian manifolds with signaure (1,3)), for which the Chern–Pontryagin scalar
\begin{equation}
K_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}_{\beta\mu\...

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### "Classifying" causally closed sets in Minkowski space

Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ ...

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### Diagonalization of symmetric matrices of functions

I asked this question some time ago in MSE but I didn't recieved any feedback.
https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions
This problem arised ...

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### Transporting a Cauchy foliation of Minkowski space

Consider a spacetime $(\zeta^{3,1},g)$
where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (...

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### Definition of the conformal metric

On page 16 of this lecture notes and in this lecture, at some point (somewhere at 1:37:45), Rod Gover defined the $\text{conformal metric}$ $\mathbb{g}$ on a conformal manifold $(M, [g])$. He ...

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### A question on future Cauchy developement

Let us consider the Minkowski spacetime $\mathbb R^{1+2}$ equipped with the metric
$$ \eta(t,x) = -(dt)^2+ (dx^1)^2+(dx^2)^2.$$
Let $\Omega$ be a bounded simply connected domain in $\mathbb R^2$ with ...

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### Conformal equivalence degenerate metric tensors

Assume that $g$ and $g'$ are metric tensors with one dimensional kernel and the same signature.
Does the classical results of Weyl (dim >3) or of Cotton (dim=3) generalise to that case, i.e. $g$ ...

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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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### Proof of equivalence between Lie triple systems and totally geodesic submanifolds

In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a ...

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### Gauge groupoid of Lorentz group & complexification

I'm learning about Lie groupoids and was inspired (by Mackenzie's book) to consider the following problem.
Consider first a principal bundle $P\xrightarrow G M$; we can construct the quotient manifold
...

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### Examples of curvature-adapted subgroups of semi-Riemannian groups

Let $G$ be a semi-Riemannian group, i.e., a Lie group equipped with a bi-invariant semi-Riemannian metric. I am looking for examples of Lie subgroups that are curvature adapted to $G$.
First, allow me ...

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### Completeness of infinitely intersecting causal geodesics in strongly causal spacetimes

Let $(M,g)$ be a connected, smooth, strongly causal Lorentzian manifold, and consider an inextendible causal geodesic $\sigma : [0,b) \to M$ (a priori, $b$ may be $\infty$) with the following property:...

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### On certain umbilic surfaces

Let $(M,g)$ be a three dimensional Lorentzian manifold with signature $(-,+,+)$ and let $p\in M$ and let $U$ be a small neighborhood of $p$. Suppose there is a smooth timelike surface $S$ containing $...

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### Smooth closed Riemannian manifolds with quasi-analytic metrics

I haven't found a reference for this exact definition in the literature so I wonder if what I want to pose here makes sense. I basically want to consider a smooth close Riemannian manifold that ...

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### Searching for a type of geometric flow in Lorentzian geometry

Let $(N,g)$ be a globally hyperbolic Lorentzian manifold. Given any smooth hypersurface $\Sigma$ in $(N,g)$ we define $\|\Sigma\|= \sup_{p \in N,X \in T_p\Sigma} |h(X,X)|$ where $h$ is the second ...

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### A question on Levi-Civita connection and a fixed hyper surface

Suppose $(M,g)$ is a three dimensional smooth compact simply connected Riemannian manifold with boundary and suppose that $\Sigma$ is a smooth simply connected hypersurface in $M$ with a smooth ...

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### A question on null geodesics in Lorentzian geometry

Suppose that $(M,g)$ is a smooth Lorentzian manifold, that $I\subset \mathbb R$ is a closed finite interval and that $\gamma: I \to M$ is a smooth null geodesic on $M$, that is to say, it satisfies
$$ ...

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### A question on light cones in Lorentzian manifolds with timelike boundary

Suppose $M= \mathbb R \times M_0$ with a Lorentzian metric $g(t,x)=-dt^2+ g_0(t,x)$ where
$M_0$ is a compact manifold with a smooth boundary and $g_0$ is a family of smooth Riemannian metrics on $M_0$ ...

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### Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds

A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. ...

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### Conformal changes of metric and Ricci curvature

Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...

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### On intersection of null geodesics

Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that
$p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the ...

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### Smoothness of conformal transformations

Given a smooth pseudo-Riemannian manifold $(M,g)$ one can define the conformal group as the set of smooth diffeomorphisms $\varphi:M\to M$ such that there is a positive smooth function $u$ with $\...

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### Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths

While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any ...

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### Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis? [closed]

Edit : (I didn't intend this as an insult or a debate discussing which way is best or better for what, I'm just asking a question for my interest and I believe in the interest of science, at least for ...

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### Globally hyperbolic spacetimes and future Cauchy developement

Let $(M,g)$ be a globally hyperbolic Lorentzian spacetime with a non-compact Cauchy hypersurface $S \subset M$. Let $ \Omega \Subset S$ be an open subset. Is it true that the future Cauchy development ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### $(M,g)$ 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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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...