Questions tagged [semi-riemannian-geometry]

It is the study of smooth manifolds equipped with a non-degenerate metric tensor, not necessarily positive-definite (and hence a generalisation of [riemannian-geometry]). Included in this are metric tensors with index 1, called "Lorentzian", which are used to model spacetimes in (general-relativity).

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Length minimizing property of geodesics

I want to show that isometries map geodesics to geodesic using that an isometry of two riemannian manifolds also implies an isometry of metric spaces. I have found proofs using the length minimizing ...
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Polar coordinates, Gauss Lemma

I don't understand the following statement from the Wikipedia page "Normal coordinates": I don't see how this follows from the Gauss Lemma. The statement of the Gauss Lemma I know is: Let $(M,g)$ be ...
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35 views

Riemannian polar coordinates

Let $(M,g)$ be a Riemannian manifold, $\phi: \mathbb{R}^n \setminus \{0\} \rightarrow (0, \infty) \times S^{n-1}$, $\phi(v)=(\vert v \vert, \dfrac{v}{\vert v \vert})$ and $(U, \psi=(x^1,...,x^n)$ ...
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What is the difference/relation between Lorentzian and Finsler geometry?

I'm kind of lost among many similar concepts. What is the difference/relation among Lorentzian manifold, Finsler manifold, Minkowsky manifold (and I just came across such a Randers space, related to ...
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2answers
29 views

Vector field with constant length

Is it correct, that for some pseudo-Riemannian manifold $M$, $X \in \mathfrak{X}(M)$, if $g(X,X)=1$, then the integral curves for $X$ are geodesics? I have the following explanation: Let $\gamma$ be a ...
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2answers
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Euler characteristic on torus

I'm trying to compute $\chi(T^2)$: I know that the sectional curvature of $T^2$ is $\dfrac{\cos(t)}{2+\cos(t)}$ with the parametrization: $F(t,s)=((2+\cos(t))\cos(s),(2+\cos(t))\sin(s),\sin(t))$ Now ...
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Three dimensional Riemann tensor on a manifold with degenerate metric

Disclaimer: In this post I'm using abstract index notation. In particular brackets around indices indicates anti-symmetrization: $$A_{[a}B_{b]}=\frac{1}{2}(A_aB_b-A_bB_a).$$ Let $\mathcal{I}$ be a ...
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Reconstruct curvature tensor from sectional curvature in semi-Riemannian Manifold

Is it possible to reconstruct the curvature tensor from the sectional curvature in the semi-Riemannian case? In Kuhnel's book there is a proof using $k(X,Y)=\left\langle R(X,Y)Y,Y\right\rangle$, but ...
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16 views

Subspace is a semi-riemannian submanifold

In my lecture we have stated something like: Non-degenerate subspaces $V \subset \mathbb{R}^n_{\nu}$ are obviously geodesically complete, totally geodesic submanifolds. But I don't see how any of ...
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1answer
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¿Does a timelike non-orientable surface exist?

Consider the Lorentz-Minkowski space of dimension $3$, $\mathbb{L}^3 = (\mathbb{R}^3,\langle \: \cdot \: \rangle)$ $$ \langle u,v\rangle=u_1v_1 + u_2v_2 - u_3v_3 $$ We say that a surface $S$ is ...
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Cones is smooth hypersurface

Let $H_c:=\{x \in \mathbb{R}^n_{\nu} \vert <x,x>=c \}$. I want to proof the following statements: 1) $H_c$ is a pseudo-Riemannian hypersurface $\Leftrightarrow$ $c \neq 0$. 2) $H_0 \setminus \{...
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Lorentz Transformation and Inverses

I'm dealing with the Lorentz Transformation for the first time and I'm not sure I got it right. The task is this: Determine all the tuples ($\alpha, \beta, \gamma,\delta $) of real numbers so that $ ...
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Is the Lie derivative along the normal well defined?

This question is cross-posted at the physics stack exchange at https://physics.stackexchange.com/q/488358/83357 Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. ...
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Connected Cauchy-hypersurface

Let $(M,g)$ be a connected globally hyperbolic spacetime with noncompact Cauchy hypersurface $S$. Can we always follow, that $S$ is also connected?
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Penrose Singularity Theorem Proof

I have problems understanding the first part of the proof for the Penrose singularity theorem in the book "Leonor Godinho José Natário An Introduction to Riemannian Geometry": I know that $\langle\...
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21 views

Unit normal vector field on Lorentzian manifold

Following situation: Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit ...
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38 views

Complete is equivalent to Misner-complete for Riemannian manifolds.

I'm trying to prove that, if $M$ is a smooth Riemannian manifold, then completeness of $M$ is equivalent to Misner-completeness. A pesudo-Riemannian (or semi-Riemannian) manifold $M$ is Misner-...
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32 views

Local coordinates on Lorentzian manifold

Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit normal vector field $\nu$. ...
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1answer
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Metric on Lorentzian manifold

Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?
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Understanding of Cauchy hypersurfaces

Let $(M,g)$ be a stably causal spacetime. Then we have a global time function $t:M \rightarrow \mathbb{R}$. A set $S_a:=t^{-1}(a)$ is said to be a Cauchy-hypersurface, if the domain of dependence of $...
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Construction of local coordinates

I am currently wokring through the book "An Introduction to Riemannian Geometry" by Leonor Godinho, José Natário. I have issues understanding the following I can't understand this way of ...
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1answer
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What does it mean for a geodesic to be orthogonal to a submanifold?

I mean obviously a geodesic $c:I \rightarrow M$ is orthogonal to some tangent vector $v\in T_pM$, if $g(v, c^{'}(0))=0$ right (if $c(0)=p$) but if I have a submanifold $S \subset M$, what does it mean,...
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1answer
50 views

Critical values of exponential function

Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit normal vector field $\nu$. ...
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There is it an easier proof that if $g(X,X)$ is a constant $\nabla_X X=0$?

In the book General Relativity for matematicians by R.K.Sachs and H.Wu there is a problem which say: Let $(M,g)$ be a Lorenzian manifold,$\nabla$ the Levi-Civita conexion, $f\colon M\to \mathbb{R}$...
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1answer
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Tangent vector to a geodesic

What exactly is a Tangent vector to a geodesic? I see this term is used often but I just can't find a definition for that. I got stuck at this place in a book:
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Question on definition of a local parameterization

In my lecture we have never defined this term and the book I'm working with uses it but didn't define it either. Do I understand it right, that for some n-dimensional manifold $N$, if $\varphi:V \...
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1answer
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Question on definition of time-orientability and future-directed curves

I can't properly understand the definitions for a future directed tangent vector. Now I know the following definitions: A spacetime $(M,G)$ is called time-orientable, if there exists a vector field $...
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1answer
67 views

Uses for volume form on a pseudo-Riemannian manifold [closed]

I was thinking about a possible use for the volume form on pseudo-Riemannian manifold. In Riemannian context we can use it to define minimal surfaces. Is there some physical or geometric ...
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1answer
59 views

Manifolds that admit Lorentzian metrics?

John Lee says in "Riemannian Manifolds: An Introduction to Curvature": With some more sophisticated tools from algebraic topology, it can be shown that every noncompact connected smooth manifold ...
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Existence of timelike curve in a particular set up.

I encountered online the following exercise: Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will identify ...
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Adapted coordinates in the entire open set where a congruence is defined

Context: Null congruences that define Kundt spacetimes. However my question is a little bit more general, since the metric is not relevant. Consider an $n$-dimensional differentiable manifold $M$, an ...
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Prove that $U^\sigma \nabla_\sigma U^\mu = g^{\mu\nu}\nabla_\nu \ln V$

Let $U^\mu$ be a vector in 4-dimensional Minkowski space with norm $-1$ and $K^\mu = V(x)U^\mu$ a vector proportional to it. We can write $V(x) = \sqrt{-K_\nu K^\nu}$. (This setup comes from physics ...
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Proving that $\langle X,V\rangle=\langle Y,V\rangle$ for all vectors $V\in T_pM$ $\implies$ $X=Y$

I'm trying to prove the following statement: Let $(M,g)$ be a semi-Riemannian manifold. For $X,Y\in T_pM$, prove that if $\langle X,V\rangle=\langle Y,V\rangle$ for all vectors $V\in T_pM$, then $...
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35 views

Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it. Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null ...
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conformal invariance of Jacobi field

Is it true that the Jacobi field $J$ of a null-geodesic $\gamma$ in a Lorentzian manifold is conformally invariant? In other words if $J$ solves the Jacobi equation on $\gamma$ with respect to $g$ it ...
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34 views

Prove that a geodesic under certain hypothesis has a conjugate point

If $(M,g)$ is a riemannian manifold, and $N$ a submanifold in $M$ with $\sigma$ a geodesic normal to $P$ at $p=\sigma(0)$ under the hypotheses: -$H(\sigma'(0))=g\left(\sigma'(0)),\vec{H_p}\right)>...
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1answer
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Flow of vector field on semi-Riemannian manifold

Consider $\mathbb{R}^{n+1}$ with the metric given by $$ g(x,x) = 2x_1x_2 + \sum_{i=3}^{n+1}x_i^2 $$ and $M$ the set of $x$ such that $g(x,x)=1$. Further, take a basis $e_1,...,e_n$ for $\mathbb{R}^{n+...
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1answer
107 views

Klein bottle with constant curvature is flat

A torus (equipped with a Riemannian or Lorentzian metric) which has constant curvature must be flat because of Gauss-Bonnet theorem. Is it true that a Klein bottle (equipped with a Riemannian or ...
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1answer
81 views

Isometry group of a lorentzian metric which preserves a Riemannian metric

I'm wondering if it is possible for the isometry group of a Lorentzian metric to preserve a Riemannian metric. So if $(M,g)$ is a Lorentz manifold and $Iso(M,g)$ its isometry group, I ask myself if ...
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A question concerning boundaries of semi-Riemannian manifolds

I am reading a proof and one line says "As $M$ has no boundary..." . To the best of my knowledge, all we know about $M$ is that it is a compact semi-Riemannian manifold with constant (sectional) ...
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1answer
69 views

Computing induced metric of semi Riemannian manifold

We have the ambient space $\mathbb{R}^{n+1}$ with the metric $ds^2 = -dx_0^2+dx_1^2+...+dx_n^2$ and the submanifold $M = \{ x\in\mathbb{R}^{n+1} | -x_0^2+x_1^2+...+x_n^2=1\}$. I would like to compute ...
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Canonical projections for $\mathbb{H}^2_1$ and $\mathbb{S}^2_1$

We know that there exists a canonical projection $\pi:\mathbb{S}^2\to\mathbb{R}\mathbb{P}^2$. Are there similar transformations for hyperbolic plane $\mathbb{H}^2_1$ or pseudosphere $\mathbb{S}^2_1$ ...
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41 views

Shape operator of pseudo-spheres

In the following, I will refer by O'Neil to Barrett O'Neil's book Semi-Riemannian Geometry With Applications to Relativiy. Consider the $M=\{ x \in \mathbb{R}^n_s | f(x)=r^2\}$ with $f(x)=g_s(x,x)$ ...
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On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
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1answer
80 views

Derivative of metric along curve

Let $(M,g)$ be a semi-Riemannian manifold with Levi-Civita connection $D$. Let $\alpha : [a,b] \rightarrow M$ be a smooth curve on $M$, and let $\frac{D}{dt}$ be the induced covariant derivative on $\...
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$GL(n,G)$ General Linear group of a group $G$?

I know you can take the General linear group of some vector space $V$: $GL(V)$. For example, suppose I have a three-sphere, elements of which I might represent as $SU(2)$ since they're diffeomorphic....
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Regarding the complete reducability of the Lorentz group

I was just reading that the Lorentz group has the property of complete reducability, that is any representation can be written as the direct sum of irreducible representations. This reminds me ...
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43 views

Uniqueness of a time-orientable covering spacetime

On a standard treatment of time-orientability in Lorentzian spaces, it is usual to see a remark of the form: "If a space-time (M,g) is not time-orientable, then it has a double covering space which is....
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1answer
22 views

Zero constant mean curvature in Minkowski space versus in Euclidean space

There's a famous result in $\mathbb{R}^3$ which goes as: there are no compact minimal surfaces in $\mathbb{R}^3$.So the mean curvature cannot be zero in compact surfaces in $\mathbb{R}^3$. Now what's ...
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1answer
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There are no $3$ linearly independent lightlike vectors such that $u+v+w = 0$.

Consider the Lorentz-Minkowski space $E^n_1$, also known as $\mathbb{L}^n$. I want to prove that there are not lightlike linearly independent vectors $u, v, w \in E^n_1$ such that $u + v + w = 0$. ...