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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Interpreting the scalar curvature in a semi-Riemannian manifold

Background: Let $M$ be a smooth Riemannian manifold of dimension $n$ and scalar curvature $R$ (with respect to the Levi-Civita connection). Let $m \in M$ and let $B$ be the geodesic ball of radius $...
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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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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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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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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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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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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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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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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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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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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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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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Uniqueness of connection $1$-forms and curvature $2$-forms

If $(M,\langle\cdot,\cdot\rangle)$ is a pseudo-Riemannian manifold and $\nabla$ denotes the Levi-Civita connection of the metric, we can define the connection $1$-forms and curvature $2$-forms ...
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Anti-de Sitter space: Is the universal cover of $\text{AdS}_n$ a submanifold of $\text{AdS}_{n+1}$?

$$\text{AdS}_n = \{\vec x\in\mathbb R^{n-1,2}\mid\vec x\cdot\vec x=-1\}$$ Using an orthonormal basis $\{e_i\}$ for $\mathbb R^{n-1,2}$, with ${e_1}^2={e_2}^2=-1$ and $(e_{\geq3})^2=1$, we can ...
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Lorentz Transformations Vs Coordinate Transformations

I'm really confused about Lorentz transformations at the moment. In most books on QFT, Special Relativity or Electrodynamics, people talk about Lorentz transformations as some kind of special ...
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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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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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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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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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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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Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm (pragmatically)...
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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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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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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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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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(Strong) causality condition

I'm studying the causality in Lorentz Manifold with the book "Semi-Riemann Geometry", B.O'Neill. I have the followig problem: He says that, picked a subset A of a Lorentzian Manifold M, the causality ...
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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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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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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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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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Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm ...
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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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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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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 $u + v + w = 0$. ...
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Smooth approximation of polyhedral metric!

do someone have an idea, how to prove that : For every polyhedral metric $(S,d)$ of curvature $\leq -1$, (S=surface), there exists a sequence of smooth metrics, converging uniformly to $(S,d)$. ...
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Show that Lorentzian metric restricts to Riemannian metric on hyperbolic space

Define the Lorentzian metric $\langle \ ,\ \rangle$ on $\mathbb{R}^{n+1}$ by $\langle x, x \rangle = - x_0^2 + x_1^2 + \cdots + x_n^2$. Let $$\mathbb{H}^n = \{ x \in \mathbb{R}^{n+1} | \langle x , ...
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Index (or signature) of a pseudo-riemannian metric on manifold

Let $(M , g)$ a $n$-dimensional pseudo-riemannian manifold. As $g(p)$ is a bilinear mapping from $T_pM \times T_pM$ to $\mathbb{R}$, we can get a basis $$ {\left\{{\left(\frac{\partial}{\partial x_i}\...
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Finding local orthonormal frame on a Pseudo-Riemannian Manifold

Suppose we have a semi-Riemannian manifold $(M^n,g)$ with metric signature $(n-k,k)$. By definition, each $p \in M$ the map $g_p : T_pM \times T_pM \to \Bbb{R}$ is a non-degenerate, symmetric, ...
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Isometries in Minkowski space

Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that: Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...
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Existence of Orthonormal Basis of a Metric in a Manifold

Definition: A metric $ g$ on a manifold $ M$ is a tensor field of type $ (0,2)$ such that (1) it is symmetric, i.e. $ g(v,w)=g(w,v)$ for any $ w,v \in V_p, p\in M$, and (2) it is non-degenerate, i.e....