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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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Existence of transverse null vector bundle

Let $(M,g)$ be a Lorentzian manifold. Let $dim(M)=d$. Given a null hypersurface, i.e., a smooth embedding $(H, \phi )$ of $M$ s.t. $dim(H)=d-1$, $h:= \phi ^* g$ over $H$ is degenerate with $rank(h_p)=...
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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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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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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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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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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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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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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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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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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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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 $(e_{\geq3})^2=1$, this space can ...
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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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A non-orientable surface $S$ such that $T_pS$ is time-type.

I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $S$ in $\mathbb{R}^3_1$ which is time-type. With the ...
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Notation regarding generalized Minkowski space

In section 12 of the book Surfaces in classical geometries: A treatment by moving frames by Gary R. Jensen, Emilio Musso and Lorenzo Nicolodi (see preview here), Möbius geometry is described. They ...
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Does divergence theorem require smoothness?

Usually, one writes the divergence theorem as \begin{equation} \int_\mathcal{M} d^4x \sqrt{-g} \nabla_\mu v^\mu=\int_{\partial \mathcal{M}} d\Sigma_\mu v^\mu\\,. \end{equation} for some vector field $...
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Irreducible decomposition of Lorentz tensors with Young Tableaux

I want to understand the irreducible decomposition of Lorentz tensors by using Young tableaux. Let me start with a trivial example. Suppose we work in $n=4$ dimensions, and that we have a rank 2 ...
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Problem at proof of Cartan's theorem about the relation between metric and curvature in do Carmo's book

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 157, Cartan's theorem. My question is, why can we take a Jacobi field $J$ in such a way that $J(0)=0$ and $J(...
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Question in the appendixes A of O'Neill book

I'm reading appendixes A of O'Neill's book, "Semi-riemannian geometry" and I don't understand a something. I don't understand at the last theorem, how we construct a universal cover for $M$ and why ...
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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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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....
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Relation between lie derivative of a vector field and associated 1-form in a Lorentzian manifold

Let $(M,g)$ be a Lorentzian manifold and $X$ and $u$ represent two vector fields in $M$ such that $\mathcal{L}_X u=0$, that is, $u$ is Lie transported along the integral curve of $X$. My question is: ...
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spacelike curves, in lorentzian geometry?

i have this question : Let $(M,g)$ be a lorentzian manifold, and $\gamma:[0,1]\rightarrow M$ be a spacelike curve in $M$, between two different point $A$ and $B$, so : can: $\underset{\gamma}{inf}\...
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The projection from the time-orientable double cover preserves topological properties.

In Relativity and Singularities, Natário states that A connected time-orientable Lorentzian manifold admits a nonvanishing vector field, and hence is either noncompact or has zero Euler ...
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Invariance of signature in semi-riemannian manifolds

The signature of a non-degenerate symmetric bilinear map $h:V\times V\to \mathbb{R}$ in a vector space $V$ is the number of negative numbers on the diagonal of the matrix $h_{ij} = h(e_i,e_j)$ when it'...
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Vacuum solutions of Einstein's field equations with $O(n)$ isometry group

In Relativity and Singularities, Natário states that Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric). ...
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Equivalent ways to refer to Lorentzian geometry

I'm a curious student about the Yamabe-type problem in Lorentzian geometry. I found a single article about this topic (2015). I don't know much about Lorentzian geometry and I wonder if one may ...
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The Fundamental Theorem of Geometric Calculus in a lorentzian manifold

I am trying to understand geometric calculus and apply it to physics. In this sense, I was reading Alan Macdonald's book "Vector and Geometric Calculus", and stumbled upon the Fundamental Theorem of ...
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Constant sectional curvature and unit normal vector to a hypersurface totally geodesic

I was reading about hypersurfaces totally geodesic when I found the next proposition: The sectional curvature $K$ of $M$ is constant at $p$ if and only if every unit vector in $T_{p}(M)$ is normal to ...
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For $0<\nu<n$ there are no compact semi-Riemannian hypersurfaces in $\mathbb{R}_{\nu}^{n}$

I'm reading a proof of the following proposition: For $0<\nu<n$ there are no compact semi-Riemannian hypersurfaces in $\mathbb{R}_{\nu}^{n}.$ The proof is as follows: Suppose that there is $M\...
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Integrating along a null geodesic

I was studying the details of integration on null hypersurfaces through A Relativist's Toolkit by Eric Poisson. If we consider a null hypersurface in 4-dimensional spacetime, then the surface element,...
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Intuition for warped product manifold.

I am reading about wave equations in manifold and encountered the term warped product manifold. More specifically, in my case it is defined as follows, $$N:=[0,\phi^*) \times_g \mathbb S^{k-1}$$ ...
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Gaussian curvature of $\exp_{p}(P)$

I'm reading about Gauss equation. During the reading there is a proof of a proposition that is not clear to me. It's the following: Let $\Pi$ be a tangent plane to $M$ at $p.$ If $P$ is a small ...
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Book recomendation on semi Riemannian geometry other than O'Neill?

I find very hard to follow O'Neill's book because it's not very formal on it's statements and proofs. Is there any modern treatment on semi Riemannian geometry with better statements and proofs?
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Connected semi-riemannian manifold is an Einstein manifold.

I'm reading the proof of the next proposition: A semi-riemannian manifold $M$ is an Einstein manifold provided $Ric=cg$ for some constant $c.$ If $M$ is connected, $n=dim(M)\geq 3$ and $Ricfg=fg,$ ...