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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Conjugate points and expansion of the geodesic congruence

I am working in a Lorentzian manifold $(M, g)$ (but I think the problem would be quite similar in a Riemannian manifold) and I am considering a timelike geodesic whose tangent vector field is denoted ...
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Pullback Connection/Riccati-Equation

I'am currently trying to understand the paper: https://epub.uni-regensburg.de/23578/1/MP171.pdf The point where I'am stucked at is in the proof of Theorem 4.6. You dont need to read the whole article, ...
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Extendibility of a curve in a compact subset (Lemma 14.2 in O'Neill's book)

Lemma $14.2(5)$ of O'Neill's book Semi-Riemannian geometry with applications to relativity states the following: If $\mathcal{C}$ is a convex open set in $M$, then, a causal curve $\alpha$ contained ...
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Shape operator as a tensor field defined in a submanifold

Definition $4.18$ in O'Neill's book Semi-Riemannian geometry, with Applications to Relativity states that given a unit vector field $U$ normal to a hypersurface $M\subset \overline M$, then, there is ...
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Contraction of a lightlike vector with another vector (free cordinate)

I'm trying to prove that there exist always a vector w whose contraction with a lightlike vector u ($g(u,u)= 0$) it's always different from zero: $$g(u,v)\ne 0$$ I know how to do this with coordinates,...
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Non-degenarate subspace with indefinite inner product.

I'm little overshadowed with this problem. Let $S = (v_1, · · · , v_m) ⊂ V$ be a linearly independent set which spans a non-degenerate subspace W = L(S) and put $W_k = L(v_1, . . . , v_k)$ for every $...
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Gram-Schmidt process to construct orthonormal base in a finite vector space with indefinite scalar product.

Im choking with this exercise because of the indefinite scalar product. I know the process for the definite one. The first thing I'm asked to do is to check GS is still valid for indefinite scalar ...
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Is Lorentzian geometry an open research field?

I have been studying Lorentzian geometry and there are numerous approaches to it, both physical and mathematical. I am interested on the mathematical theory of Lorentzian geometry and more ...
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Completeness of projection of vector fields

Let $X$ and $Y$ be complete commuting vector fields on a (semi-) Riemannian manifold. Denote by $\pi X$ the component of $X$ orthogonal to $Y$, i.e. $\pi X = X - \frac{\langle X,Y\rangle}{\langle Y,Y\...
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Doubt in the Second fundamental form of a immersed manifold

Let $f \colon M^{n} \rightarrow \overline{M}^{n+m}$ be an immersion. Identifying $TpM$ with $Im(df_{p})$, we have the following decomposition $$T_{p}\overline{M} = T_{p}M \oplus (T_{p}M)^{\perp}$$ ...
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Integrals over a space with Lorentz signature metric

$$\int_{spacetime}\frac{d^4x}{(x^2)^2}=\int_{-\infty}^{\infty} dt\int_{-\infty}^{\infty} dx\int_{-\infty}^{\infty} dy\int_{-\infty}^{\infty} dz\frac{1}{(t^2-x^2-y^2-z^2)^2}$$ To show that $\int_{...
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Wrong sign in the energy method on Minkowski space

Given any vector field $X$ in Minkowski space, we can use Stokes' theorem to derive $$ \int_{t_0}^{t_1}\text{div} X \text d V_g =\int_{\{t=t_0\}} g(X,\partial_t)-\int_{\{t=t_1\}} g(X,\partial_t), $$ ...
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Partial derivative in O'Neill's book

May be this is a stupid question but... in Definition $10$ of O'Neill's book Semi-Riemannian Geometry With Applications to Relativity, it is stated that the partial derivative of a function $f\in \...
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Understanding differential geometry result

I'm studying differential topology from Barrett O'Neill, Semi Riemannian Geometry, in specific, integral curves and flow. There is a lemma that I just can't figure out its proof: If V is a vector ...
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Reading off connection 1-forms from Cartan's structural equation $de=-\omega\wedge e$

Suppose we have a Lorentzian metric of the form \begin{align} g&=-f(r)^2\,dt^2+ h(r)^2(dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\phi^2) \end{align} Where $f,h$ are say strictly positive functions. We ...
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Are there alternate geometric interpretations of the Riemann tensor based on its symmetries?

In Riemannian geometry, when we think of the Ricci tensor merely as $Ric(\vec a, \vec b)$, then it's hard to describe it other than saying it's the contraction of the Riemann tensor. But when we ...
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Changing the signs in the metric and the curvature

Suppose we have two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ and consider their product $C:=B\times F$ endowed with the metric tensor $g_C:=-\pi_B^*(g_B)+\pi_F^*(g_F)$, where $\pi_B$ and $\pi_F$ ...
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visualizing conformally compactified Minkowski immersed in $\mathbf L^3$ [closed]

Boy's surface is an immersion of the real projective plane in $\Bbb R^3.$ And the real projective plane is a compactification of $\Bbb R^2$. I've seen images of Boy's surface on the web. I'm ...
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intersecting *semi-riemannian* manifolds

I read this interesting post: Intersection of manifolds. In geometry/topology one can consider intersection of manifolds (which need not be a manifold). I'm curious if one can do this in Semi-...
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Does the Nash embedding theorem extend to pseudo-Riemannian manifolds?

Does the Nash embedding theorem extend to pseudo-Riemannian manifolds? More precisely, can any ($\mathcal{C}^k$ for some specified $k$) pseudo-Riemannian manifold be isometrically embedded into some ...
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Gaussian Curvature for Poincare upper half plane embedded in Minkowski

I started with the mapping of Poincaré half plane in Minkowski space $\mathbb{R}^{(1,2)}$ by the mapping $f(x,y)\to(X_0,X_1,X_2)$ via $$ X_0=\frac{x^2+y^2+1}{2 y}\notag\\ X_1=\frac{x}{y}\notag\\ X_2=\...
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What topology has that static spherically spacetime?

For a static spherically symmetric metric below: \begin{equation} ds^2=(r/r_{0})^4~c^2dt^2-dr^2-r^2(d\theta^2+sin^2\theta~d\phi^2),\tag{1} \end{equation} where $r$ is sectional curvature radius and $...
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The meaning of diffeomorphism invariance in general relativity

I have been studying semi-riemannian geometry from the Barrett O'Neill's book and general relativity from different sources, but I haven't been able to understand what physicist mean by diffeomorphism ...
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4 votes
1 answer
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Can an incomplete geodesic intersect a point infinitely many times?

Consider a Lorentzian manifold $(M,g)$. Must a maximal, affinely parameterized geodesic $\gamma: [0,b) \to M$ which intersects a point $p \in M$ infinitely many times be complete (i.e., have $b = \...
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Calculating the laplacian for a metric

For a manifold of dimension $n+1$, consider a metric $\tilde{g}=s^{2}\left(g-d s^{2} / s^{2}\right)$, where $g$ is the metric of a co-dimension $1$ submanifold. I calculate its laplacian to be $$\...
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Manifolds with zero Chern term

Has any study been made of manifolds where the integrand $e(\Omega)$ from the Chern-Gauss-Bonnet theorem (which I gather is called the Euler class) is identically zero? If so, is there any broader ...
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If $[[V,V^{\perp}],V^{\perp}] \subset V$, does it then follow that $V^{\perp}$ is a Lie subalgebra?

Let $g$ be a Lie algebra equipped with an ad-invariant nondegenerate symmetric bilinear form $\langle \cdot{,}\cdot \rangle$, let $V$ be a nondegenerate subspace, and let $V^{\perp}$ be the orthogonal ...
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Gauge groupoid of Lorentz group & complexification of the latter

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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Show that $[\nabla_{\mu}, \nabla_{\nu}]w_\lambda = -R^{\rho}_{\lambda \mu \nu} w_{\rho}$ using the Levi-Civita Connection.

How do I show that $[\nabla_{\mu}, \nabla_{\nu}]w_\lambda = -R^{\rho}_{\lambda \mu \nu} w_{\rho}$, given that $[\nabla_{\mu}, \nabla_{\nu}]V^\lambda = R^{\rho}_{\lambda \mu \nu} V^{\rho}$? I have ...
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5 votes
1 answer
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Is every semi-Riemannian group geodesically complete?

I recently found out from this answer that every Lie group equipped with a left-invariant Riemannian metric is a (geodesically) complete Riemannian manifold. I wonder whether the same holds also for a ...
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What is the space of pseudo-Riemannian metrics?

I want to know the structure of the space of pseudo-Riemannian metrics of definite signature (in particular I want $diag(-1,1,1,1)$). Is it possible to identify it with $R^{n(n+1)/2}$ which is the ...
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Conjugate Points and their neigbourhood – Riemannian geometry

Let $M$ be a Riemannian or pseudo-Riemannian manifold , $p,q \in M$ conjugate points. Is it true that there exists an open neigbourhood of $p$, $U \subseteq M$ such that $\forall x \in U$, $x$ has a ...
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Completeness of infinitely intersecting geodesics

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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Is every three-dimensional pseudo-Riemannian group a space form?

In an earlier post, I asked whether every three-dimensional Lie group equipped with a bi-invariant Riemannian metric has constant curvature. The answer is yes. Now I am curious to know what happens if ...
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transversely intersecting spacetimes

A maximally symmetric causal diamond is a solution to Einstein's equation with a cosmological constant. Consider a causal diamond in a maximally symmetric spacetime for a ball-shaped spacelike region $...
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Perturbation Theory Ambiguity?

I am trying to solve problem 2.1 in Schwartz, which is to derive the transformations $x \rightarrow \frac{x+vt}{\sqrt{1-v^2}}$ and $t \rightarrow \frac{t+vx}{\sqrt{1-v^2}}$ in perturbation theory. ...
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5 votes
2 answers
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Significant differences between Riemannian and pseudo-Riemannian submanifolds

In Chapter 8 of his Introduction to Riemannian Manifolds, Jack Lee discusses (Riemannian) submanifolds. In particular, in the first section, which is titled The Second Fundamental Form and presents ...
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extension problem with motivation

I am motivated by the Lorenz curve used in economics and statistics - a proper multivariate generalization will help researchers (Arnold, Taguchi, etc.) assess multidimensional risk and inequality (...
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3 votes
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Computation of Lie Derivative using Cartan's Magic Formula

In section 2.6 of the notes by Natario, he uses Cartan's magic formula $$\mathcal{L}_V\omega = i_V(d\omega) + d(i_V\omega)$$to compute the second fundamental form of timelike hypersurface $\sigma = \...
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Mixed type surfaces and type changing metrics

I'm trying to understand the paper: "Isometric deformations of mixed type surfaces in Lorentz-Minkowski space" by A. Honda. Why does Honda embed four mixed type surfaces and arrange them so ...
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Metric decay under linear transformation

Motivation: If you take a metric space which is also an analytic manifold, embed it in Minkowski $3$-space and perform a linear map (squeeze map/lorentz boost) i.e. $(ax,y/a)$ in $2D$ for real ...
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Christoffel symbols/ Christoffel forms

Let $(M,g)$ be a smooth manifold with connection $\nabla$. Then the definition for Christoffel symbols I am familiar with is $\nabla_{\partial_i} \partial_j = \Gamma^h_{ij} \partial_h$ where $\{\...
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$SO(2, 1)$ and $Spin(2, 1)$ in terms of $SL(2, R)$ and $GL(1, R)$

We know that $Spin(3)$ is a double cover of $SO(3)$, which means it is a double cover of $RP^3$. So both $Spin(3)=S^3$ and $SO(3)=RP^3$. They are not isomorphic. Now, how do we show that $SO(2, 1) = ...
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$SO(1, 1)$ and $Spin(1, 1)$ in terms of $GL(1, R)$

We know that $\mathsf{Spin}(2)$ is a double cover of $\mathsf{SO}(2)$, which means it is a double cover of $\mathsf{U}(1)$. So both $\mathsf{Spin}(2)$ and $\mathsf{SO}(2)$ are $\mathsf{U}(1)$. They ...
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3 votes
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Cauchy surface for sphere

If we have a Lorentzian manifold $\mathbb{R} \times S^n$ with metric $g= -dt^2 + ds^n$ where $ds^n$ is just the standard round metric for spheres. Does this manifold have a Cauchy surface, i.e. is it ...
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Triangles of light

This exercise is taken from the book Introduction to Lorenztian Geometry: Curves and Surfaces. Exercise 1.2.13. (Triangles of light). Prove that there are not three vectors $u,v,w\in \mathbb{L}^n$ all ...
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subspace $U \subset \mathbb{L}^n$ is of type light

This is an exercise from the book "Introduction to Lorentzian Geometry: curves and surfaces". Exercise 1.2.4. Show that a subspace $U \subset \mathbb{L}^n$ is of type light if and only if ...
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1 vote
1 answer
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Main difference between Riemannian geometry and Pseudo-Riemannian geometry

First of all I want to know: what is the main focus of each subject, what we study in each one? And secondly, why and how they are different? Thirdly, what's the connection between them? The answers I ...
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2 votes
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conformal compactification $\overline G$

Construct a conformal compactification, $\overline G$ of $G:=\Bbb R^{1,1}_*$ and/or provide a diagram of the conformal compactification of $G?$ conformal compactification Let $G$ have the metric ...
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Narrow convergence and convergence of points

I'm reading a paper about the curvature-dimension condition in Lorentzian geodesic spaces, and there's this line: "Let $\mathfrak{m}_j\rightharpoonup\mathfrak{m}_\infty$ be probability measures ...
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