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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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Orthogonal lightlike (null) vectors are colinear

I want to prove the statement that in a Lorentzian vector space, i.e., vector space with a scalar product whose index is 1, if lightlike vectors are orthogonal, then they are colinear. Equivalently, ...
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What math paper is this from?

I tried to write it in latex, but apparently it's too complicated for this website to read the code. Here is the link to it: https://ibb.co/dA4cV5 Anyway, it has been 2 years since I got it. Now, I ...
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81 views

Mesh smoothing problem and normalized mean curvature flow

I am trying to implement mesh smoothing algorithm mentioned here But I got the problem with the delta of the vertex position (formula 13). The mean curvature flow is divided by the sum of the ...
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143 views

Pseudo Riemann metric and Riemann metric

I have two questions The first question : Who can give me an example of a manifold is not paracompact and we can't define a Riemann metric on it ? The second one I want read about pseudo ...
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Justifying “determinant” of second fundamental form using the definition of sectional curvature

Question: if $M\subseteq \widetilde{M}$ is a non-degenerate hypersurface in a pseudo-Riemannian manifold $M$, then is it true that $$\langle X, \widetilde{\nabla}_XN\rangle =\langle Y, \widetilde{\...
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Lorentz Group. Positive determinant ?!

I really hope someone can help with that.. $ c>0 \in \mathbb{R}. \mathbb{R^2}$ is a vector space with a symmetric form s with $ s((x,t),(x',t')) := xx'-c^2tt'$. to Show is that all elements ...
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257 views

Universal Cover of the Lorentz Group in $n$ Dimensions

I'm interested in learning some Lorentzian geometry and Spin geometry. From what I've heard, the idea is to lift the psuedo-orthonormal frame bundle, an $O(1,n)$-bundle, to a double cover, a principal ...
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126 views

Construction of a naturaly adapted coordinate system to a reference frame

Let $(M,g)$ be a four-dimensional Lorentzian manifold representing spacetime. We define a reference frame on the open subset $U\subset M$ to be a unit timelike future-directed vector field $Z$. We ...
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43 views

Projection from $SO^+(1,4)$ into $\mathbb{S}^3$

This question is related with the projective light cone model of the conformal geometry of $\mathbb{S}^3$. Let me denote the Minkowski space $\mathbb{R}_1^5$ as $\mathbb{R}^5$ equipped with a ...
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References request about infinitesimal rigidity of surfaces in Minkowski space.

I am reading an article about rigidity of surfaces in Minkowski space time and i am having big trouble for finding references about the topic. I have been working with the material and definitions ...
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49 views

Reparametrisation of geodesic

Suppose that we have a lorentzian manifold M, with a geodesic $\gamma_1:I_1\to M$ that is not constant. Here $I_1$ is an open interval in the real line. Now suppose that $\gamma_2:I_2\to M$ (where $...
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Definition of gradient of a function $f$ in Riemannian manifold

I'm reading Semi Riemannian Geometry with applications to relativity by Barret Oneill and I'm trying understand the definition of gradient of a function $f$ in Riemannian Manifold. I know that ...
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117 views

Inertial frames: from General Relativity to Special Relativity

here I come with a question about inertial frames as defined in General Relativity, and how to prove that the general definition is consistent with the particular case of Special Relativity. So to ...
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Deducing tensorial structure of a tensor

Consider an expression of the following form: $$I^{\mu\nu}(r) = \int d^{3}k\ \ d^{3}l\ \ \delta^{4}(r-k-l)\ (g^{\mu\nu}k\cdot{l}+k^{\nu}k^{\mu}-k^{\mu}l^{\nu})$$ $I^{\mu\nu}$ must be of the form $$...
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45 views

When can a cone be a lightcone?

Suppose I am given a cone of vectors in $\mathbb{R}^n$ with $n\geq3$, which we can take to be an $(n-2)$-parameter family of vectors $V^a(\theta_1,\ldots,\theta_{n-2}$) along with all scalar multiples ...
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193 views

How to prove that a sphere cannot carry a Lorentz metric?

An exercise in "Riemannian Geometry" by Gallot, Hulin, Lafontaine (p. 53): [Check that] For instance, there is no Lorentzian metric on the sphere $S^2$. I am aware of this question and also of ...
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Timelike straight line maximize energy for normal variations in Minkowski Space

First some nomenclature: Let $Q(s,t): [0,1]\times[0,1]$ $\rightarrow$ $L^4$ be a smooth map. Let $x^i$ be affine coordinates in $L^4$. For each $t$ we define a longitudinal curve of $Q$ by $\gamma_t(...
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Why is an interval invariant on coordinate transformations?

I learned that in (semi-)Riemannian geometry an interval $ds^{2}=g_{ab}dx^{a}dx^{b}$ is invariant on coordinate transformations. But there is something I do not get in the concept. Consider the simple ...
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Does an observer on an expanding three-sphere naturally have a hyperbolic sense of time?

On can consider the 3-sphere of radius a as being embedded in a four-dimensional Euclidean space. One has in this view the condition for any coordinate system with origin at the center of the 3-...
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What are conditions on existence of local orthogonal coordinate system on two-dimensional Lorenzian manifold?

Consider two-dimensional Lorenzian manifold i.e. of signature $(1,1)$. Does there always exist on every point local coordinate chart so that the corresponding coordinate one-forms are orthogonal i.e. $...
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64 views

Surface orientation when integrating a 2-form in Minkowski space

Let $\bf F$ be differential 2-form on a 4-dimensional Lorentzian pseudo-Euclidean manifold $M$ with signature (3, 1) endowed with coordinate functions (t, x, y, z), where t increases in the dimension ...
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How can we plot a parametric curve in a Lorentzian space using Mathematica.

We have the following informations: The components of a metric tensor are given by $g_{11}=-2z, g_{22}=4x^{2}+2z, g_{33}=1,g_{23}=g_{32}=2x$, where $(x,y,z)$ are standard cartesian coordinates where $...
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Unparametrized lightlike geodesics are invariant under conformal change. How does the parametrization change?

Let $(M,g)$ be a pseudo-Riemannian manifold and $\gamma: I \rightarrow M$ be a lightlike geodesic with $0 \in I$. Let $\hat{g}=e^{2 \sigma} g$ be a conformally equivalent metric. Maybe having to make $...
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Gram-Schmidt process in Minkowski space $\Bbb L^n$.

I'm trying to prove a version of Gram-Schmidt orthogonalization process in Minkowski space $\Bbb L^n$ (for concreteness, I'll put the sign last). I am not interested in the existence of orthonormal ...
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120 views

Gauss' theorem for null boundaries

Note: I have solved this problem on my own, mostly while actually typing it in here, as I was stuck with this problem previously. This is however quite important for my research, so I nontheless would ...
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283 views

Failure of geodesic uniqueness - what does it say about the manifold?

I am more of a physicist than a mathematician, but this question is properly mathematical rather than physical, even though it is motivated by a physical application; please assume mathematical ...
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133 views

About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems to ...
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Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
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364 views

Stereographic projection with de Sitter space and hyperbolic plane

How can we do stereographic projection using de Sitter space $\Bbb S^2_1$ and the hyperbolic plane $\Bbb H^2$, in Lorentz-Minkowski space $\Bbb L^3$. For $\Bbb S^2_1$ it is not clear what point ...
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A General Question on Semi-Riemannian Geometry and Finsler Geometry

I’m a first-year graduate math student on my way to earn my master's. I like geometry and topology and our faculty has two geometers and hence I have two options for the thesis. First is walker ...
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Positive Definite Matrix Induced by Lorentz Matrix

Assume $G$ is a Lorentzian matrix, which means it has signature $(+,-,\cdots,-)$, and $v$ is a unit timelike vector, i.e. $v^TGv=1$. So do we have that matrix $2Gvv^TG-G$ is positive definite? Any ...
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Is every hyperbolic isometry the restriction of an orthochronous Lorentz transformation?

I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where $$...
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Physical Object to Pseudo-Riemannian Manifold

It is well known that Lorentzian mainfold is studied in general relativity. So this raises my curiosity about How about the classical mechanics? Does it correspond to the manifold $\mathbb{R}\times M$...
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Example of a degenerate metric which doesn't have the Levi-Civita connection

The proof of existence of the Levi-Civita connection for pseudo-Riemannian manifolds uses heavily the fact that the metric is non-degenerate - so that $\nabla_XY$ is characterized by all the values $\...
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138 views

The null cone is not a proper subspace

I'm stuck on an exercise in Barret O'Neill's book on Semi-Riemannian Geometry(ex. 12 ch. 2). "Let b be a symmetric bilinear form on V.[...] The null cone of b is the set $\Lambda$ of all null vectors ...
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On a proof that the metric volume form is parallel wrt to the Levi-Civita connection

In the context of (semi-)Riemannian geometry, the following fact is well-known: if a (semi-)Riemannian manifold $(M,g)$ is oriented, then the unique volume form $\epsilon = \mathrm{vol}_g$, induced by ...
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Why such coordinates are still called “isothermal” in the Lorentz case?

We know that in a two dimensional Riemannian manifold there always is, at least locally, a coordinate system in which $g_{ij} = \lambda^2 \delta_{ij}$ - such coordinates are called isothermal. There ...
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Inclusions maps, parameterisation and charts

So it makes sense that an inclusion map $$\iota : S \longrightarrow S \subset M $$ maps $$ p \mapsto p $$ But how do you construct these guys in the context of manifolds? To my understanding, if $S$ ...
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278 views

Calculating Christoffel symbols using variational geodesic equation

Given the line element $$ds^2 = e^v dt^2 - e^{\lambda} dr^2 - r^2 d \theta^2 - r^2 \sin^2 \theta d \phi^2$$ we wish to compute the Christoffel symbols $\Gamma^{a}_{bc}$ using the geodesic equation. ...
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368 views

Why a sphere cannot have a Lorentzian Metric?

I was listening to a Lecture and the lecturer said that a sphere cannot have a Lorentzian Metric. Is that accurate? If so, why?
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139 views

Existence of a Lorentzian manifold

Does there exist a smooth compact Lorentzian manifold $M^n$ with n > 2, that has constant curvature, is simply connected and has Euler number (Euler characteristic) $\chi(M)=0$?
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Hopf-manifolds are complete

Let $M=\mathfrak{R}^2-{0}$ be a manifold equipped with the metric \begin{equation} g=\frac{\langle,\rangle}{x^2+y^2}, \end{equation} where $\langle,\rangle$ is the standard Euclidean metric. Let ...
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55 views

Compact GX-manifolds

Let $M=(G,X)$ be a compact smooth Lorentzian manifold with constant sectional curvature, where $X$ is any of the well-known spaceforms $\mathcal{M}^n$, de-Sitter och Anti-de-Sitter and G is their ...
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145 views

Incompleteness of Lorentzian manifolds

Let $\tilde{M}$ be a simply connected Lorentzian manifold and suppose that $\tilde{M}$ admits some Riemannian metric. Question: What can be said about the relation between the geodesic completeness ...
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How to do calculus with split-complex (hyperbolic) numbers?

TL;DR: how do I define a "split-holomorphic" function? As far as I've heard, there is a notion of split-complex numbers: $z = x+uy$, with $u\not\in \Bbb R $ and $u^2 = 1$. One defines the conjugate ...
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156 views

Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a semi-...
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258 views

Local diffeomorpism is a covering?

$D:\tilde{M}\rightarrow \Omega$ is a local isometry (developing map) onto a connected manifold $\Omega$. $\tilde{M}$ is simply connected, has constant sectional curvature and there exists a deck group ...
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71 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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285 views

Volume form as eigenform of the Lie derivative

Suppose (M,g) is a homogeneous Lorentzian manifold and $Y$ a vector field on it. $G$ is a transitively acting Lie group. It is stated that the volume form $\omega$ is $G$-invariant. It is also stated ...
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Coordinates on de Sitter space

I am trying to use a certain parametrization on de-Sitter space $dS^n$ and I am getting both the wrong scalar curvature and metric determinant. The formal definition of $dS^n$ in my work is $-x_1^2+...