# 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).

219 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$\langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}... 2answers 364 views ### Stereographic projection with de Sitter space and hyperbolic plane How can we do stereographic projection using de Sitter space \Bbb S^2_1 and the hyperbolic plane \Bbb H^2, in Lorentz-Minkowski space \Bbb L^3. For \Bbb S^2_1 it is not clear what point ... 0answers 32 views ### A General Question on Semi-Riemannian Geometry and Finsler Geometry I’m a first-year graduate math student on my way to earn my master's. I like geometry and topology and our faculty has two geometers and hence I have two options for the thesis. First is walker ... 0answers 32 views ### Positive Definite Matrix Induced by Lorentz Matrix Assume G is a Lorentzian matrix, which means it has signature (+,-,\cdots,-), and v is a unit timelike vector, i.e. v^TGv=1. So do we have that matrix 2Gvv^TG-G is positive definite? Any ... 1answer 87 views ### Is every hyperbolic isometry the restriction of an orthochronous Lorentz transformation? I know that every isometry of the sphere \Bbb S^2 is the restriction of some A \in {\rm O}(3,\Bbb R): namely, if A_0:\Bbb S^2\to \Bbb S^2 is an isometry, then A_0 = A\big|_{\Bbb S^2} where$$...
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 $\mathbb{R}\times M$...