# 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).

232 questions
Filter by
Sorted by
Tagged with
49 views

### Length minimizing property of geodesics

I want to show that isometries map geodesics to geodesic using that an isometry of two riemannian manifolds also implies an isometry of metric spaces. I have found proofs using the length minimizing ...
61 views

### Polar coordinates, Gauss Lemma

I don't understand the following statement from the Wikipedia page "Normal coordinates": I don't see how this follows from the Gauss Lemma. The statement of the Gauss Lemma I know is: Let $(M,g)$ be ...
35 views

### Riemannian polar coordinates

Let $(M,g)$ be a Riemannian manifold, $\phi: \mathbb{R}^n \setminus \{0\} \rightarrow (0, \infty) \times S^{n-1}$, $\phi(v)=(\vert v \vert, \dfrac{v}{\vert v \vert})$ and $(U, \psi=(x^1,...,x^n)$ ...
6 views

### What is the difference/relation between Lorentzian and Finsler geometry?

I'm kind of lost among many similar concepts. What is the difference/relation among Lorentzian manifold, Finsler manifold, Minkowsky manifold (and I just came across such a Randers space, related to ...
29 views

### Vector field with constant length

Is it correct, that for some pseudo-Riemannian manifold $M$, $X \in \mathfrak{X}(M)$, if $g(X,X)=1$, then the integral curves for $X$ are geodesics? I have the following explanation: Let $\gamma$ be a ...
65 views

### Euler characteristic on torus

I'm trying to compute $\chi(T^2)$: I know that the sectional curvature of $T^2$ is $\dfrac{\cos(t)}{2+\cos(t)}$ with the parametrization: $F(t,s)=((2+\cos(t))\cos(s),(2+\cos(t))\sin(s),\sin(t))$ Now ...
20 views

### Three dimensional Riemann tensor on a manifold with degenerate metric

Disclaimer: In this post I'm using abstract index notation. In particular brackets around indices indicates anti-symmetrization: $$A_{[a}B_{b]}=\frac{1}{2}(A_aB_b-A_bB_a).$$ Let $\mathcal{I}$ be a ...
16 views

### Reconstruct curvature tensor from sectional curvature in semi-Riemannian Manifold

Is it possible to reconstruct the curvature tensor from the sectional curvature in the semi-Riemannian case? In Kuhnel's book there is a proof using $k(X,Y)=\left\langle R(X,Y)Y,Y\right\rangle$, but ...
16 views

### Subspace is a semi-riemannian submanifold

In my lecture we have stated something like: Non-degenerate subspaces $V \subset \mathbb{R}^n_{\nu}$ are obviously geodesically complete, totally geodesic submanifolds. But I don't see how any of ...
29 views

### ¿Does a timelike non-orientable surface exist?

Consider the Lorentz-Minkowski space of dimension $3$, $\mathbb{L}^3 = (\mathbb{R}^3,\langle \: \cdot \: \rangle)$ $$\langle u,v\rangle=u_1v_1 + u_2v_2 - u_3v_3$$ We say that a surface $S$ is ...
27 views

38 views

### Is the Lie derivative along the normal well defined?

This question is cross-posted at the physics stack exchange at https://physics.stackexchange.com/q/488358/83357 Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. ...
15 views

### 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?
32 views

30 views

### 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 ...
37 views

### 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,...
50 views

### 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 $\nu$. ...
32 views

### 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, $f\colon M\to \mathbb{R}$...
62 views

### 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:
27 views

67 views

### 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 ...
59 views

### 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 ...
16 views

### 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 ...
16 views

### 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 ...
20 views

### 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 ...
24 views

38 views

126 views

### $GL(n,G)$ 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....
10 views

### 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 ...
43 views

### 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....
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 ...
### 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 $u + v + w = 0$. ...