# 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).

379 questions
Filter by
Sorted by
Tagged with
21 views

### 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 ...
• 535
1 vote
26 views

### 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, ...
18 views

### 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 ...
• 43
35 views

### 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 ...
• 43
1 vote
14 views

### 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,...
• 111
1 vote
13 views

• 93
1 vote
53 views

### 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 $$T_{p}\overline{M} = T_{p}M \oplus (T_{p}M)^{\perp}$$ ...
23 views

• 43
1 vote
67 views

### 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 ...
• 121
111 views

### 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 ...
• 975
52 views

### 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 ...
• 133
52 views

### 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 $\pi_F$ ...
50 views

### 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 ...
• 725
1 vote
30 views

### 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-...
• 725
41 views

### 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 ...
• 5,329
87 views

• 121
45 views

### 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 ...
• 133
76 views

### 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 ...
• 1,700
1 vote
18 views

### 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 ...
• 289
38 views

### 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 ...
72 views

### 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 ...
• 1,700
55 views

### 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 ...
35 views

### 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 ...
192 views

### 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:...
• 2,007
76 views

### 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 ...
• 1,700
37 views

1 vote
37 views

### 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 ...
• 725
1 vote
57 views

### 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 ...
• 725
1 vote
82 views

1 vote
45 views

### $SO(1, 1)$ 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 ...
51 views

### 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 ...
• 152
35 views

### 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 ...
• 109
15 views

### 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 ...
• 109
1 vote
95 views

### 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 ...
• 1,871
### 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 ...
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 ...