# 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).

96 questions
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### Vectors that geodesically generate the same surface

Suppose that $\langle M,g \rangle$ is a complete, simply connected Riemannian symmetric space. The surface geodesically generated by a vector $\xi$ in $T_pM$ is the set of points lying on geodesics ...
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### Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm (pragmatically)...
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### Integration on manifold using the flow of a vector field

In a Physics paper the author states without proof something that seemed quite strange to me. The paper is on General Relativity, so he assumes a Lorentzian manifold $(M,g)$ is given. His hypothesis ...
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### Constant sectional curvature and unit normal vector to a hypersurface totally geodesic

I was reading about hypersurfaces totally geodesic when I found the next proposition: The sectional curvature $K$ of $M$ is constant at $p$ if and only if every unit vector in $T_{p}(M)$ is normal to ...
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### Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
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### Local coordinates on Lorentzian manifold

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 $\nu$. ...
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### Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it. Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null ...
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### On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
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### $GL(n,G)$ 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....
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### Is there a natural choice of cone in Minkowski space as a real affine space?

Minkowski space is a homogeneous affine real space and under this translationally invariant perspective that doesn't have privileged points it seems easier to consider its one-sheet 3-hyperboloid ...
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### A non-orientable surface $S$ such that $T_pS$ is time-type.

I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $S$ in $\mathbb{R}^3_1$ which is time-type. With the ...
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### Irreducible decomposition of Lorentz tensors with Young Tableaux

I want to understand the irreducible decomposition of Lorentz tensors by using Young tableaux. Let me start with a trivial example. Suppose we work in $n=4$ dimensions, and that we have a rank 2 ...
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### Question in the appendixes A of O'Neill book

I'm reading appendixes A of O'Neill's book, "Semi-riemannian geometry" and I don't understand a something. I don't understand at the last theorem, how we construct a universal cover for $M$ and why ...
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### Invariance of signature in semi-riemannian manifolds

The signature of a non-degenerate symmetric bilinear map $h:V\times V\to \mathbb{R}$ in a vector space $V$ is the number of negative numbers on the diagonal of the matrix $h_{ij} = h(e_i,e_j)$ when it'...
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### Vacuum solutions of Einstein's field equations with $O(n)$ isometry group

In Relativity and Singularities, Natário states that Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric). ...
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### Equivalent ways to refer to Lorentzian geometry

I'm a curious student about the Yamabe-type problem in Lorentzian geometry. I found a single article about this topic (2015). I don't know much about Lorentzian geometry and I wonder if one may ...
I see two definitions of a Cauchy surfaces $\Sigma$ of a spacetime $M$. a closed achronal subset such that the domain of dependence is $M$ ...