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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Intuition for warped product manifold.

I am reading about wave equations in manifold and encountered the term warped product manifold. More specifically, in my case it is defined as follows, $$N:=[0,\phi^*) \times_g \mathbb S^{k-1}$$ ...
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Connected semi-riemannian manifold is an Einstein manifold.

I'm reading the proof of the next proposition: A semi-riemannian manifold $M$ is an Einstein manifold provided $Ric=cg$ for some constant $c.$ If $M$ is connected, $n=dim(M)\geq 3$ and $Ricfg=fg,$ ...
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Group of positive Lorentz matrices acts transitively on hyperbolic space

We call a vector $x \in \mathbb{R}^{n+1}$ positive if $x_{n+1} > 0$. $A \in M_{n,n}(\mathbb{R})$ is called a Lorentz matrix if $J = A J A^t$ where \begin{align} J = \begin{pmatrix} 1 & & &...
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Relation of commutators $[A_{\xi},A_{\eta}]$ with the normal bundle

Let $(\overline{M}^{n+m}, \newcommand\pair{\left\langle #1 \right\rangle} \pair{\cdot,\cdot})$ be a pseudo-Riemannian manifold and $M^n \subseteq \overline{M}$ a non-degenerate submanifold. We have ...
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Computation of exterior derivative of an $(n-1)$ form

Suppose $(M,g)$ is a Lorentzian manifold of dimension $n$. Let $V$ be a one-form on $M$ and define the $(n-1)$ form $\omega = \ast V$ where $\ast$ is the Hodge dual. In a chart $(U,x)$, if we have ...
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The one-parameter curves generated by Lie dragging a geodesic by a Killing field are geodesics

Let $(M,g)$ be a manifold with Riemannian or Lorentzian metric $g$ (the signature doesn't matter for our purposes). Consider $\gamma : (a,b)\to M$ a geodesic and $\phi^\xi_s : M\to M$ the flow of a ...
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Estimating the product of timelike vectors in $\Bbb R^n_\nu$

Consider pseudo-Euclidean space $\Bbb R^n_\nu$ with $\nu$ negatives first $(-,\ldots,-,+,\ldots,+)$. Split $\Bbb R^n_\nu = \Bbb R^\nu_\nu \times \Bbb R^{n-\nu}$. $\renewcommand\vec{{\bf #1}}$Assume ...
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What are examples of the use of countable choice axiom and DC in (pseudo)riemannian geometry results

What are some specific results in (pseudo)riemannian geometry that require either the axiom of countable choice or the axiom of (countable) dependent choice (DC)? For instance, does the definition of ...