# 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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### On the definition of pregeodesics as stationary points of the length functional in (pseudo-)Riemannian geometry

On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways: curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is ...
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### Proof of commutativity of trace and covariant derivative in orthonormal frame

Suppose we have a symmetric (0,2) tensor $k$. (The original setting is that $k$ is the scalar second fundamental form for a Riemannian hypersurface, but I don't think it matters in my specific ...
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### Proving $\phi$ is a smooth map and constructing an explicit isometry

Consider a Lorentzian manifold $(\zeta,g)$ with metric: $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$ For $u,v,w,r,>0$. Suppose we take a Cauchy foliation of $\zeta,$ called $\mathcal F$,...
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### Laplace-Beltrami operator in coordinates

I know that on a Lorentzian manifold the Laplace-Beltrami operator of a function $\phi\in C^\infty(M)$ is defined as $$\triangle_g \phi = \text{div}(\text{grad}\phi).$$ Now I've come across the ...
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### Pullback of a Lorentzian metric is non-degenerate

Background: Recently I've read this post of one person trying to prove that the pullback $F^*g$ of a riemannian metric $g$ is a Riemannian metric iff $F: N \to (M,g)$ is a smooth immersion. Question: ...
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### Pullback of lorentzian metric is degenerate at the boundary of embedded submanifold

Let $\Sigma = \mathbb{R} \times [0,1]$ be a smooth manifold with boundary $\partial \Sigma = \mathbb{R} \times \{0,1\}$.Let $M = R^{1,D-1}$ be the $D$-dimensional Minkowski space with scalar product (...
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### A global view-point on pseudo-Riemannian manifolds [closed]

A pseudo-Riemannian manifold $(M,g)$ is a generalisation of the concept of a Riemannian manifold where we relax positive-definiteness to non-degeneracy. $\alpha$) Non-degeneracy is still enough to ...
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### What is Higher Teichmuller Theory?

I am interested in representation of surface group. So, I am studying the book Lectures on Representations of Surface Groups by Francois Labourie. The main goal of representation of surface groups is ...
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