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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Vector invariant under the flow

I'm studying a paper by Demir N. Kupeli, On submanifolds in spacetimes, and during a proof of a proposition, the author say: "Extend $X\in T_p S$ by making it invariant under the flow generated by ...
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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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Lagrange's Identity in $\Bbb L^3$.

I'm working in $\Bbb L^3 = (\Bbb R^3, \langle \cdot, \cdot \rangle_L)$, with $\langle {\bf x},{\bf y}\rangle_L = x_1y_1+x_2y_2-x_3y_3$. Let $\|\cdot\|_L = \sqrt{|\langle \cdot, \cdot \rangle_L|}$ be ...
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Example of a connected semi-Riemannian manifold

Give an example of a connected semi-Riemannian manifold that is complete at one point but not complete.
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Definition of the second fundamental form in $\Bbb L^3$, following Kühnel.

I'm having trouble understanding a few comments is Kühnel's "Differential Geometry: curves - surfaces - manifolds", in page $118$, about the definition of the second-fundamental form in Lorentz-...
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Prove: Gravitation operator is invertible.

Let $(M,g)$ be a Riemannian manifold and $\Gamma(S^2M)$ the space of symmetric 2-covariant tensors. Define the gravitation operator as the map \begin{align*} G:\Gamma(S^2M)&\rightarrow\Gamma(S^2M)...
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Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm ...
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Searching for a thesis.

In several articles about curves, etc, in Minkowski spaces $\Bbb L^3$ and $\Bbb L^4$, there is Walrave, J., Curves and surfaces in Minkowski space. Doctoral Thesis, K.U. Leuven, Fac. of Science, ...
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No Lorentzian metric on $S^{2}$

In the book Riemannian Geometry by Gallot et al there is a remark at the beginning that there is no Lorentzian metric on $S^{2}$. Is it a difficult theorem? Or there is an easy solution? Any hint/idea ...
Helices in Lorentz-Minkowski space $\Bbb L^3$.
Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$ (which I'll denote just by $\langle \cdot, \cdot \rangle$) If $\alpha$ is spacelike and \${\bf ...
I want to compute the mean curvature of a reaction diffusion equations of the form $$u_t=u_{xx}-u^2$$ Any suggestions is appreciated!