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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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How to visualize rotation on a hyperbola?

I am studying Lorentz transform and I do not quite get what it means to use the hyperbolic matrix to rotation a point on a hyperbola, mainly it is because the hyperbola consists of two divergent ...
114 views

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 ...
174 views

Space of $G$-invariant Riemannian metrics contractible?

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ ...
408 views

Covariant derivative of a constant inner product and how it decomposes into local coordinates

I wanted to confirm that my explicit (symbolic) computations of the covariant derivative of a constant inner product is (carefully) done right and correct. For the physicists (includes me), this is ...
111 views

Example of locally symmetric spaces

A locally symmetric manifold is a manifold with parallel curvature tensor $\nabla R=0$. Can you give an example except spheres, projective spaces and hyperbolic spaces?
1k views

Degenerate subspace

A null vector is a nonzero vector that is orthogonal to itself. If W is a subspace of V,let $W^{\perp}$ = [$v{\in}$ W : $v{\perp}$W]. $W^{\perp}$ is a subspace of V called W perp. A subspace W of ...
88 views

Volume element and orientabality

A volume element on an $n$-dimensional semi-Riemannian manifold $M$ is a smooth $n$-form $w$ such that $w(e_1,\cdots, e_n) = \pm1$ for every frame on $M$. How do I prove A semi-Riemannian ...
52 views

When are geodesically generated surfaces everywhere spacelike?

Suppose that $\langle M, g\rangle$ is a Lorentzian manifold, and that $\xi$ is a timelike vector in $T_pM$, at some point $p \in M$. Let $S$ be a surface consisting of points that lie on some ...
289 views

Index notation.

I have a very basic question concerning index notation, normally used in physic papers. I have never used index notation and it is very difficult to me to translate from index free notation, even in ...
163 views

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mean curvature of a nonlinear parabolic PDE

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!
291 views

(Strong) causality condition

I'm studying the causality in Lorentz Manifold with the book "Semi-Riemann Geometry", B.O'Neill. I have the followig problem: He says that, picked a subset A of a Lorentzian Manifold M, the causality ...
140 views

Pseudo-scalar product on Manifold

I'm trying to study the Semi-Riemannian Manifold and the relativity (I use the book Semi-Riemannian Manifold- O'Neill). But I don't understand the following thing: In a Semi-Riemannnian Manifold, I ...
86 views

Semi-Riemannian Manifold

I don't understand the principal idea of Semi-Riemannian Manifold. Why is that if I have a metric tensor $g$ on a smooth manifold $M$ that is a symmetric nondegenerate $(0, 2)$-tensor field on $M$ of ...
376 views

When do isometries commute with the compatible derivative operator on a semi-Riemannian manifold?

Let $M$ and $\tilde{M}$ be smooth manifolds, each with a metric $g_{ab}$ and $\tilde{g}_{ab}$, assumed here to be smooth symmetric invertible tensor fields, which are non-degenerate but not ...
111 views

Caracterization of isometries that preserve time-orientation in $\Bbb L^3$

First of all, I'm considering $\Bbb L^3$ with the convention: $$\langle (x_1,y_1,z_1),(x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ Let $\Lambda = (\lambda_{ij})$ be an isometry of $\Bbb L^3$. I ...
Let $G$ be a connected Lie group acting isometrically on a pseudo-riemannian manifold, $M$. I need a practical way to calculate the sectional curvature of the orbits at any point. Actually, I have ...