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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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3
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3answers
280 views

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 ...
6
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0answers
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 ...
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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 $G$ ...
3
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1answer
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 ...
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0answers
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?
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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 ...
1
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1answer
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 ...
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0answers
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 ...
2
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1answer
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 ...
4
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1answer
163 views

Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with points $x\...
6
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0answers
381 views

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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1answer
126 views

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 ...
3
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0answers
99 views

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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0answers
213 views

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-...
3
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0answers
179 views

Second structural equations in lorentzian space $\Bbb L^3$.

I'm rewriting O'Neill's "Elementary Differential Geometry"'s section on connection forms in Lorentz-Minkowski space $\Bbb L^3$, and I'm having trouble proving the second structural equations $${\rm d}\...
3
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1answer
194 views

Moving frame in a semi-Riemannian manifold

Can someone point me some reference for the moving frame theory in semi-Riemannian manifolds, using differential forms? In special, I'm looking for a version of Cartan's structural equations. I've ...
7
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1answer
332 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid x^2+y^2-z^...
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3answers
97 views

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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4answers
997 views

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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1answer
395 views

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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2answers
346 views

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 ...
8
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1answer
266 views

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 ...
2
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0answers
50 views

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!
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2answers
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 ...
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0answers
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 ...
0
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1answer
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 ...
6
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1answer
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 ...
0
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1answer
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 ...
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0answers
37 views

Sectional curvature of orbits generated by an isometric action

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 ...
2
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0answers
219 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let $\...
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0answers
53 views

Schwarzschild solution question

Since we set the Ricci tensor to be zero everywhere, why is it still a solution if it doesn't apply to the point where the point mass exists? Shouldn't it apply also to that point as well, or am I ...
3
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1answer
95 views

About reparametrization of timelike curves in $\mathbb{L}^3$ (Lorentz-Minkowski space)

I think there is something wrong with the proof this text gives of Lemma $2.1.5$, in pages $19$ and $20$, for timelike curves. I used another function, and it seems to work. Either I'm wrong, or he is....
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2answers
165 views

Causal character of a surface (Lorentz-Minkowski space $\mathbb{L}^3$)

I'm trying to analyze the causal character of the surface $x^2 + y^2 - z^2 = -1$ in Lorentz-Minkowski space $\mathbb{L}^3$, with the convention $\mathrm{diag[1,1,-1]}$, that is $$\langle \left(x_1, ...
3
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0answers
85 views

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 ...
2
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4answers
210 views

Boosts in Lorentz-Minkowski space $\mathbb{L}^3$ (or $E_{1}^3$) (+ material)

Can someone give me examples of boosts in $\mathbb{L}^3$? I understand that boosts are isometries that leave pointwise fixed a straight line $\mathcal{L}$. The only thing I can think of, until now, ...
6
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1answer
794 views

Show isometry of flow on a compact Riemannian manifold where the vector field is Killing

Let $(M,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection of $g$. A vector filed $V$ on $M$ is called a Killing field if for every $p\in M$ and every $X,Y\in T_p M$, $$ g(\nabla_X V, Y)...
0
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1answer
58 views

precise meaning of connected manifold

what does it mean for a manifold to be "connected" precisely? what is the difference between a connected riemannian manifold and a nonconnected one. (i know what a riemannian manifold is a manifold ...
0
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1answer
624 views

connected complete totally geodesic sub manifold of $S^n$

Let $M$ and $N$ be manifolds with Riemannian metrics $g$ and $h$ respectively. A diffeomorphism $F: M\to N$ is an isometry if \begin{equation*} h_{F(x)}(T_x F(u), T_x F(v))=g_x(u,v) \end{...
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0answers
136 views

Metric space not a vector space

Can anyone show that the space of all metrics of a pseudo-Riemannian manifold is not a metric space? Even if the space of all metrics of a pseudo-Riemannian manifold is a metric space, can anyone give ...
2
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1answer
121 views

Proving invariance of $ds^2$ from the invariance of the speed of light

I've started today the book of Landau "Field theory". He starts from the invariance of the speed of light, expresses it as the fact that $c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$ is ...
4
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3answers
1k views

What is the exact motivation for the Minkowski metric?

In introductory texts about Lorentz Geometry, one always learns about the Minkowski space, i.e. $R^4$ with the Minkowski metric $$ m(x, y) := -x_0 y_0 + x_1y_1 + x_2y_2+ x_3 y_3 $$ Using this ...
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2answers
496 views

Spacelike curves definitions

Well, I am looking for the definition, if there is any, of points separated by a spacelike curve in a Lorentzian or more generally in Semi-Riemannian space?
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0answers
532 views

Interpreting the scalar curvature in a semi-Riemannian manifold

Background: Let $M$ be a smooth Riemannian manifold of dimension $n$ and scalar curvature $R$ (with respect to the Levi-Civita connection). Let $m \in M$ and let $B$ be the geodesic ball of radius $...