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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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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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364 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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Integration on manifold using the flow of a vector field

In a Physics paper the author states without proof something that seemed quite strange to me. The paper is on General Relativity, so he assumes a Lorentzian manifold $(M,g)$ is given. His hypothesis ...
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Constant sectional curvature and unit normal vector to a hypersurface totally geodesic

I was reading about hypersurfaces totally geodesic when I found the next proposition: The sectional curvature $K$ of $M$ is constant at $p$ if and only if every unit vector in $T_{p}(M)$ is normal to ...
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Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
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How to do calculus with split-complex (hyperbolic) numbers?

TL;DR: how do I define a "split-holomorphic" function? As far as I've heard, there is a notion of split-complex numbers: $z = x+uy$, with $u\not\in \Bbb R $ and $u^2 = 1$. One defines the conjugate ...
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How can the notion of “two curves just touching” (vs. “two curves intersecting”) be expressed for a given metric space?

A popular introductory description of a "tangent (in geometry)" is presented as "the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the ...
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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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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}\...
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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 ...
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Local coordinates on Lorentzian manifold

Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit normal vector field $\nu$. ...
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Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it. Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null ...
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On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
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$GL(n,G)$ General Linear group of a group $G$?

I know you can take the General linear group of some vector space $V$: $GL(V)$. For example, suppose I have a three-sphere, elements of which I might represent as $SU(2)$ since they're diffeomorphic....
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Is there a natural choice of cone in Minkowski space as a real affine space?

Minkowski space is a homogeneous affine real space and under this translationally invariant perspective that doesn't have privileged points it seems easier to consider its one-sheet 3-hyperboloid ...
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Diagonalization of $2\times 2$ Lorentz-symmetric matrix

$\newcommand\pair[1]{\left\langle #1 \right\rangle}$ Consider Minkowski plane $\Bbb L^2 = (\Bbb R^2, \pair{\cdot,\cdot}_L )$, where$\renewcommand\vec[1]{{\boldsymbol #1}}$ $$\pair{\vec{u},\vec{v}}_L\...
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Timelike straight line maximize energy for normal variations in Minkowski Space

First some nomenclature: Let $Q(s,t): [0,1]\times[0,1]$ $\rightarrow$ $L^4$ be a smooth map. Let $x^i$ be affine coordinates in $L^4$. For each $t$ we define a longitudinal curve of $Q$ by $\gamma_t(...
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What are conditions on existence of local orthogonal coordinate system on two-dimensional Lorenzian manifold?

Consider two-dimensional Lorenzian manifold i.e. of signature $(1,1)$. Does there always exist on every point local coordinate chart so that the corresponding coordinate one-forms are orthogonal i.e. $...
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Gauss' theorem for null boundaries

Note: I have solved this problem on my own, mostly while actually typing it in here, as I was stuck with this problem previously. This is however quite important for my research, so I nontheless would ...
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32 views

Positive Definite Matrix Induced by Lorentz Matrix

Assume $G$ is a Lorentzian matrix, which means it has signature $(+,-,\cdots,-)$, and $v$ is a unit timelike vector, i.e. $v^TGv=1$. So do we have that matrix $2Gvv^TG-G$ is positive definite? Any ...
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Integral curves in null hypersurfaces

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p S$ ...
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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!
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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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Penrose Singularity Theorem Proof

I have problems understanding the first part of the proof for the Penrose singularity theorem in the book "Leonor Godinho José Natário An Introduction to Riemannian Geometry": I know that $\langle\...
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Unit normal vector field on Lorentzian manifold

Following situation: Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit ...
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Understanding of Cauchy hypersurfaces

Let $(M,g)$ be a stably causal spacetime. Then we have a global time function $t:M \rightarrow \mathbb{R}$. A set $S_a:=t^{-1}(a)$ is said to be a Cauchy-hypersurface, if the domain of dependence of $...
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There is it an easier proof that if $g(X,X)$ is a constant $\nabla_X X=0$?

In the book General Relativity for matematicians by R.K.Sachs and H.Wu there is a problem which say: Let $(M,g)$ be a Lorenzian manifold,$\nabla$ the Levi-Civita conexion, $f\colon M\to \mathbb{R}$...
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Existence of timelike curve in a particular set up.

I encountered online the following exercise: Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will identify ...
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Prove that $U^\sigma \nabla_\sigma U^\mu = g^{\mu\nu}\nabla_\nu \ln V$

Let $U^\mu$ be a vector in 4-dimensional Minkowski space with norm $-1$ and $K^\mu = V(x)U^\mu$ a vector proportional to it. We can write $V(x) = \sqrt{-K_\nu K^\nu}$. (This setup comes from physics ...
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Prove that a geodesic under certain hypothesis has a conjugate point

If $(M,g)$ is a riemannian manifold, and $N$ a submanifold in $M$ with $\sigma$ a geodesic normal to $P$ at $p=\sigma(0)$ under the hypotheses: -$H(\sigma'(0))=g\left(\sigma'(0)),\vec{H_p}\right)>...
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Canonical projections for $\mathbb{H}^2_1$ and $\mathbb{S}^2_1$

We know that there exists a canonical projection $\pi:\mathbb{S}^2\to\mathbb{R}\mathbb{P}^2$. Are there similar transformations for hyperbolic plane $\mathbb{H}^2_1$ or pseudosphere $\mathbb{S}^2_1$ ...
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39 views

Shape operator of pseudo-spheres

In the following, I will refer by O'Neil to Barrett O'Neil's book Semi-Riemannian Geometry With Applications to Relativiy. Consider the $M=\{ x \in \mathbb{R}^n_s | f(x)=r^2\}$ with $f(x)=g_s(x,x)$ ...
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Uniqueness of a time-orientable covering spacetime

On a standard treatment of time-orientability in Lorentzian spaces, it is usual to see a remark of the form: "If a space-time (M,g) is not time-orientable, then it has a double covering space which is....
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Anti-de Sitter space: Is the universal cover of $\text{AdS}_n$ a submanifold of $\text{AdS}_{n+1}$?

$$\text{AdS}_n = \{\vec x\in\mathbb R^{n-1,2}\mid\vec x\cdot\vec x=-1\}$$ Using an orthonormal basis $\{e_i\}$ for $\mathbb R^{n-1,2}$, with ${e_1}^2={e_2}^2=-1$ and $(e_{\geq3})^2=1$, we can ...
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A non-orientable surface $S$ such that $T_pS$ is time-type.

I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $S$ in $\mathbb{R}^3_1$ which is time-type. With the ...
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Irreducible decomposition of Lorentz tensors with Young Tableaux

I want to understand the irreducible decomposition of Lorentz tensors by using Young tableaux. Let me start with a trivial example. Suppose we work in $n=4$ dimensions, and that we have a rank 2 ...
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40 views

Question in the appendixes A of O'Neill book

I'm reading appendixes A of O'Neill's book, "Semi-riemannian geometry" and I don't understand a something. I don't understand at the last theorem, how we construct a universal cover for $M$ and why ...
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Invariance of signature in semi-riemannian manifolds

The signature of a non-degenerate symmetric bilinear map $h:V\times V\to \mathbb{R}$ in a vector space $V$ is the number of negative numbers on the diagonal of the matrix $h_{ij} = h(e_i,e_j)$ when it'...
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29 views

Vacuum solutions of Einstein's field equations with $O(n)$ isometry group

In Relativity and Singularities, Natário states that Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric). ...
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35 views

Equivalent ways to refer to Lorentzian geometry

I'm a curious student about the Yamabe-type problem in Lorentzian geometry. I found a single article about this topic (2015). I don't know much about Lorentzian geometry and I wonder if one may ...
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The equivalence of two definitions of Cauchy surfaces.

I see two definitions of a Cauchy surfaces $\Sigma$ of a spacetime $M$. a closed achronal subset such that the domain of dependence is $M$ ...
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209 views

Inner product on pseudoriemannian manifold

In this topic: Scalar product on manifold. Henry said $(\omega, \eta)=\int_M \omega \wedge \star \eta=\int_M \left<\omega, \eta \right>\mathrm{d vol}$ is an inner product on forms, where $\left&...
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610 views

How does the hodge codifferential operator act over a wedge product?

We have that the exterior derivative acts over a wedge product in the following manner. Let $\alpha,\beta$ be $p,q$ forms respectively. Then we have that \begin{equation} d(\alpha\wedge \beta) = (d\...
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71 views

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 ...
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81 views

Mesh smoothing problem and normalized mean curvature flow

I am trying to implement mesh smoothing algorithm mentioned here But I got the problem with the delta of the vertex position (formula 13). The mean curvature flow is divided by the sum of the ...
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143 views

Pseudo Riemann metric and Riemann metric

I have two questions The first question : Who can give me an example of a manifold is not paracompact and we can't define a Riemann metric on it ? The second one I want read about pseudo ...
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20 views

References request about infinitesimal rigidity of surfaces in Minkowski space.

I am reading an article about rigidity of surfaces in Minkowski space time and i am having big trouble for finding references about the topic. I have been working with the material and definitions ...
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21 views

Does an observer on an expanding three-sphere naturally have a hyperbolic sense of time?

On can consider the 3-sphere of radius a as being embedded in a four-dimensional Euclidean space. One has in this view the condition for any coordinate system with origin at the center of the 3-...
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32 views

A General Question on Semi-Riemannian Geometry and Finsler Geometry

I’m a first-year graduate math student on my way to earn my master's. I like geometry and topology and our faculty has two geometers and hence I have two options for the thesis. First is walker ...
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44 views

Physical Object to Pseudo-Riemannian Manifold

It is well known that Lorentzian mainfold is studied in general relativity. So this raises my curiosity about How about the classical mechanics? Does it correspond to the manifold $\mathbb{R}\times M$...