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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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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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Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ...
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Integral curves and null geodesics

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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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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433 views

If $\alpha''$ and $\alpha'$ are collinear, then $\alpha$ is a pre-geodesic.

I'm trying to solve exercise $19$ in pages $95$ and $96$ in O'Neill's Semi-Riemannian Geometry book. There are four items and I'm having trouble with the last one. Let $(M,\langle \cdot,\cdot\rangle)$...
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Composite killing vector field

Let $f$ be a smooth function and $\nabla_V V = 0$. Prove that $fV \neq 0$ is a killing vector field, if and only if $f$ is a constant and $V$ is killing. If I assume that $f$ is constant and $V$ is ...
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Riemannian geometry - worldline meets nullcone.

I've been studying the book "Semi-Riemannian Geometry" by B. O'Neill and doing some of the excersises. Chapter 6 (special relativity) includes the following one: If $p$ is an event not on the world ...
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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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The Hessian at a critial point $p$

In my studies I've come across the hessian in the context of Riemannian geometry. I use the following definition of the hessian $$ H^f(X,Y)=XYf-(\nabla_XY)f=\langle \nabla_X(\operatorname{grad} f),T\...
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Affine parameterization of null geodesics

How does one find an affine parameter for a null geodesic? I found this advice on planetmath.org: Take s as an arbitrary parameter; Set $$u^\mu=\frac{dx^\mu}{ds}$$ Then $$u^\mu \nabla_\mu u^\nu = f(...
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Conformal Killing vectors fields on Minkowski spacetime

As is well known Minkowski spacetime (which is four dimensional vector space with scalar product $\eta _{\mu \nu}$ of signature $-+++$) is maximally symmetric, which manifests itself in presence of ...
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Metric of Lorentzian signature on a compact homogeneous space.

Can anyone give an example of a metric of Lorentzian signature on a compact homogeneous space.
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Difficulty calculating velocity after lorentz transformation

I'm working on understanding Lorentz transformations via a text by Garrity, "Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang-Mills". On pages 43 and 44 he ...
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Looking for a biholomorphism [closed]

Is there a biholomorphism between $\mathbb{H}_2 \times S^1$ and a half-space of $\mathbb{R}^3$?
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ricci tensor of 2-sphere $S^2$

Hi could someone show me explicitly how to compute the ricci tensor $g_{ij}$?
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Are lightlike curves in the De Sitter space straight lines?

I think that every lightlike curve in $\mathbb{S}_1^2 \subseteq \mathbb{L}^3$ must be a line. But I'm having trouble concluding it. Let $\alpha\colon I \subseteq \Bbb R \to \Bbb S^2_1 \subseteq \Bbb ...
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How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
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Accounting for signs in divergence thm. on Lorentzian manifold

I am trying to learn about integration in Lorentzian manifolds (I will use signature -+++) and have some problems. Oft quoted (in books for GR) form of divergence theorem is: $\int _U div( X )vol_M=\...
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Isometry from warped product onto the base.

Let $B$ and $F$ be semi-Riemannian manifolds with metric tensors $g_B$ and $g_F$, and consider the warped product $B \times_f F$ by a smooth map $f: B \to \Bbb R$, with metric tensor: $$g = \pi^\ast ...
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60 views

Proving that this metric tensor is Riemannian

Let $(M,g)$ be a Riemannian $n$-manifold, and $\varphi: M \to \Bbb R$ be a smooth map. Define another metric tensor by: $$\widetilde{g} = g - {\rm d}\varphi \otimes {\rm d}\varphi$$I know that $\...
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Master's Exploration in General Relativity

just throwing a query out to the Math community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My main ...
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Statement about the isometries of a product manifold

I'm studying Minkowski spacetime $\Bbb{M}$, and I would like to make the following statement about its symmetry transformations. Since $\Bbb{M}$ is the product manifold of time and space, it inherits ...
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How to justify that the following system of PDE only admits linear solutions?

I have the following system of partial differential equations: $$ \frac{\partial\Lambda^{\mu}}{\partial x^{\nu}}= $$ $$ \small \begin{pmatrix} \cosh\phi_{0}&\sinh\phi_{1}&\sinh\phi_{2}&\...
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168 views

Hodge self-duality in Minkowski spacetime

I was computing the dual map for $k$-forms in Minkowski spacetime, and I found that any $2$-form is either self-dual or anti-self-dual if and only if it is the null form. Does this result make any ...
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On geodesics in Schwarzschild spacetime

I am required to show that a circular lightlike geodesic exists in the Schwarzschild spacetime, and to find its radius. What's the best way to start this?
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About timelike surfaces with non-diagonalizable shape operator.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, with $${\rm d}s^2 = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2.$$ Take a differentiable surface $M \subset \Bbb L^3$, ...
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Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = \int_{t_1}^{t_2}...
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3answers
266 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 ...
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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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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 $G$ ...
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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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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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978 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 ...
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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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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 ...
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269 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 ...
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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\...
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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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1answer
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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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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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1answer
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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 ...
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324 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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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 ...
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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 ...
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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!