Questions tagged [homogeneous-spaces]

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Universal cover of homogeneous space $G/H$.

Let $G$ be a compact connected Lie group with finite fundamental group and $H \leq G$ be a closed subgroup. Consider the homogeneous space $G/H$. Let $H'=\pi^{-1}(H)$ and $\bar{H}=(H')_0$ be the ...
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The associated Klein geometry of a manifold

I'm trying to understand the equivalence between the definition of a Klein geometry and the definition of geometric structures on manifolds. I have read that a geometry is a pair $(X,\mathrm{Isom}(X))$...
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The projective geometry contains the affine geometry

I am currently reading some papers about geometric structures on manifolds and most of them state, without any details, that the affine geometry is contained in the projective geometry. So I want to ...
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Haar measure on Homogeneus groups

I would like to have a reference where it is proved that the haar measure of a Homogeneus group is the push forward of the lebesgue measure on the lie algebra through the exponential map. I have some ...
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Relationship between distances on homogeneous spaces and their Lie groups

Consider the (round) sphere $M=\mathbb{S}^{n-1}$ as a homogeneous $O(n)$-space. Then for $x,y\in\mathbb{S}^{n-1}$ there is $g\in O(n)$ such that $y=g\cdot x$. Denote the Riemannian distance on $\...
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Understanding projection of vector field in homgeneous spaces

I'm studying homogeneous spaces from the book of A.Arvanitoyeorgos, "An introduction to Lie groups and the geometry of homogeneous spaces". Consider $G/H$ be a homogeneous space, and let $\pi:G\...
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Killing vector fields in reductive homogeneous spaces

I'm studying homogeneous spaces from the book of Arvanitoyeorgos, "An introduction to Lie groups and the geometry of homogeneous spaces", but I have a doubt at page 79 (the section about the ...
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Definition of reductive homogeneous space

I'm studying homogeneous spaces from the book "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces" by A. Arvanitoyeorgos. I have some problems understanding the definition of ...
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Why homogenise these vectors?

I was looking for a solution for finding two intersecting lines. I am aware that this can be done by finding a vector product of two lines. I have stumbled upon this example here: https://...
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If $G$ is a Lie group and $H$ is a closed Lie subgroup, then $G\to G/H$ is a principal- $H$ bundle.

Let $G$ be a Lie group and $H$ be a closed Lie subgroup of $G$. Let $G/H$ has the quotient topology. Then $ p: G\to G/H$ is a principal-$H$ bundle. I was reading the above theorem from the book ...
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The expected distance from a point to its neighbors in homogeneous spatial poisson process

Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and $R$ be the distance from a point to its neighbors within a distance $r$. What is ...
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Dimension of an orbit in a $G$-variety

Let $X$ be a $G$-variety and $x \in X$. The dimension of the $G$-orbit of $x$, denoted by $\operatorname{dim}(G.x)$, is defined to be the algebraic dimension of the variety $\overline{G\cdot x}^{Z}$; ...
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Finding the linear mapping between homogeneous coordinates of affine camera

If I have an affine camera with a projection relationship governed by: \begin{equation} \begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b \end{...
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How locus of points of parallel lines in homogeneous coordinates, forms infinity?

In the diagram shown in link, what does writer mean when he says "locus of these points forms the line r". In the diagram "r" is curved, why is it called a line. I am facing difficulty in grabbing the ...
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expected value from some points in continuous homogeneous spatial Poisson point process

Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and assume that $4$ fixed points are located at $(a/2,a/2)$, $(-a/2,a/2)$, $(a/2,-a/2)$ ...
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How to visualize a 2D cartesian point in homogeneous coordinates?

Geometric primitives Hi, I am trying to understand below text from a book on Computer Vision by Szeliski. I am not able to visualize it combining cartesian and homogeneous coordinates. Below is my ...
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Does high rank difference imply large amounts of non-positive curvature for homogeneous spaces?

Let $G$ be a compact Lie group and $H$ a closed subgroup of $G$. Consider the homogeneous space $G/H$ with some $G$-invariant metric, and let $\sec$ denote the sectional curvature function associated ...
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Compact simply-connected homogeneous symplectic manifold

I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold have non-zero Euler characteristic. They prove it by quoting a theorem by ...
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Systematically determine the generators of a Lie algebra given the structure constants

I'm currently studying group theory to delve into the Bianchi classification of 3-dimensional spaces, to then learn more about homogeneous spaces. Instead of going for the quite orthodox way that ...
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Why are Klein geometries $G/H$?

The idea behind Klein geometries is simple, clear and beautiful. Simply we have a manifold M and a Lie group G who acts on it and we study the properties that remain invariants under this action. But ...
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Homogeneous space construction theorem and expected dimensions of quotient maps

I am reading the notes linked here. On page 3 we read at Theorem 2 that the left coset space $G/H$ is a topological manifold of dimension $dim(G)-dim(H)$. Here $G$ is a Lie group and $H$ a closed ...
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Classifying bundles with homogeneous space as fibers

A principal $G$-bundle over a space $X$ is classified by the homotopy classes of maps $[X,BG]$, where $BG$ is the classifying space of the group $G$. My question is what can we do about this when the ...
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Why is the homogeneous line through all points at infinity (1:0:0) and not (0:0:0)?

So I just had a geometry lecture that introduced me to homogenous coordinates. To be clear with notation let me recap: Homogenous coordinates in $\mathbb R^n$ space are described as $$(x_0:x_1: ... : ...
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The quotient space of U(n)/SO(m)

Is there some good way to understand a quotient space of $$ U(n)/SO(m)=? $$ say $n=16$ and $m=10$? Can it be some kind of more familiar manifold? What it is? The Lie algebra generators of $U(n)$ ...
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U(n) v.s. SO(n) and their quotient / homogeneous spaces

By counting the number of generators, it is easy to see that unitary group U(n) has much more generator than SO(n). So it would make sense to consider the mod out group or the quotient space of U(n)/...
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Curvature of a homogenous manifold.

I was a reading a paper and it seemed to me that in one of the equations the authors used the fact if $M$ is a homogenous Riemannian manifold (i.e., the group of isometries of $M$ act transitively on $...
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Various transformation in Homogeneous Coordinates

Today, I learned homogeneous coordinates which solve the notation problem of translation. By homogeneous coordinates, I can became to use translation as matrix notation. Along this knowledge, there ...
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relationship between the point m and line in projective space

a 3-vector [a b c]T can be either a point in projective space of dimension 2(p2) or a line in P2. What is the relationship between the point m ~ [a b c]T and the line l ~ [a b c]T ?(explain) (Hint: ...
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The set of left cosets, homogeneous space, and higher homotopy groups 2

Following this, we consider a more advanced question, below we take $N=3$ for all $N$. Consider the group: $$\mathcal{G}=\frac{SU(N)_A \times SU(N)_{B_1} \times SU(N)_{B_2} \times U(1)}{(\mathbb{...
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1answer
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The set of left cosets, homogeneous space, and homotopy group

Here are statements that I attempt to make sure I am doing things on the right track. Below we take $N=3$ for all $N$. Consider the group: $$G=\frac{SU(N)_A \times SU(N)_B \times U(1)}{(\mathbb{...
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Homeomorphisms which switch 2 points

Let $X$ be a space and suppose: For every two points $x,y\in X$ there is a homeomorphism $h$ that maps $X$ onto itself and such that $h(x)=y$ and $h(y)=x$. Is there a name for this property? ...
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Homogeneous space and quotient space for spin groups

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. These are in some sense spheres. If we embed the spin group $Spin(n)$ into $Spin(n+1)$, ...
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Homogeneous space and quotient space for real projective spaces

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space. These are in some ...
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Homogeneous space and quotient space for complex projective spaces

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space. These are in some ...
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Homogeneous space and nice manifolds

We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space. Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space. Do we have similar ...
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Is there a class of non-symmetric spaces $G/SO(n)$ such that $Hol(G/SO(n))=SO(n)$?

Is there a certain class (or unique construction of a set) of simply connected non-symmetric spaces $G/SO(n)$ such that the Riemannian holonomy is $Hol(G/SO(n))=SO(n)$? (That is, is there a way to ...
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How and when do principal homogeneous spaces form a group?

Here's what I understand: Given a group $G$, a principal homogeneous space for $G$ consists of a set $X$ together with a free and transitive action of $G$ on $X$. Free in this context means that for ...
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Does the space of matrices above rank $k$ admit a transitive Lie group action?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector space ($d \ge 4$). Let $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \text{End}(...
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Good Introductory Sources for Grassmannian, Flag, and Stiefel Manifolds

I am looking to gain a deeper understanding of the Grassman, Stiefel, and Flag manifolds but finding good introductory sources so far has eluded me. I would prefer sources which have: (1) Concrete ...
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Finding the transformation matrix given points in space and their projections on a plane

Description of plane and points Given a set of non-planar points ($\rm\color{green}{P_i}$) in $\color{green}{\text{Coordinate System 1}}$, the projections of those points ($\rm\color{blue}{S_i}$) on ...
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If $H$ acts transitively on $G/K$, does it contain a copy of $AN$?

This is the question that I actually meant to ask in Which groups $H$ act transitively on a noncompact symmetric space $G/K$? I got confused about the definition of parabolic subgroups, so the answer ...
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Which groups $H$ act transitively on a noncompact symmetric space $G/K$?

All Lie groups here are assumed to be real. Let $M=G/K$ be a symmetric space of noncompact type and $H \subset G$. $H$ acts on $G/K$ by left-multiplication. If $H$ is a parabolic subgroup, then $H$ ...
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Homogeneous representation of the line passing through the points

I got the task calculate the homogeneous representation of the line passing through the points (-4, 0) and (−2, 2) and to visualize the situation by drawing the lines and points in a 2D cartesian ...
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Even Dimensional Spheres and Lie Algebra Inclusions

The even dimensional spheres are homogeneous spaces of the form $S^{2n} = SO(2n+1)/SO(2n)$. What is the inclusions of Lie algebras $\frak{so}(2n) \hookrightarrow \frak{so}(2n+1)$ dual to the inclusion ...
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Connected locally homogenous space which is not globally homogenous?

I'm using following definitions: Definition 1. A topological space $X$ is (globally) homogenous if for any two points $x,y\in X$ there exists a homeomorphism $f:X\to X$ such that $f(x)=y$. and the ...
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Different expressions of $\mathbf{CP}^n$

I have seen several different expressions of complex projective spaces $$ \mathbf{CP}^n =\frac{SU(n+1)}{S(U(1) \times U(n ))} \tag{1} $$ $$ \mathbf{CP}^n = \frac{SU(n+1)}{U(1) \times SU(n)} \tag{2} $$...
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Finite measure fundamental domain for a discrete group implies it's a lattice

Here $G$ is a locally compact second countable topological group with left haar measure $\mu$, and $\varGamma$ is a discrete subgroup with a borel subset $\varOmega \subseteq G$ s.t. $G=\biguplus_{\...
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An example of a homogeneous, non-symmetric space

A Riemannian manifold $M$ is said to be homogeneous if the group of isometries $Isom(M)$ acts transitively on $M$. A Riemannian manifold is said to be symmetric if it is connected, homogenuous, and ...
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Cohomology and homology of $SU(N)/SO(N)$

My question is: What is the singular homology and cohomology of $SU(N)/SO(N)$ i.e. $H_n(SU(N)/SO(N),\mathbb{Z})$ and $H^n(SU(N)/SO(N),\mathbb{Z})$? Thank you!
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Real Hyperbolic Plane $\mathbb{RH}^2$ as Homogenous Space

Let us define the real hyperbolic plane $\mathbb{RH}^2$ to be the dual symmetric space of non-compact type for the compact symmetric space $S^2=SO(3)/SO(2)$. For a symmetric space $G/K$ with Cartan ...