Questions tagged [homogeneous-spaces]

The tag has no usage guidance.

150 questions
54 views

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 ...
46 views

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 $(X,\mathrm{Isom}(X))$...
35 views

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 ...
23 views

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 ...
55 views

31 views

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 ...
21 views

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 ...
55 views

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://...
15 views

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 ...
32 views

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 ...
17 views

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 $\overline{G\cdot x}^{Z}$; ...
12 views

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{...
15 views

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 ...
25 views

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 $(a/2,a/2)$, $(-a/2,a/2)$, $(a/2,-a/2)$ ...
24 views

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 ...
27 views

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 ...
41 views

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 ...
29 views

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 ...
61 views

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 ...
15 views

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 ...
63 views

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 ...
104 views