Questions tagged [homogeneous-spaces]

This tag is for questions relating to "homogeneous-spaces", a particular class of manifolds that behave per construction very symmetrically under the action of some groups, and they can be fully reconstructed just by looking at their behaviour under curtain actions.

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Does every smooth homogeneous manifold have a Lie group acting simply transitively?

I have some smooth manifold $\mathcal{M}$. If there exists a Lie group $G$ that acts transitively on $\mathcal{M}$ does this imply that there exists a Lie group $H$ that acts simply transitively on $\...
Jannis's user avatar
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Understanding $\mathbb{P}^2$ and rational Bézier curves

I've never taken a projective geometry course, and I'm trying to understand the real projective plane $\mathbb{P}^2$ and its description using homogeneous coordinates, and how these relate to rational ...
bubba's user avatar
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(Homogeneous spaces) Moduli Space of Lattices

I'm looking at the "moduli space of $n$-dimensional lattices", which should be the double quotient $$ \mathcal{M} = \text{GL}_n(\mathbb{Z}) \backslash \text{GL}_n(\mathbb{R}) / (\mathbb{R}^\...
Johann Birnick's user avatar
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1 answer
41 views

How to define periodicity of orbits for general group actions?

Let $H$ be a topological group acting on a topological space $X$. Is there a general definition of periodicity in this case? Write $X=G/\Gamma$ and consider the orbit $Hg\Gamma$, what does it mean for ...
taylor's user avatar
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1 answer
86 views

When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact?

My question is exactly that on the title. I'm interested in the action of some (discrete) subgroup $H$ on $\operatorname{GL}_n(\mathbb{R})$ by left multiplication. For example, $H$ can be $\...
Bumblebee's user avatar
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55 views

For $(G,X)$ - manifolds can we assume $X$ is simply connected?

This is based on my rough understanding, so let me know which part if any is wrong. Suppose $M$ is a $(G,X)$-manifold, where $X$ is a homogeneous $G$-space. Pullback a $(G,X)$ structure from $X$ to ...
subrosar's user avatar
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1 answer
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Realizing $\mathbb{CP}^n$ as a coadjoint orbit of $SU(n+1)$

The natural action of $SU(n+1)$ is transitive on $\mathbb{CP}^n$. This suggests that the latter can be realized as a coadjoint orbit of the former. I am trying make this explicit. I have been able to ...
A.D.'s user avatar
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Under what geometric conditions is the image of an irrep of $H$ isomorphic to the image of an irrep of the fundamental group of $G/H$?

Let $G$ be a Lie group and $H$ a closed subgroup. Let $\pi_1(G/H)$ denote the fundamental group of the homogeneous space $G/H$. If $\lambda$ is an irrep of $H$, then under what conditions does there ...
Eric Kubischta's user avatar
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2nd Homotopy group of homogeneous space constructed from SO(n)

Let $G=\mathrm{SO}(n)$ for $n\geq 3$ and consider some element $x\in\mathfrak{so}(n)$. Let $H\subseteq G$ be the subgroup $H:=\{g\in G\mid gxg^T=x\}$. I know that $H$ is a compact, connected Lie ...
Chistlo's user avatar
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Is the total space of a $ G $ principal bundle over a compact homogeneous space always itself homogeneous?

Let $ G \to E \to B $ be a $ G $ principal bundle, where $ G $ is a compact Lie group. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the ...
Ian Gershon Teixeira's user avatar
3 votes
0 answers
40 views

Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space

I eventually want to check how the Hopf fibration $$ S^{2n+1}\to {\mathbb C}P^n $$ satisfies the Riemannian submersion formula, but I am frustrated that I could not even see how things work for $S^{2n+...
Three aggies's user avatar
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Confused by part of the proof of Stein's maximal principle

I'm reading through the proof of Theorem 1 in Stein's "On limits of sequences of operators", and I'm confused at a step. I'll try to replicate all the parts I think are relevant. At the end ...
AJY's user avatar
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Smooth volume form on quotient of Lie group

I consider a compact Lie group $K$ and a closed subgroup $K'$. Let $\mu_K$ be the probability Haar measure on $K$ and let $\omega_{K}$ be the associated left-invariant volume form. I consider the ...
user6453's user avatar
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1 answer
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Is torus a symmetric space?

Let $\Lambda$ be a lattice in $\mathbf{R}^{n}$. The flat torus $T:= \mathbf{R}^{n}/\Lambda$ is a homogeneous space. I have two questions: Is $T$ a symmetric space? If so, is it irreducible for any $\...
shikao's user avatar
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What does it mean that homogeneous spaces "look the same everywhere"?

In many places, including Wikipedia, a homogeneous space is informally described as "a space that looks the same everywhere, as you move through it, with movement given by the action of a group&...
CBBAM's user avatar
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Ways to check a homogeneous space $G/H$ is Riemannian homogeneous

When a homogeneous space $G/H$ is given, a differential geometer may ask Is this homogeneous space also Riemannian homogenous? Now many sources I read interpret this question as whether this space ...
Math The Novice's user avatar
2 votes
0 answers
86 views

Why can't a Lie group act transitively on a finite volume hyperbolic manifold?

In the comments on Is Seifert-Weber space homogeneous for a Lie group? it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (constant negative sectional curvature) ...
Ian Gershon Teixeira's user avatar
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Euclidean first fundamental form as vector bundle valued, first order differential invariant

I'm self studying the book Cartan for beginners and i'm trying to solve the exercise 2.5.2. The setup is the following: Let $\operatorname{ASO}(n+s)$ the set of rototranslation in the $n+s$ ...
Marco's user avatar
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Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?

Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
Ian Gershon Teixeira's user avatar
7 votes
4 answers
712 views

Does the Mobius Strip have an homogenous embedding?

So in this question I’m trying to do two things at once. 1. Define what a “homogenous embedding” is by describing what it is like and then furthermore ask if such an embedding exists for the mobius ...
Sidharth Ghoshal's user avatar
1 vote
1 answer
96 views

Geometry of homogeneous spaces $G/T$

Let us work over $\mathbb C$. It is well known that if $G$ is a (connected) complex reductive group and $B$ a Borel subgroup, then the homogeneous space $G/B$ is a smooth projective variety. For ...
bernardorim's user avatar
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A parallelizable holonomy bundle

I am following the proof of Ambrose-Singer theorem for Riemannian homogeneous spaces in this book https://www.cambridge.org/core/books/homogeneous-structures-on-riemannian-manifolds/...
Fumihiro Ueno's user avatar
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Homogenous geodesic metric spaces

Consider a metric space with a path between any two points, so a real line segment of some length between them, and the length of this line is the same as the distance between the two points in the ...
Electro-blob's user avatar
7 votes
1 answer
145 views

Homogeneity of Topological Manifolds: Are they all quotients of topological groups?

I am trying to prove that every connected topological manifold $M$ (assume Hausdorff and second countable) is homogeneous in the sense defined by Bourbaki's General Topology Ch.3 Sec.2.5 pg 232. The ...
Daniel Grimmer's user avatar
2 votes
0 answers
55 views

A cocompact quotient of one of the 8 geometries comes from only one geometry?

We know by Thurston's Geometrization Conjecture, that every closed 3-manifold admits a prime decomposition: it must be the connected sum of prime 3-manifolds. My question is: if $\mathbb{X}$ is one of ...
Odylo Abdalla Costa's user avatar
1 vote
0 answers
44 views

Uniqueness of Smooth Structure Given a Smooth Transitive Action (of an Infinite-Dimensional Lie Groups)

Let $S$ be a naked set (later this will be equipped with various smooth structures, making it into various smooth manifolds, $\mathcal{M}_0$, $\mathcal{M}_1$, $\dots$). Let $H_1\subset\text{Perm}(S)$ ...
Daniel Grimmer's user avatar
3 votes
0 answers
46 views

When is the Quotient of Two Infinite-Dimensional Lie Groups a Smooth Manifold?

It is well-known under what conditions the quotient of two finite-dimensional Lie groups forms a smooth manifold. It is also well-known what smooth structure the resulting quotient manifold might have....
Daniel Grimmer's user avatar
1 vote
1 answer
129 views

How to describe all metrics invariant by $O(3)$ on a given space?

I want to describe all pseudo-Riemannian metrics (symmetric covariant non degenerate 2-tensor fields) invariant by $O(3)$ on different manifolds ($\mathbb R^3$, the sphere $S^2$, and $]-\epsilon,\...
ZHYMRA's user avatar
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1 vote
1 answer
33 views

Classifying the core-free Lie subgroups $G$ or some Lie group $H$

Let $H$ be a finite-dimensional Lie group and let $G\subset H$ be a closed Lie subgroup thereof. Let us say that $G$ is a core-free Lie subgroup of $H$ if and only if $G$ contains no normal subgroups ...
Daniel Grimmer's user avatar
2 votes
1 answer
54 views

Decomposition of $L^2(G)$ related to Peter Weyl Theorem involving induced representations

Let $G$ be a compact Lie group and $H$ a subgroup. In The geometry of inequivalent quantizations (right under equation 2.11) the authors seem to claim that $$ L^2(G) = \bigoplus_{\mu \in \widehat{...
Eric Kubischta's user avatar
0 votes
1 answer
102 views

Definition of principal homogeneous space for elliptic curves

Definition of principal homogeneous space for elliptic curves, for example, in $p322$ of 'The arithmetic of elliptic curves'. Let $E/K$ be an elliptic curve defined over a field $K$. Principal ...
Pont's user avatar
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3 votes
1 answer
68 views

Equivariant map to homogeneous space is a fiber bundle

I have a Lie group $G$, an orbit manifold $Y = G/H$ for some closed subgroup $H$, and a $G$-equivariant map $p : X \to Y$, where $X$ is a manifold with a smooth $G$-action (not assumed to be ...
ronno's user avatar
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1 vote
0 answers
42 views

Inverse of Riemannian exponential in locally symmetric space

Let $\mathfrak{g}$ a real Lie algebra and $s$ an involutive automorphism of $\mathfrak{g}$. Let $\mathfrak{f}$ the set of fixed point of $s$ and let $G$ a Lie group with Lie algebra $\mathfrak{g}$ and ...
Marco's user avatar
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2 votes
1 answer
115 views

Is every compact, homogeneous, totally disconnected space a power of a finite discrete space?

In a comment of this question, t.b. suggests that every compact, homogeneous, totally disconnected, second countable space is either a finite discrete space or the Cantor set; equivalently, it must be ...
Steven Clontz's user avatar
2 votes
1 answer
229 views

Lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$: Stiefel-Whitney class and non/spin manifold

Define the lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$. What is the property of Stiefel-Whitney class $w_1(TM)$ and $w_2(TM)$ for $M= L^k(n)$? What is the spin or nonspin manifold property? Is ...
wonderich's user avatar
  • 5,909
5 votes
0 answers
73 views

Lifting the quotient homogeneous space to the group and define the measure

Let $G$ be a Lie group equipped with a Haar measure $\mu$ and $H$ be a closed subgroup of $G$, equipped with a Haar measure $\nu$. We do not assume that there exist a $G$-invariant measure on $H\...
taylor's user avatar
  • 559
1 vote
1 answer
51 views

Erlangen-Style approach to Homogneoues Fiber Bundles

In connection with a philosophy of physics project I have recently been looking at fiber bundles from an Erlangen-inspired perspective. My inspiration is the following well-known result: Every smooth ...
Daniel Grimmer's user avatar
1 vote
0 answers
33 views

Lagrangian Grassmanian as Homogeneous Space

This is part of Exercise 2.14 in Kirillov's "An Introduction to Lie Groups and Lie algebras". The reader is supposed to prove that $Sp(2n,\mathbb{R})$ acts transitively on the space $L_n$ of ...
topolosaurus's user avatar
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1 vote
1 answer
113 views

Condition of two orthogonal lines in homogeneous coordinates

Let $l_1$ and $l_2$ be the representations in homogeneous coordinates of two lines in the plane. How could you express the fact that these two lines are orthogonal? Deduce that, in general, the image ...
Esteban's user avatar
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1 vote
1 answer
111 views

Free actions and double coset manifolds

Double coset spaces are often manifolds. For example, locally symmetric spaces and more generally Clifford-Klein space forms. A familiar family of examples are the surfaces of genus $ g $ $$ \Gamma_g \...
Ian Gershon Teixeira's user avatar
1 vote
0 answers
36 views

Two different maximum symmetry metrics

Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
80 views

Standard representative for a coset space

Motivation: finding convenient real coordinates for the full complex flag manifold $ \mathrm{Flag} \left( 1, \ldots, n \right) $. Details: Consider the Lie group $ \mathrm{SU} \left( n \right) $ and ...
smitke6's user avatar
  • 162
0 votes
1 answer
77 views

Finding all rational (closed and connected) subgroups of $\text{SL}(2,\mathbb R)$ with respect to $\text{SL}(2,\mathbb Z)$

A subgroup $H$ of $\text{SL}(2,\mathbb R)$ is called rational w.r.t. the lattice $\text{SL}(2,\mathbb Z)$ if $H \cap \text{SL}(2,\mathbb Z)$ is a lattice in $H$ (namely the quotient homogeneous space ...
taylor's user avatar
  • 559
4 votes
0 answers
154 views

Homogeneous Space $G_2/P$ is a Quadric Hypersurface inside $\mathbb{P}^6 $

Let $G_2$ be one of the exceptional simple Lie groups and $\mathfrak{g}_2$ it's Lie algebra. On pges 355/356 in Fulton & Harris' Representation Theory is shown that the stadard representation of ...
user267839's user avatar
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1 vote
1 answer
163 views

Correspondence between Parabolic groups and highest Weights for complex algebraic groups

Let $G$ be a simple algebraic group over $\mathbb{C}$ and $ T \subset G$ the Cartan sub-Lie-group (ie the maximal torus) and $\text{Hom}(T, \mathbb{C}^*)$ it's dual space. The lattice of weights $\...
user267839's user avatar
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4 votes
2 answers
245 views

If the pushforward of $\mu$ under the group action is invariant, then $\mu$ must be the Haar measure.

Let $G=\text{PU}(d)$ the projective unitary group, acting on the complex projective space $X=\mathbb{P}(\mathbb{C}^d)$, in the usual way. Let $\mu:\mathcal{B}(G)\to[0,\infty]$ be a Radon measure, ...
Saúl Pilatowsky-Cameo's user avatar
1 vote
1 answer
219 views

Understanding distance from point to line in homogeneous coordinates

I would like to find the distance between A 2d point given as the homogeneous $P=(x, y, z)$, and a 2d line given as $L=(a, b, c)$. I found an algorithm that does this, I can't figure out why it is ...
Gulzar's user avatar
  • 190
1 vote
1 answer
88 views

Can two vector spaces be defined as heterogeneous?

In the field of Natural Language Processing the term "heterogeneous vector spaces" is often used when describing two different embedding spaces that are created from different types of data, ...
Sepfins's user avatar
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1 vote
0 answers
55 views

Decomposition of Induced Representations like Peter-Weyl

Let $G$ be a compact Lie group. Part of the Peter-Weyl theorem is a decomposition result: $$ L^2(G) = \bigoplus_\ell V_\ell \otimes V_\ell^*. $$ Here $V_\ell$ is an irrep of $G$ and the sum is ...
Eric Kubischta's user avatar
1 vote
2 answers
90 views

How does positive homogeneity of degree $1$ yield linearity?

The problem I came across is as follows: Suppose $E, F$ are Banach spaces over $\mathbb{K}$. Show that if $f\in C^1(E,F)$ (continuously differntiable) is positively homogeneous of degree $1$, then $f\...
erpxyr2001's user avatar

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