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2
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0answers
17 views

Who first computed the Euler characteristic of a generalized flag manifold?

Let $G$ be a compact Lie group and $H$ a closed subgroup containing a maximal torus $T$ of $G$. Then the Euler characteristic of $G/H$ satisfies $$\chi(G/H) = \frac{|W_G|}{|W_H|},$$ where $W_G = ...
0
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0answers
20 views

Vanishing Points using Homogeneous Coordinates

I am reading a document http://www.robots.ox.ac.uk/~ian/Teaching/CompGeom/lec2.pdf In which the following is stated: Question 1: I understand that we can write the line equation in homogeneous ...
2
votes
1answer
38 views

Identifying the orbit space of the unitary group $U(n)$ in the compact symplectic group $Sp(n)$

Let $Sp(n)$ be the compact symplectic group. Let $U(n)$ the unitary group, and $O(n)$ the orthogonal group. What is $Sp(n)/U(n)$? What is $U(n)/O(n)$? I obtain that ...
1
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0answers
19 views

When does a homogeneous space define a fibration?

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we ...
0
votes
1answer
31 views

Trying to understand homogeneous coordinates

I am trying to understand how homogeneous coordinates work, and think I have an explanation but need to check it is correct. For a homogeneous coordinate $[x,y,1]$, or $[x,y,3]$ does the last number ...
1
vote
1answer
36 views

Compute the isotropy representation

Suppose $SU(1,1)$ acts on the open unit disc $\mathbb{D}$ in the natural way, by linear fractional transformations. The isotropy group is $U(1),$ since it stabilizes the point $0.$ I am trying to ...
0
votes
1answer
50 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
4
votes
1answer
77 views

Canonical connection on $CP^n$

I have heard something along the lines of "There is a canonical $U(1)$ connection on $CP^n$" and I am trying to understand what that means. First I suppose that the sentence refers to a line bundle ...
1
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0answers
27 views

Liesubgroups given by representations

Maybe it's a little odd question, but I encountered a classification of homogeneous spaces, and I'm stuck with the following description (c.f. Onishchick, "Topology of transitive transformation ...
4
votes
0answers
77 views

When do invariant measures arise from smooth differntial forms?

It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq ...
1
vote
1answer
47 views

projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is projective variety.

I am looking for a proof(or refference) for this fact A projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is parabolic subgroup.
2
votes
2answers
93 views

Total dimension of the cohomology of a homogeneous space (or of a graded Tor)

I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let $G$ be a compact Lie group, $T$ a maximal torus, ...
2
votes
2answers
73 views

Homogeneous space question: a quotient $U(n)/U(n-1)$

One can block-diagonally embed a copy $H$ of the unitary group $U(n-1)$ into $U(n)$ by $$A \mapsto \begin{bmatrix}\det(A)^{-1}&0\\0& A\end{bmatrix}.$$ According to a remark in the example ...
1
vote
1answer
27 views

Is there any definition for homogeneous rotations?

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin. ...
0
votes
0answers
11 views

Why is $LG(n) \cong U(n)/O(n)$?

Let $LG(n)$ be a Lagrangian Grassmanian manifold. That is, $LG(n)$ is the set of Lagrangian subspaces of a symplectic vector space of dimension $2n$. Why is $LG(n)$ can be identified with $U(n)/O(n)$? ...
2
votes
1answer
84 views

Basic question on almost complex structures and Chern classes of homogeneous spaces

Toward the end of "Characteristic Classes and Homogeneous Spaces, III," Borel and Hirzebruch prove that given a compact Lie group $G$ and toral subgroup $T$ (no restriction on rank), one has $w(G/T) = ...
1
vote
1answer
37 views

Projective Co-ordinate Geometry

I am learning projective geometry in my computer vision course. So, we represent a co-ordinate point in an image as a homogeneous co-ordinate as $(x,y,1)$. My professor says that if we are given two ...
3
votes
1answer
86 views

Is there an easy way to tell if these two SO(2)s in SO(4) are conjugate?

I am currently interested in quotients of Lie groups by submaximal tori. $G = Sp(1) \times Sp(1)$ double-covers $SO(4)$, as noted at The Quaternions and $SO(4)$. Define a circle subgroup $T = \{1\} ...
1
vote
0answers
60 views

3D Animation of object flying straight towards a surface

Lets say we have the following the orthogonal(?) 4x4 matrix, which represents a world space transformation in a right-handed coordinate system. ...
2
votes
0answers
94 views

How often is a torus in a compact Lie group nullhomologous?

Minor nomenclature question: What is the standard name for the ring structure induced on the homology of an H-space by its multiplication? I've seen "homology ring" and "Pontrjagin ring." Hopefully ...
6
votes
0answers
47 views

Two definitions of real flag manifolds: do they coincide?

Let $G$ be a real semisimple Lie group with finite center. Definition 1 A real flag manifold is a homogeneous space $G/Q$ where $Q$ is a parabolic subgroup of $G$. Definition 2 Let $K$ be a ...
0
votes
0answers
17 views

Closed $U$-orbits in the space of plane lattices

Let $G=\mathbf{SL}_n(\mathbf{R})$ and $\Gamma=\mathbf{SL}_n(\mathbf Z)$, so $G/\Gamma$ is the space of unimodular plane lattices. Let $U$ be the upper unipotent group, that is the set of ...
3
votes
1answer
74 views

Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G ...
1
vote
2answers
110 views

Vector field invariant under transitive action: restricts to free transitive action?

Thinking about how you can put vector fields on homogeneous spaces that respect the homogeneity, I'm interested in the following situation: Let $V$ be a nonzero vector field on a manifold $M$, let ...
5
votes
1answer
160 views

Computing characteristic numbers of homogeneous spaces

I apologize in advance if this is a bad question. I would like to prove or refute a conjecture about the vanishing of characteristic numbers of homogeneous spaces, and to this end am looking (and ...
2
votes
2answers
37 views

Can I identify $S_k(V)$ with an homogeneous space?

I'm in trouble with a question: Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can I identify the manifold, $$S_k(V):=\{(X_1, \ldots, X_k): X_1, \ldots, X_k\in V\ \textrm{are linearly ...