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2
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1answer
27 views

Reductive homogeneous spaces

If $G$ is a connected Lie group and $K$ is a closed subgroup of $G$ then $G/K$ is a homogeneous space. If $\frak g,k$ are the lie subalgebras of $G,K$ resp. Then under the projection $\pi:G\rightarrow ...
2
votes
1answer
28 views

The projective space as a homogeneous space

I want to understand why the projective space $\mathbb RP^n$ is diffeomorophic to $SO(n+1)/O(n)$? and why we can write the latter as $O(n+1)/O(n)\times O(1)$?
1
vote
1answer
53 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
1
vote
0answers
51 views

Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...
0
votes
0answers
19 views

open set in tangentspace induces open set in tangentbundle (for homogeneous spaces)

Let $M=G/K$ be a homogeneous space with a $G$-invariant riemannian metric $<.,.>$. Then $G$ defines an action on $TM$ by derivatives. Let $p=eK \in M$. Assuming I have a set $V_p \subset T_pM$ ...
1
vote
1answer
36 views

proper action on homogeneous space

Let $M = G/K$ be a homogeneous space. It is easy to show, that the left action of $G$ on itself by multiplication is a free and proper action. My question is, if the induced action $$G \times G/K \...
0
votes
1answer
15 views

Homogeneous functions and inner products

I am having trouble with understanding how homogeneous functions are related to the inner product. I'm trying to prove the following: For a functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$, $f$ is ...
0
votes
0answers
7 views

Two-point functions and spatial homogeneity

Consider a two-point function $f(\mathbf{r}_{1},\mathbf{r}_{2})$. If one requires homogeneity, then this implies that for a constant vector $\mathbf{a}$ we must have $$f(\mathbf{r}_{1},\mathbf{r}_{2})=...
0
votes
1answer
33 views

Line clipping in 2D perspective transformation

Situation I have two 2D spaces which are related one to other by a transformation matrix - 3*3 homography matrix for homogeneous coordinates: The first space is "map" and the second one is "camera ...
1
vote
1answer
83 views

Luna-Vust theory for embeddings of homogenous spaces

I'm interested in the theory of Luna and Vust of embeddings of homogenous spaces like presented in D. Luna, Th. Vust: Plongements d'espaces homogènes, Comment. Math. Helvetici 58 (1983) 186-245. ...
2
votes
1answer
34 views

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
2
votes
2answers
42 views

What is the interpretation of homogeneous line intersection?

I understand homogeneous coordinate systems. I read the intersection of lines in homogeneous coordinate can be computed by taking a cross products of lines $l_1(a_1,b_1,c_1)$ and $l_2(a_2,b_2,c_2)$. ...
1
vote
0answers
29 views

Quotients by simply connected closed subgroups

I have come across an exercise asking for a proof of something that is definitely false: If $G$ is a Lie group, $H$ a connected closed subgroup and $G/H$ simply connected, then $G$ is itself simply ...
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0answers
13 views

Two nilmanifolds of the same Lie group

By a nilmanifold I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group G by the left action of a maximal lattice 􀀀, i.e. a discrete cocompact subgroup. ...
1
vote
1answer
32 views

Examples of quotient manifolds which are not locally trivial fibrations?

Let $X$, $Y$ be differentiable manifolds, and $f : X \to Y$ a smooth surjection. Then $Y$ is said to be a quotient of $X$ if 1) $Y$ has the quotient topology 2) A function $g : Y \to \mathbb{R}$ is ...
2
votes
1answer
41 views

Spheres as Homogeneous Spaces

Any odd dimensional sphere $S^{2n+1}$ can be expressed as an homogenous space of $SU(n)$ by $S^{2n+1} \simeq SU(n)/SU(n-1)$. Any even dimensional sphere $S^{2n}$ sphere can be expressed as an ...
2
votes
1answer
43 views

Spheres as Symplectic Homogeneous Spaces

Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a ...
1
vote
1answer
48 views

Homogeneous space minus a point

If $X$ is homogeneous and $p\in X$, then is $X\setminus \{p\}$ necessarily homogeneous? This seems to work with all the simple examples I've tried. I would be interested in any counterexamples. Or ...
1
vote
2answers
53 views

Riemannian metrics on homogeneous spaces

Let G be a Lie group and H be a compact subgroup. The (left) coset space G/H is, up to an isomorphism, equivalent to the smooth homogeneous manifold M. My question is, is it possible to impose an ...
2
votes
1answer
45 views

Can the cohomology of the Grassmannian identified with the cohomology of a specific dense open subvariety?

Let $(\mathbb{C}^{2p},Q)$ be a $2p$-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form $Q$ where $p\geq 3$. Let $l\leq p-2$. You may assume that $l$ is odd if this ...
4
votes
1answer
64 views

Is every Hausdorff homogeneous space also regular?

Every Hausdorff topological group is regular (completely regular, in fact). Is this true if I replace topological group with homogeneous space? This is not obvious to me because there are Hausdorff ...
1
vote
1answer
71 views

Is the inverse image of an irreducible variety under the natural projection irreducible (in the setting of homogeneous spaces)?

Let $p\colon Z\to X$ be a morphism between irreducible varieties (= reduced schemes of finite type over $\mathbb{C}$). Assume that every fiber of $p$ over a closed point of $X$ is also irreducible. ...
1
vote
0answers
36 views

Commutator of Lie sub-algebra

I have a problem understanding the proof of Proposition 8.20, page 211, in Besse's Einstein Manifolds. He considers a semi-simple Lie algebra $\mathfrak{g}$ with $\operatorname{Ad}$-invariant scalar ...
1
vote
0answers
18 views

Homogeneous Space structures of the sphere

I'm reading through these lecture notes and trying to apply the following Theorem: A Lie group $G$ acts globally and transitively on a manifold $M$ if and only if $M \cong G/H$ is isomorphic to the ...
1
vote
0answers
80 views

Why is $G/H\to G/K$ proper and projective where $H\subseteq K\subseteq G$?

Let $G$ be a (linear) algebraic group (over an algebraically closed field of characteristic zero - if you like you can assume $\mathbb{C}$). Let $K$ and $H$ be closed subgroups of $G$ such that $H\...
0
votes
0answers
34 views

Action of Lie group on vector fields (of homogeneous space)

1) If I have a Lie group $G$ acting smoothly on a manifold $M$, how does this gives also an action of $G$ on the vector fields on $M$? 2) When I read that the group G acts canonically on the vector ...
0
votes
1answer
27 views

Schubert cell decomposition and full flags

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.
0
votes
1answer
23 views

low-dim unitary groups and their actions

I need someone to explain for me the unitary groups $U(1)$, $U(2)$ and $U(3)$ and their actions: Specifically: $U(3)/U(2)$ $U(3)/U(2)\times U(1)$ $U(3)/U(1)\times U(1) \times U(1)$ I have seen ...
0
votes
0answers
16 views

2D Homogeneous Transformation : Reflection vs Mirroring

I have two questions: (1) Is there any difference between the terms Reflection and Mirroring in 2D Transformation? (2) What are their Transformation matrices with reference to an arbitrary line?
0
votes
1answer
18 views

Extracting sign of scaling from modelView matrix

I want to retrieve the sign of scaling for each axis from modelview matrix. Right now I am able to extract the sign only if all 3 signs are same but it fails when one of them is different. Here is the ...
4
votes
1answer
210 views

Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
1
vote
0answers
47 views

Orbit under Lie group and projective variety

On https://en.wikipedia.org/wiki/Generalized_flag_variety#Highest_weight_orbits_and_homogeneous_projective_varieties there is a section which says Blockquote If G is a semisimple algebraic group ...
0
votes
1answer
30 views

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.
3
votes
1answer
50 views

Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
2
votes
1answer
74 views

What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
3
votes
1answer
75 views

Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where $\...
2
votes
0answers
48 views

How to Induce a Metric on Homogeneous Space $G/H$ by the Metric from Bundle G

I am having a question on how to induce a metric $g$ on homogeneous space $G/H$, if one is given a ${\rm Ad}_H$-invariant metric $\bar{g}$ on G. More specifically and simply, consider principal ...
1
vote
1answer
59 views

The action of the orthogonal group $O_n(\mathbb{R})$ on the Stiefel manifold $V_{k,n}(\mathbb{R}) $ .

I'm trying to prove that $O_n(\mathbb{R}) / O_{(n-k)}(\mathbb{R}) \cong V_{k,n}(\mathbb{R})$ where $ O_n(\mathbb{R}) = \left\lbrace A \in M_n(\mathbb{R}) / A A^t = I_{n}\right\rbrace $ is the ...
2
votes
0answers
41 views

Try to use Homogeous space Characterize the space of all lines in the plane .

The problem is just to gives a smooth manifold structure of all straight lines in $\mathbb{R^2}$ ( not just those which pass through the oringin).Moreover, identify it with a well-known manifold! My ...
1
vote
1answer
28 views

Converting to homogenuous coordinates

Let's assume elliptic curve $E$ over $\mathbb{R}$: $y^2 = x^3 + x + 1$ How to convert this equation to homogeneous coordinates? My notes say it's $zy^2=x^3+xz^2+z^3$. Unfortunately, I have no idea ...
6
votes
1answer
209 views

Fibre bundles for homogeneous spaces

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then it is known that $G/H$ can be equipped with a unique differentiable structure such that $G\xrightarrow{\pi} G/H$ is a locally ...
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vote
3answers
31 views

Why cannot the homogeneous coodinates be zero?

Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. Why can't Z be zero?
3
votes
1answer
69 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 = N_G(T)...
2
votes
1answer
57 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 $Sp(1)/U(1)=S^3/U(1)=\mathbb{C}P^1\...
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0answers
59 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
51 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
116 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 ...
3
votes
2answers
175 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
138 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
vote
0answers
29 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 groups"...