The tag has no usage guidance.

learn more… | top users | synonyms

0
votes
0answers
6 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 ...
0
votes
1answer
24 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
82 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
29 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
39 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
23 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 ...
0
votes
0answers
12 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
29 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
38 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
40 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
47 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
47 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
42 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
59 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
58 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
32 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
17 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
79 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 ...
0
votes
0answers
32 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
26 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
22 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
15 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
15 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
208 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
46 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
26 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
48 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
71 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
44 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
52 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
23 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
182 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 ...
1
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
61 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 = ...
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 ...
1
vote
0answers
52 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
50 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
105 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 ...
2
votes
2answers
164 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
136 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 ...
4
votes
0answers
92 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
57 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
108 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
87 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
34 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. ...
2
votes
1answer
35 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 can $LG(n)$ be identified with $U(n)/O(n)$? ...
2
votes
1answer
178 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) = ...