# Tag Info

Accepted

• 17.9k
Accepted

• 85.1k

### Can a homogeneous space be a manifold with boundary?

Being a boundary point of a manifold can be characterised purely topologically (in terms of local homology groups say). So for each group element $g$ as $x\mapsto gx$ is a homeomorphism of $M$ to ...
• 154k
Accepted

### Is it possible to realize $\mathbb{RP}^2$ as a linear group orbit?

We can embed $\mathbb{RP}^2$ into the space $M_3$ of $3\times 3$ matrices via the map which takes a line $[v]$ to the orthogonal projection onto $[v]$: $$[v]\mapsto\frac{vv^T}{\|v\|^2}$$ This gives ...
• 13.5k

### Rotation Matrix of rotation around a point other than the origin

These matrices are left-side multiplicated with vector positions, so the order of multiplication is from right to left - on the right side is the first operation, on the left side - the last one.
• 4,845
Accepted

• 302k

### Riemannian metrics on homogeneous spaces

The answer to your question is no for several reasons. First, just because you can write $M = G/H$ does not mean that any of the natural metrics on $M$ have isometry group $G$ - it is sometimes ...
• 46.7k
Accepted

### Spheres as Homogeneous Spaces

The first part of the question has a positive answer, since we have $$S^n\cong SO(n+1)/SO(n)$$ also for odd $n$. For the case $SU(n)$ this is not possible in this way, however it might be realized ...
• 120k
Accepted

### 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.

Your argument has a significant issue in the paragraph after the theorem you quote: $H$ almost never acts transitively on $G/H$ by any action, and the induced action of $H$ on $G/H$ coming from the ...
• 46.7k
Accepted

### Finding the orbits of $SL(n+1)$ and a parabolic subgroup $P$ in a representation?

Even for reductive Lie groups $G$, the general problem of determining the $G$-orbit decomposition of a linear representation of $G$ is very difficult---or even “hopeless”, see this MathOverflow post. (...
• 85.4k
Accepted

### Finding explicit elements in $\text{SL}(3,\mathbb{R})/\text{SO}(3)$ whose simultaneous stabilizer in $\text{SL}(3,\mathbb{R})$ is the identity

You can (equivariantly) identify the quotient $SL(3,R)/SO(3)$ with the space of positive-definite quadratic forms in 3 variables with determinant (of the associated symmetric matrix) equal to $1$. The ...
• 85.4k
Speaking of a connected Riemannian manifold $(M, g)$, the structure is Homogeneous if, for every pair of points $p$ and $q$, there exists an isometry $f:M \to M$ such that $f(p) = q$. (The ...