All Questions
9 questions
13
votes
2
answers
3k
views
What's the Lipschitz constant of the exponential map for $\mathrm{SO}(n,R)$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\so{\mathfrak{so}}$Consider the Lie algebra $\so(n)$ equipped with the metric $\langle e_i \wedge e_j, e_k \wedge e_l \rangle = \delta_{i,k} \delta_{j,l}...
- 4,466
11
votes
1
answer
726
views
Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?
$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
- 6,741
9
votes
2
answers
715
views
Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$The exceptional isomorphism $\Spin(5,1)\simeq \SL(2,\mathbb{H})$ is well-known, and I can find references that ...
- 35.5k
7
votes
1
answer
271
views
Closest point in $SU(n) \otimes SU(n)$ to $SU(n^2)$
What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. minimize over $V_1, V_2$:
$\min_{V_1, V_2} | V_1 \otimes V_2 - U|$ in ...
- 2,099
3
votes
1
answer
494
views
A question on Grassmannian
Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...
- 119
3
votes
0
answers
82
views
Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?
$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
- 6,741
3
votes
0
answers
83
views
Particular decomposition of $SU(n)$
Given $a,b \in \mathfrak{su(n)}$ which generate the full algebra, it is possible to write and $G \in SU(n)$ as:
$G = \exp(\alpha_1 a)\exp(\beta_1 b) \ldots \exp(\alpha_m a)\exp(\beta_m b)$
for some ...
- 2,099
2
votes
1
answer
274
views
Isomorphism of Complex Stiefel manifold and Homogeneous space of unitary group, and the Stiefel logarithm problem
It is well known that $U(n)/U(n-k) \cong V_k(\mathbb{C}^n)$ where $U(n)$ is the unitary group, and $V_k(\mathbb{C}^n)$ is the appropriate Stielfel manifold.
I further understand that $V_k(\mathbb{C}^...
- 2,099
1
vote
1
answer
330
views
On Euler angles decomposition of $\mathrm{SU}(N)$
$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion:
$$
\SU(N)\ni m = a\, u \, b
$$
where $a,b$ are independent ...
- 41