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visits member for 3 years, 5 months
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University lecturer.

11h
comment Tori in Compact Riemannian Symmetric Spaces
@IgorBelegradek: Thanks for the reference.
1d
comment Tori in Compact Riemannian Symmetric Spaces
I guessed that's what you meant but just wanted to make sure. Thanks again for the nice answer.
1d
comment Tori in Compact Riemannian Symmetric Spaces
Thanks for the answer Robert but there's something puzzling me. The flats you describe above need not be tori, right? For example, if we have a symmetric space of noncompact type then the exponential map is a diffeomorphism and so the space can't contain a torus. Have I misunderstood something?
1d
accepted Tori in Compact Riemannian Symmetric Spaces
2d
asked Tori in Compact Riemannian Symmetric Spaces
Jul
2
awarded  Curious
Jun
24
comment Decomposition of Lie subspaces
I can check my claim with $S^7=Spin(7)/G_2$.
Jun
24
comment Decomposition of Lie subspaces
For a symmetric space, the canonical connection has zero torsion and so $\frak{m}$$_1=0$. So the result is trivially true, isn't it?
Jun
24
comment Decomposition of Lie subspaces
Let me be a little more specific. Is it possible to view $\frak{m}$$_1$ as the Lie algebra of some subgroup of $G$?
Jun
23
comment Decomposition of Lie subspaces
I think this answers my question, Jose. Thanks. However, I was wondering whether there were any results of this nature in the literature. The Lie group result is well-known and I would have thought that it was generalized to reductive spaces.
Jun
23
accepted Decomposition of Lie subspaces
Jun
23
comment Decomposition of Lie subspaces
@JoséFigueroa-O'Farrill What I meant was $T_o(\frak{m}_1,\frak{m}_1)\ne 0$. Also, in the decomposition, $\frak{m}_0$ is a subspace and $\frak{m}_1$ is a semi-simple subalgebra of $(\frak{m}$,$T_o)$.
Jun
23
asked Decomposition of Lie subspaces
May
21
comment Stiefel manifolds and polar decompositions
Thanks for the suggestions; I'll look into it.
May
20
comment Decomposition of $S^7=Spin(7)/G_2$
@ClaudioGorodski If you view $Spin_7$ as $8\times 8$ real matrices, what do you get for $\text{exp}\> {\frak p}$?
May
20
comment Decomposition of $S^7=Spin(7)/G_2$
@JoséFigueroa-O'Farrill I think the situation is the same. By $SO(3,1)$ I mean the proper orthochronous Lorentz group $SO^+(3,1)$, which is connected.
May
19
comment Decomposition of $S^7=Spin(7)/G_2$
@Deane: An example of what I'm talking about. Take hyperbolic space $H^3=SO(3,1)/SO(3)$. This is a Riemannian symmetric space of noncompact type. Elements of the Lorentz group $SO(3,1)$ can be decomposed into a product of a rotation and a boost. This is by virtue of the fact that we have a Cartan decomposition at the Lie algebra level. The boosts $B({\bf u})$ can be parameterized by elements ${\bf u}\in H^3$ and it can be shown that $B({\bf u})B({\bf v})$=$B({\bf u}+{\bf v})O$ where $O$ is a rotation and ${\bf u}+{\bf v}$ is the relativistic velocity addition.
May
19
awarded  Critic
May
19
comment Decomposition of $S^7=Spin(7)/G_2$
@Andre: We can consider elements of $Spin(7)$ as $7\times 7$ real matrices. What I'm looking for is a type of polar decomposition in terms of $G_2$. This doesn't have to be complicated.
May
19
comment Decomposition of $S^7=Spin(7)/G_2$
@Andre: I'm making no such assumption. I'm not even approaching the problem from the bundle point of view. As I already mentioned, the result I'm using is in Helgason and it uses the exponential map, not bundles. Results of this type are generally not easy; see my earlier post on Stiefel manifolds: <mathoverflow.net/questions/139542/>;. However, in the case of $S^7$, I wonder if the octonionic structure doesn't help.