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University lecturer.
11h

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Tori in Compact Riemannian Symmetric Spaces
@IgorBelegradek: Thanks for the reference. 
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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. 
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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? 
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accepted  Tori in Compact Riemannian Symmetric Spaces 
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asked  Tori in Compact Riemannian Symmetric Spaces 
Jul 2 
awarded  Curious 
Jun 24 
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Decomposition of Lie subspaces
I can check my claim with $S^7=Spin(7)/G_2$. 
Jun 24 
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Decomposition of Lie subspaces
For a symmetric space, the canonical connection has zero torsion and so $\frak{m}$$_1=0$. So the result is trivially true, isn't it? 
Jun 24 
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Decomposition of Lie subspaces
Let me be a little more specific. Is it possible to view $\frak{m}$$_1$ as the Lie algebra of some subgroup of $G$? 
Jun 23 
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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 wellknown and I would have thought that it was generalized to reductive spaces. 
Jun 23 
accepted  Decomposition of Lie subspaces 
Jun 23 
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Decomposition of Lie subspaces
@JoséFigueroaO'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 semisimple subalgebra of $(\frak{m}$,$T_o)$. 
Jun 23 
asked  Decomposition of Lie subspaces 
May 21 
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Stiefel manifolds and polar decompositions
Thanks for the suggestions; I'll look into it. 
May 20 
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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 
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Decomposition of $S^7=Spin(7)/G_2$
@JoséFigueroaO'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 
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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 
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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 
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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. 