# Questions tagged [symmetric-spaces]

A symmetric space is a connected Riemannian manifold in which at every point there exists a global self-isometry whose differential at the given point is minus identity.

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### De Rham product decomposition theorem in a particular setting

Let $G$ be a Lie group, and $H$ a Lie subgroup of $G$ such that $G/H\sim \mathcal M$ is a homogeneous space diffeomorphic to $\mathbb{R}^n$, equipped with an invariant Riemannian metric $g$. If this ...
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### References for $K$-orbits in $G/B$

Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
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### Spherical functions in the space of functions on real Grassmannians

Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$. Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
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### Jacobi fields on non-symmetric spaces

I am searching for examples of manifolds which are not symmetric spaces but where Jacobi fields can be computed in closed form. For now, I am aware of Gaussian distribution with the Wasserstein ...
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### Showing an action of a higher rank lattice on hyperbolic space has a fixed point

In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
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I am interested in the manifold of affine subspaces of dimension $k$ of $\mathbb{R}^n$, which can be viewed as the homogeneous space $$E(n)/(E(k)\times O(n-k)),$$ where $E$ refers to rigid motions ...