# Questions tagged [symmetric-spaces]

A symmetric space is a differentiable manifold with the additional structure of a pseudo-Riemannian metric and which has many isometries.

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### Conjugacy action of $SO(2m)$ on $O(2m)/U(m)$

I seek intuition about the symmetric space $S$, the set of orthogonal complex structures in $\mathbb{R}^n$ for even $n=2m$. I am finding J.H.Eschenburg's Lecture Notes on Symmetric Spaces very helpful....
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### Does isometry on PSD matrices preserve eigenvalues?

Let $S$ be the set of symmetric matrices and $T: S\rightarrow S$ be a linear isometry. Moreover, let $T$ be a bijection from the space of PSD matrices to the set of PSD matrices. Must $T$ preserve ...
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### (locally) symmetric spaces where every conformal transformation is an isometry

By an argument using Liouville's theorem for conformal maps Conformal automorphism of $H^n$ it can be shown that every conformal automorphism of a hyperbolic manifold is an isometry. Are there any ...
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### Some confusion about Riemannian symmetric space

Recently I'm learning some basic theories about symmetric spaces from "Differential geometry, Lie groups, and symmetric space" written by Sigurdur Helgason, and I have some confusion. I Hope ...
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### Computation of the LES of homotopy groups associated with compact symmetric spaces

I am looking for an efficient way to compute the homotopy groups, as well as morphisms between them, of certain matrix groups and compact symmetric spaces. To be specific, I want to determine the long ...
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### Symmetrization map over the polynomial ring of a vector space.

Let $V$ be a finite-dimensional complex vector space. Is the morphism \begin{gather*} \mathrm{Sym}^{\bullet}(V \oplus V^*) \to D(V) \cong \frac{\mathrm{T}^{\bullet}(V\oplus V^*)}{I} \,, \\[0.5em] (v_1,...
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### Maximum symmetry metric on Cayley Plane $F_4/Spin(9)$

The maximum symmetry metric on real projective space $\mathbb{RP}^n$ is the round metric. The maximum symmetry metric on complex projective space $\mathbb{CP}^n$ is the Fubini-Study metric. https:/...
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### Why is the bilinear form $B_{\theta}$ (defined via Killing form) positive definite?
Let $\frak{g}$ be a matrix Lie algebra with Cartan involution $\theta : X\mapsto -X^*$ (negative conjugate transpose). Let $B$ be the bilinear symmetric Killing form \$B(X,Y)=\operatorname{tr}(\...