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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Is $\mathbb{CP}^2-\mathbb{RP}^2$ a Stein manifold?
The space $\mathbb{CP}^2-\mathbb{RP}^2$ looks plausibly Stein to me, because it's a complex manifold with the homotopy type of $S^2$, so it has the correct homological dimension. Is it actually, ...
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Characterisation of symmetric spaces via dimension
Let $(M^n,g)$ be a simply-connected, complete Riemannian manifold. It is well known that the dimension of the isometry group of $(M,g)$ is bounded by $n(n+1)/2$, and this bound is attained if and only ...
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Conjugate of a positive-definite quaternionic matrix
Let $\mathbb H = \text{span}_{\mathbb R}\{1, i, j, k\}$ denote the quaternions. Recall that, given $q = a+ bi + cj + dk \in \mathbb H$ where $a,b,c,d \in \mathbb R$, we define the conjugate of $q$ as $...
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Computing the Connection $\nabla_X Y$ in the SPD Manifold
I am studying the geometry of the space of symmetric positive definite (SPD) matrices, denoted $SPD_n(\mathbb{R})$, and I am trying to properly compute the connection $\nabla_X Y $, where $X, Y$ are ...
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Explicit formula for Helgason Fourier Transform on Hyperbolic Plane
I'm quite new to abstract harmonic analysis and have been reading about the Helgason-Fourier transform on non-compact spaces. I'm trying to understand a simple rank-one example, the hyperbolic plane, ...
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Cartan subspace
We know that in the classification of Cartan for Symmetric spaces (spaces admitting an involution), Cartan subalgebras and Cartan subspaces (residing in the tangent space of the space) play an ...
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The symmetric space $\operatorname{SO}(n,\mathbb C) / \operatorname{SO}(n)$
I am looking for a reference for the symmetric space $\operatorname{SO}(n,\mathbb C) / \operatorname{SO}(n)$; I haven't been able to find any references on about it online. In particular, I would ...
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Automorphism group of a real simple Lie algebra
Question: Let $\mathfrak g$ be a real finite-dimensional simple Lie algebra of non-compact type.
What is the automorphism group $\text{Aut}(\mathfrak g)$ of $\mathfrak g$?
Non-compact above means that ...
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Are the Siegel upper half-spaces complex/Kähler submanifolds of one another?
Let us write $G_r=\mathrm{Sp}(2r,\mathbb{R})$ and $K_r=G_r\cap\mathrm{U}(2r)$. Then the Siegel upper half-space of rank $r$ is given (as a smooth manifold) by $X_r=G_r/K_r$. In this post it is shown ...
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Block Embeddings of Classical Groups Inducing Embeddings of Symmetric Spaces
Families of certain classical groups induce families of (irreducible) symmetric spaces.
Let $(G_r)_r$ be such a family of classical groups, e.g. $G_r=\mathrm{Sp}_{2r}(\mathbb{R})$ and let $\iota_r:G_r\...
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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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Different geometry implies different topology for compact (locally) symmetric spaces
All spaces in this question are assumed to be connected.
It seems to me that if $ (M_1,g_1) $ and $ (M_2,g_2) $ are non-isometric, compact, symmetric spaces then the underlying manifolds $ M_1 $ and $ ...
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Is the space $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/\mathrm{SU}(2)$ a symmetric space?
My question is: is the homogeneous space $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/\mathrm{SU}(2)$ a symmetric space?
Definitions: Let $G$ be a connected Lie group and $H$ a closed subgroup. The pair $(...
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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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