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9 votes
1 answer
646 views

Explicit construction of a (the?) dual symmetric space

I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type. ...
S.T.'s user avatar
  • 113
8 votes
1 answer
673 views

Classification of compact globally symmetric spaces

It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
shrinklemma's user avatar
7 votes
0 answers
509 views

Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space

Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
B K's user avatar
  • 1,942
6 votes
2 answers
540 views

Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement. Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...
HenrikRüping's user avatar
5 votes
1 answer
530 views

Geodesic distance on $\mathrm{SO}(n)$

$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
Math_Newbie's user avatar
5 votes
0 answers
276 views

Fundamental group of compact globally symmetric spaces

The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient $$(*) \quad \pi_1(...
Lucas Seco's user avatar
  • 1,123
3 votes
2 answers
1k views

Reference for homogeneous spaces

I am a graduate student of differential geometry. I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
Nick's user avatar
  • 79
3 votes
0 answers
108 views

$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and ...
user43326's user avatar
  • 3,051