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 Riemannian homogeneous space is a quotient $G/H$ of a Lie group $G$ modulo a closed subgroup $H$ together with an $\mbox{Ad}(H)$-invariant scalar product on a complement to the tangent space of $H$. As far as I know many results that are difficult to obtain for general manifolds are much easier to get for homogeneous spaces, since one can reduce the problem to a Lie algebraic problem. I would like to get an overview about the way how this method is used. How does one reduce the problem to a Lie algebraic one? Do you know any references in the internet about this method? Thank You in advance.
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2$\begingroup$ Your sentence "By definition" is unclear. This is not the definition. The definition is a $G$-invariant Riemannian metric on $G/H$. This is equivalent to the datum of a $Ad(H)$-invariant scalar product on $\mathfrak{g}/\mathfrak{h}$. When $H$ is connected the latter is the same as an $ad(\mathfrak{h})$-invariant scalar product on on $\mathfrak{g}/\mathfrak{h}$. $\endgroup$– YCorCommented Oct 14, 2016 at 1:44
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$\begingroup$ I recommend the chapter "Homogeneous Spaces" in the book "Comparisons theorems in Riemannian geometry" by Cheeger&Ebin. Two classical books are Helgason's "Differential Geometry etc" and Vol II of Kobayashi&Nomizu "Foundations of Differential Geometry". $\endgroup$– HolonomiaCommented Oct 14, 2016 at 18:31
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An Introduction to Lie Groups and the Geometry of Homogeneous Spaces written by By Andreas Arvanitogeōrgos is a useful reference Read online here.
Another good reference for physic and math students is Differential Geometry and Lie Groups for Physicists written by Marián Fecko Read on google book.
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A. L. Onishchik, Topology of transitive transformation groups is an excellent book for the study of compact homogeneous Riemannian manifolds.
Gorbatsevich, V. V.; Onishchik, A. L. Lie groups of transformations is an excellent book about more general homogeneous spaces.
answered Oct 14, 2016 at 11:52