# 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 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.

• 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 $\mathfrak{g}/\mathfrak{h}$. – YCor Oct 14 '16 at 1:44
• 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". – Holonomia Oct 14 '16 at 18:31

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.

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.