# 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