Textbooks on the Riemannian geometry of homogenous manifolds

Question

I'm curious to learn more about homogenous manifolds, particularly the Riemannian geometry. As you know, homogenous manifolds are quotient manifolds of Lie groups. As a result, I'm curious what properties a metric on the parent Lie group induces on the quotient metric on the homogenous manifold. What can a geodesic on the parent Lie group say on the homogeneous manifold, and vice versa? I would appreciate any suggestions.