TITLE: Geodesics for a Homogeneous Space?
QUESTION [9 upvotes]: Is there a specific formula/method to find geodesics for a Homogeneous space? (excluding general methods applicable to arbitrary riemannian manifold)

REPLY [10 votes]: One method is to use the generalization of the principle of conservation of momemntum and
angular momentum.  If $\phi_t$ is any one parameter group of isometries of any Riemannian manifold, with time derivative expressed as a vector field $X$, then for any geodesic $g$,
the inner product
$\left \langle X, \dot g \right \rangle $ is constant along the geodesic, where $\dot g$ denotes the unit tangent vector to $g$  (assuming $g$ is parametrized by arclength).  In Euclidean space,
applied to 1-parameter groups of translations, this gives the principle of conservation of
momemntum for a geodesic; applied to 1-parameter groups of rotations, it gives the principle
of conservation of angular momentum.
In a homogeneous space, there are at least enough such vector fields  to form a basis at each point.  In an $n$-dimensional homogeneous space, if there are more than $n$ linearly independent infinitesimal isometries, then the equations, for a typical initial vector, are consistent only in a proper subset, and on that subset, they typically define a vector field.
This may give you all the information you need, without even integrating the vector field.
But in any case, it cuts the differential equation down to many fewer degrees of freedom.