Is there a specific formula/method to find geodesics for a Homogeneous space? (excluding general methods applicable to arbitrary riemannian manifold)



Geodesics for a Homogeneous Space?
I assume you mean riemannian homogeneous space and you are talking about geodesics relative to the Levi-Civita connection.


If so, you can always try to find geodesics which are **homogeneous**; that is, geodesics which correspond to the orbits of one-parameter subgroups.


Let $M = G/H$ and let $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ be a reductive split (always possible for riemannian homogeneous spaces, since $H$ is compact) of the Lie algebra of $G$. By homogeneity, you can look for geodesics passing through the identity coset $o \in M$ and then transport them using the $G$-action to any other point. To specify a geodesic through $o$ it is necessary and sufficient to specify the tangent vector at $o$, which we can think of as element $X \in \mathfrak{g}$. Then the orbit $\exp(t X) \cdot o$ is a geodesic if and only if


$$
\left< [X,Z]\_{\mathfrak{m}},X\_{\mathfrak{m}} \right> = 0
$$


for all $Z \in \mathfrak{m}$, and where the subscript means the component along $\mathfrak{m}$.


Although this is not guaranteed to find you all geodesics, there are some homogeneous spaces (so-called **geodesic orbit** or **g.o.** spaces) for which all geodesics are of this type.


