Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.


What is the appropriate quotient metric on the homogeneous space and what are the geodesics? Clearly, for example, $K=U(n-1)$ gives $\mathbb{C}P^{n-1}$ and the Fubini-Study geodesics which can be expressed in terms of the original geodesics on $SU(n)$. Is any similar formula available for the more general case and is the resulting metric unique (upto constant positive multiple) as in the FS case?


I'm not seeking homogeneous geodesics specifically.



Geodesics on Homogeneous Spaces of $SU(n)$
I believe the answers to your and related questions are covered in Ch.X of 


[1] Foundations of Differential Geometry, vol. II, Kobayashi, Nomizu


More specifically, let $\mathfrak g:=Lie(SU(n))$ and $\mathfrak k:=Lie(K)$ be Lie algebras of the corresponding groups. Since $SU(n)$ is a compact simple Lie group there is unique (up to multiple) invariant metric $\kappa$ on $\mathfrak g$ coming from the Killing form. 


Let us fix a point $p=[eK]\in M=SU(n)/K$; any invariant tensor on $M$ is completely determined by its value at $p$. In particular, invariant metric on $M$ is determined by a metric on $T\_p M\simeq \mathfrak g/\mathfrak k$, where isomorphism is given by the infinitesimal action along $X\in\mathfrak g$. One of the ways to equip $\mathfrak g/\mathfrak k$ with a metric is to identify it with a $\kappa$-orthogonal complement of $\mathfrak k$. This way you get so-called **naturally reductive** homogeneous space. For $K=SU(n-1)$ this gives Fubini-Study metric on $\mathbb P(n-1)$; for $K=S(U(k)\times U(n-k))$ this metric makes $Gr(n,k)$ a symmetric space.


There are two important connections on $M$ with a naturally reductive metric. One is the standard Levi-Civita connection $\nabla^{LC}$ corresponding to the metric defined above, another one is **canonical** connection $\nabla$ (see Ch X.2 of [1]). Important facts are:


1. $\nabla^{LC}$ and $\nabla$ have the same geodesics (Theorem 2.10 and Theorem 3.3 in [1])
2. Geodesic of $\nabla$ in the direction $X\in \mathfrak k^\perp\simeq T\_p M$ is given by the curve $\exp (tX)p$ (Corollary 2.5 in [1]).
3. Any invariant tensor field on $M$ is $\nabla$-parallel.


Finally, I would like to mention that naturally reductive metric is not the only invariant metric on $M$ and other invariant metrics might have different geodesics.


