Given:


1) a Finsler metric $F\_p : T\_p G \rightarrow \mathbb{R}$ on $SU(N+1)$


2) a $U(N)$ subgroup of $SU(N+1)$ which is a stabilizer subgroup of some point on $CP^N$ (complex projective space) in the standard action of $SU(N+1)$ on $CP^N.$ This action is given by realizing $CP^N$ as a quotient manifold on $\mathbb{C}^{N+1}$. The appropriate equivalence relation on $\mathbb{C}^{N+1}$ is given by $u \sim v$ if $u = Zv$ for some $Z \in \mathbb{C}/\{0\}$. $SU(N+1)$ acts on $\mathbb{C}^{N+1}$ by matrix multiplication. The standard action I'm speaking of is now given by setting $U \in SU(N+1)$ acts on $CP^N$ by the formula $U \circ [v] = [Uv]$.


It is true (and well known) that $CP^N \cong SU(N+1)/U(N)$.
But what exactly is invariance the condition on $F$ for it to push forward through this isomorphism to a Finsler metric on $CP^N$ that is invariant under the action of $SU(N+1)$?



Invariant Finsler Metrics on Homogeneous Spaces
Because $\mathrm{SU}(N{+}1)$ acts transitively on the projectivized tangent space of $\mathbb{CP}^N$ (a fact that has nothing to do with metrics on $\mathbb{CP}^N$), any $\mathrm{SU}(N{+}1)$-invariant Finsler structure is determined by knowing its value on a single tangent line at a single point of $\mathbb{CP}^N$. In particular, since the Fubini-Study metric is $\mathrm{SU}(N{+}1)$-invariant, any $\mathrm{SU}(N{+}1)$-invariant Finsler structure on $\mathbb{CP}^N$ must be a constant multiple of the Fubini-Study metric. 


