Geodesic transformations of the complex projective plane Are there non-trivial diffeomorphisms (i.e., different from isometries) of the complex projective plane that map geodesics (for the canonical Riemannian metric) to geodesics? 
Same question for all other rank one symmetric spaces different from spheres and real projective spaces. 
 A: The answer  is no. The explanation of Anton is of course correct but there exist stronder statements in the literature: for example by Sinjukov  (Dokl. Akad. Nauk SSSR (N.S.) 98, (1954) 21--23) any symmetric space is locally \emph{geodesically rigid}  is the sense that any 
other metric having the same (unparameterized) geodesics with it  is affinely equivalent to if (i.e., the Levi-Civita connections coincide) which in the  irreducible  case means that the metrics are proportional. 
Actually, stronger statements hold. For example from the Lichnerowicz-Obata conjecture arXiv:math/0407337  it follows that compact Riemannian homogeneous 
 metrics  such that sectional curvature is not constant and positive 
are also geodesically rigid. Indeed, a Killing vector field for the initial metric is a infinitesimal projective transformation for the second, which must be also Killing by the projective Lichnerowicz-Obata conjecture. Then, the isometry algebras   of the metrics are the same and therefore their volume forms are the same and these already implies (short tensor calculations, see for examples eqns. (1), (4), (5) of     arXiv:0806.3169) that the metrics are affinely equivalent.  I do not know whether homogeneous metrics of nonconstant curvature are geodesically rigid locally but all examples indicate that probably they are. 
Now, in the case your metric is Kähler and not flat,
then if it is not geodesically rigid then it is locally a cone over a (sasakian) manifold which in particular implies that the manifold is not compact. 
This  statement  is pretty nontrivial and  follows from Theorem  4.6. of  Mikes (Journal of Mathematical Sciences 78(1996)  311-333)  combined with   the  Splitting Lemma  from  arXiv:0904.0535 and combined with the following statement which was explained to me by Kiosak and which is probably not published:  Warped product Kähler nonflat 
 metric  is a locally a cone over a sasakian manifold. 
A: For complex projective plane with the canonical metric you get only isometries.
Indeed, note that such map has to send complex lines to the complex lines.
It follows since, any complex line is a union of an infinite family of geodesics passing through two points and the other way arround.
It remains to check which complex projective maps send geodesics to geodesics.
Note that a Möbius transformation of a sphere (=complex projective line) sends geodesic to geodesic if and only if it is an isometry. Hence the result follows.
(The same might follow from the projective curvature tensor, but I do not know how one calculates it.)
