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 can 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.)

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.)

For complex projective plane with the canonical metric you get only isometries.

Note that such map has to send complex lines to the complex lines; i.e., it is a complex projective map. It remains to check which projective maps send geodesics to geodesics.

A Moebius transformation of a sphere (=complex line) sends geodesic to geodesic if and only if it is an isometry. Hence the result follows.

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 can 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.)