It is known that the parallel imbeddings of a complex 

or a quaternionic projective space into real space 

forms are the examples of planar geodesic submanifolds.

Namely each geodesic on these projective spaces is mapped 

to a plane curve in the ambient real space form through 

the parallel imbeddings.

Moreover, we know that some particular circles of 

positive curvature on these submanifolds are also mapped 

to plane curves.

In this paper we consider the converse of this geometric 

property of such planar geodesic immersions.