  Let N a compact complex submanifold of a compact complex manifold M. We say N
splits in M, if the holomorphic tangent bundle sequence splits holomorphically.
By a result of Mok a splitting submanifold of a Kaehler Einstein manifold with
a projective structure is totally geodesic. The classification of all splitting
submanifolds of families of fake elliptic curves given here completes the case
of threefolds M with a projective structure by a previous result of the
authors.
