On the invariance conditions of almost Hermitian structure classes under holomorphically-projective transformations

The authors relate holomorphically-projective transformations (HP-transformations) of almost Hermitian structures to the Gray-Hervella classes of almost Hermitian manifolds. par More precisely, given an almost Hermitian manifold (M,g,J), an HP-transformation of (g,J) is an almost Hermitian structure (tilde g,J) having the same holomorphically projective curves as (g,J). par The authors consider the (1,2)-tensor fields B,C on (M,g,J) such that, for any vector fields X,Y, B(X,Y)=frac12 ((nabla_{JX}J)Y-(nabla_XJ)JY), C(X,Y)=frac12((nabla_{JX}J)Y+(nabla_X J)JY), nabla denoting the Levi-Civita connection. par It is known that nabla J,B,C are invariant under HP-transformations. Let B_0 be the trace-free component of B, so that B=B_0+B_1, B_1 being determined by the Lee form beta of M. The authors prove that beta is an HP-transformation invariant. This allows them to state the HP-invariance of eight of the sixteen Gray-Hervella classes of almost Hermitian manifolds. In particular, Kähler, nearly-Kähler, quasi-Kähler, Hermitian, semi-Kähler and G_1-structures are HP-invariant. par Finally, if (M,g,J) does not belong to any of the classes considered, the authors determine the Gray-Hervella class of an HP-transformation of (g,J). For instance, if (g,J) is almost Kähler, then any HP-transformation of (g,J) is quasi-Kähler.