2000 Mathematics Subject Classification: Primary 53B35, Secondary 53C50.In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.

Let $\Phi$ be an Hermitian quadratic form, of maximal rank and index $(n,1)$, defined over a complex $(n+1)$ vector space $V$. Consider the real hyperquadric defined in the complex projective space $P^{n}V$ by $$Q=\{\left[\varsigma \right]\in P^{n}V,\Phi \left(\varsigma \right)=0\}.$$ Let $G$ be the subgroup of the special linear group which leaves $Q$ invariant and $D$ the $\left(2n\right)-$ distribution defined by the Cauchy Riemann structure induced over $Q$. We study the real regular curves of constant type in $Q$, tangent to $D$, finding a complete system of analytic invariants for two curves to be locally equivalent...

In this paper we consider holomorphically projective mappings from the compact semisymmetric spaces $A_n$ onto (pseudo-) Kählerian spaces ${\overline{K}}_n$. We proved that in this case space $A_n$ is holomorphically projective flat and ${\overline{K}}_n$ is space with constant holomorphic curvature. These results are the generalization of results by T. Sakaguchi, J. Mikeš, V. V. Domashev, N. S. Sinyukov, E. N. Sinyukova, M. Škodová, which were done for holomorphically projective mappings of symmetric, recurrent and semisymmetric Kählerian...