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We show that there is no proper CR submanifold with semi-flat normal connection and semi-parallel second fundamental form in a complex space form with non-zero constant holomorphic sectional curvature such that the dimension of the holomorphic tangent space is greater than 2.

We study affine hypersurface immersions $f:M\to {ℝ}^{2n+1}$, where M is an almost complex n-dimensional manifold. The main purpose is to give a condition for (M,J) to be a special Kähler manifold with respect to the Levi-Civita connection of an affine fundamental form.

We give a characterization of totally $\eta$-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator $A$ of a real hypersurface $M$ of a complex space form $M^n\left(c\right)$, $c\ne 0$, $n\ge 3$, satisfies $g(AX,Y)=ag(X,Y)$ for any $X,Y\in T_0\left(x\right)$, $a$ being a function, where $T_0$ is the holomorphic distribution on $M$, then $M$ is a totally $\eta$-umbilical real hypersurface or locally congruent to a ruled real hypersurface....