This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m\left(4ϵ\right)$. In the second, we give a complete classification of real hypersurfaces in $M_m\left(4ϵ\right)$ which satisfy the above geometric facts.

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type $\left(\mathrm{A}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for $M$ in $G_2\left({ℂ}^{m+2}\right)$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi {\phi }_1$ induced by two structure tensors $\phi$ and ${\phi }_1$. That is, this commuting shape operator is given by $\phi {\phi }_{1}A=A\phi {\phi }_1$. Using this condition, we prove that...

Real affine hypersurfaces of the complex space ${ℂ}^{n+1}$ are studied. Some properties of the structure determined by a J-tangent transversal vector field are proved. Moreover, some generalizations of the results obtained by V. Cruceanu are given.

We study real affine hypersurfaces $f:M\to {ℂ}^{n+1}$ with an almost contact structure (φ,ξ,η) induced by any J-tangent transversal vector field. The main purpose of this paper is to show that if (φ,ξ,η) is metric relative to the second fundamental form then it is Sasakian and moreover f(M) is a piece of a hyperquadric in ${ℝ}^{2n+2}$.

Real affine hypersurfaces of the complex space ${ℂ}^{n+1}$ with a J-tangent transversal vector field and an induced almost contact structure (φ,ξ,η) are studied. Some properties of hypersurfaces with φ or η parallel relative to an induced connection are proved. Also a local characterization of these hypersurfaces is given.

The aim of the present paper is to classify real hypersurfaces with pseudo-𝔻-parallel structure Jacobi operator, in non-flat complex space forms.

The notion of special symplectic connections is closely related to parabolic contact geometries due to the work of M. Cahen and L. Schwachhöfer. We remind their characterization and reinterpret the result in terms of generalized Weyl connections. The aim of this paper is to provide an alternative and more explicit construction of special symplectic connections of three types from the list. This is done by pulling back an ambient linear connection from the total space of a natural scale bundle over...

We introduce a conformal Sasakian manifold and we find the inequality involving Ricci curvature and the squared mean curvature for semi-invariant, almost semi-invariant, $\theta$-slant, invariant and anti-invariant submanifolds tangent to the Reeb vector field and the equality cases are also discussed. Also the inequality involving scalar curvature and the squared mean curvature of some submanifolds of a conformal Sasakian space form are obtained.

We consider an almost Kenmotsu manifold $M^{2n+1}$ with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that $M^{2n+1}$ is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that $M^{2n+1}$ is ξ-Riemannian-semisymmetric. Moreover, if $M^{2n+1}$ is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove that $M^{2n+1}$ is...