Almost gradient pseudo-Ricci solitons on 3-dimensional real hypersurfaces

On a Hopf hypersurface of a complex space form

Let M be a Hopf hypersurface in a nonflat complex space form , , of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].

We determine when the Hopf vector fields on orientable real hypersurfaces $(M,g)$ in complex space forms are minimal or harmonic. Furthermore, we determine when these vector fields give rise to harmonic maps from $(M,g)$ to the unit tangent sphere bundle $(T_1M,g_S)$. In particular, we consider the special case of Hopf hypersurfaces and of ruled hypersurfaces. The Hopf vector fields on Hopf hypersurfaces with constant principal curvatures provide examples. The minimal ruled real hypersurfaces form another class of particular examples.

In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.

In the class of real hypersurfaces M 2n−1 isometrically immersed into a nonflat complex space form $$\widetilde {{M_n}}\left( c \right)$$ of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ℂP n (c) or a complex hyperbolic space ℂH n (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in $$\widetilde {{M_n}}\left( c \right)$$ , we consider a certain real hypersurface of type (A2) in ℂP n (c) and give a geometric characterization of this Hopf manifold.

We prove that the Ricci tensor Ŝ with respect to the generalized Tanaka-Webster connection of a real hypersurface with the almost contact structure (η, ϕ, ξ, g) in a complex space form of complex dimension n ≥ 3 satisfies Ŝ(X, ϕY ) = λg(X, ϕY ) for any vector field X and Y , λ being a function, if and only if the real hypersurface is locally congruent to some type (A) hypersurface.

Let M be a real hypersurface of a complex space form , . Suppose that the structure vector field ξ of M is an eigen vector field of the Ricci tensor S, , β being a function. We study on M, a gradient pseudo‐Ricci soliton () that is an extended concept of gradient Ricci soliton, closely related to pseudo‐Einstein real hypersurfaces. When , we show that M is a Hopf hypersurface.

We classify real hypersurfaces of a non-flat complex space form of complex dimension 2 satisfying the condition that the Ricci tensor S is the transversal Killing tensor field on the direction of the structure vector field $\xi$ and $S\xi = \beta\xi$ where $\beta$ is a function.

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(4e). In the second, we give a complete classification of real hypersurfaces in M m(4e) which satisfy the above geometric facts.

We study curvature of Hopf hypersurfaces in a complex projective space or hyperbolic space. In particular, we prove that there are no real hypersurfaces in a non-flat complex space form whose Reeb-sectional curvature vanishes.

We give a characterization of pseudo-Einstein real hypersurfaces of a complex space form. We prove that if the Ricci tensor S of a real hypersurface M of a complex space form with constant principal curvatures satisfies for any , where a is a function and T 0 is the holomorphic distribution on M , then M is a pseudo-Einstein real hypersurface.

Cyclic η-parallel shape and Ricci operators on real hypersurfaces in two-dimensional nonflat complex space forms

We prove that a real hypersurface in a nonflat complex space form is of type $$(A)$$ if and only if the structure Lie operator is cyclic parallel.

Transversal Killing operators on real hypersurfaces in nonflat complex space forms

We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor.