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12 pages, 244 KiB

Open AccessArticle

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

by Daniel Guan

Mathematics 2024, 12(11), 1710; https://doi.org/10.3390/math12111710 - 30 May 2024

Viewed by 1300

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex [...] Read more.

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification

G^{C}

of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. Full article

8 pages, 221 KiB

Open AccessArticle

The Shape Operator of Real Hypersurfaces in S⁶(1)

by Djordje Kocić and Miroslava Antić

Mathematics 2024, 12(11), 1668; https://doi.org/10.3390/math12111668 - 27 May 2024

Viewed by 856

Abstract

The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere

S^{6} (1)

. More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we [...] Read more.

The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere

S^{6} (1)

. More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we classify real hypersurfaces in a nearly Kähler sphere

S^{6} (1)

whose Lie derivative of the shape operator coincides with its covariant derivative. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)

14 pages, 285 KiB

Open AccessArticle

The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric

by Rongsheng Ma and Donghe Pei

Mathematics 2023, 11(1), 90; https://doi.org/10.3390/math11010090 - 26 Dec 2022

Viewed by 1489

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds)

12 pages, 286 KiB

Open AccessArticle

Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S⁶(1)

by Miroslava Antić and Djordje Kocić

Mathematics 2022, 10(13), 2271; https://doi.org/10.3390/math10132271 - 29 Jun 2022

Cited by 16 | Viewed by 1871

Abstract

It is well known that the sphere

S^{6} (1)

admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector [...] Read more.

It is well known that the sphere

S^{6} (1)

admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field

ξ = - J N

is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to

ξ

is called the structure Jacobi operator, and is denoted by

l = R (\cdot, ξ) ξ

, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere

S^{6} (1)

. More precisely, we prove that such real hypersurfaces do not exist. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds: Volume II)

13 pages, 301 KiB

Open AccessArticle

Real Hypersurfaces in Complex Grassmannians of Rank Two

by Dehe Li and Shujie Zhai

Mathematics 2021, 9(24), 3238; https://doi.org/10.3390/math9243238 - 14 Dec 2021

Cited by 2 | Viewed by 1841

Abstract

It is known that there does not exist any Hopf hypersurface in complex Grassmannians of rank two of complex dimension

2 m

with constant sectional curvature for

m \geq 3

. The purpose of this article is to extend the above result, and [...] Read more.

It is known that there does not exist any Hopf hypersurface in complex Grassmannians of rank two of complex dimension

2 m

with constant sectional curvature for

m \geq 3

. The purpose of this article is to extend the above result, and without the Hopf condition, we prove that there does not exist any locally conformally flat real hypersurface for

m \geq 3

. Full article

15 pages, 308 KiB

Open AccessArticle

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Mathematics 2021, 9(4), 333; https://doi.org/10.3390/math9040333 - 7 Feb 2021

Viewed by 1739

Abstract

We study the semi-Riemannian geometry of the foliation

F

of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation

ω = 0

, provided that

\nabla ω = 0

and

c = ∥ ω ∥ \neq 0

( [...] Read more.

We study the semi-Riemannian geometry of the foliation

F

of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation

ω = 0

, provided that

\nabla ω = 0

and

c = ∥ ω ∥ \neq 0

(

ω

is the Lee form of M). If M is conformally flat then every leaf of

F

is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature

c / 4

, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index

2 s

,

0 < s < n

, is locally biholomorphically homothetic to an indefinite complex Hopf manifold

C H_{s}^{n} (λ)

,

0 < λ < 1

, equipped with the indefinite Boothby metric

g_{s, n}

. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

9 pages, 229 KiB

Open AccessArticle

On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

by George Kaimakamis and Konstantina Panagiotidou

Symmetry 2019, 11(4), 559; https://doi.org/10.3390/sym11040559 - 18 Apr 2019

Cited by 5 | Viewed by 2115

Abstract

In this paper the notion of

^{*}

-Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the

^{*}

-Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems [...] Read more.

In this paper the notion of

^{*}

-Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the

^{*}

-Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of

^{*}

-Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing

^{*}

-Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose

^{*}

-Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

15 pages, 232 KiB

Open AccessArticle

Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

by Dae Ho Jin and Jae Won Lee

Axioms 2018, 7(3), 68; https://doi.org/10.3390/axioms7030068 - 10 Sep 2018

Viewed by 3388

Abstract

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric [...] Read more.

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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