Special Issue "Complex and Contact Manifolds"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: 31 October 2020.

Special Issue Editors

Prof. Dr. Ion Mihai
Website SciProfiles
Guest Editor
Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania
Interests: differentiable manifolds; Riemannian manifolds; distinguished vector fields; Riemannian invariants; sectional curvature; complex manifolds; contact manifolds; affine manifolds; statistical manifolds; submanifold theory
Special Issues and Collections in MDPI journals
Assoc. Prof. Dr. Adela Mihai
Website
Guest Editor
Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Bd. Lacul Tei 122-124, 020396 Bucharest, Romania
Interests: (pseudo-)Riemannian manifolds, curvature invariants, complex manifolds, contact manifolds, submanifold theory, statistical manifolds

Special Issue Information

Dear Colleagues,

The most studied differentiable manifolds are those endowed with certain endomorphisms of their tangent bundles: almost complex, almost product, almost contact, and almost paracontact manifolds, etc. Among complex manifolds, Kaehler manifolds play the most important role via their geometrical properties. Roughly speaking, contact manifolds are the odd-dimensional version of complex manifolds; in particular, Sasakian manifolds correspond to Kaehler manifolds. There are topological obstructions to the existence of Kaehler and Sasakian structures, respectively, on compact Riemannian manifolds.

The geometry of submanifolds in such manifolds is an important topic of research. Obstructions to the existence of special classes of submanifolds in complex and Sasakian manifolds were obtained in terms of their Riemannian curvature invariants.

The purpose of this Special Issue is to collect selected review works written by well-known researchers in the field, as well as new developments in the geometry of complex and contact manifolds or/and explore applications in other areas.

Prof. Dr. Ion Mihai
Assoc. Prof. Dr. Adela Mihai
Guest Editors

Manuscript Submission Information

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Keywords

  • complex manifolds
  • contact manifolds
  • Riemannian invariants
  • complex contact manifolds
  • submanifolds in complex and contact manifolds
  • holomorphic and Sasakian statistical manifolds

Published Papers (6 papers)

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Research

Open AccessArticle
Hypersurfaces of a Sasakian Manifold
Mathematics 2020, 8(6), 877; https://doi.org/10.3390/math8060877 - 01 Jun 2020
Abstract
We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξT on the hypersurface, namely the tangential component of ξ to hypersurface, and it also [...] Read more.
We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ρ has a certain lower bound. Full article
(This article belongs to the Special Issue Complex and Contact Manifolds)
Open AccessArticle
Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds
Mathematics 2019, 7(12), 1202; https://doi.org/10.3390/math7121202 - 08 Dec 2019
Cited by 1
Abstract
We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ(2,2). [...] Read more.
We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) . Full article
(This article belongs to the Special Issue Complex and Contact Manifolds)
Open AccessArticle
Submanifolds in Normal Complex Contact Manifolds
Mathematics 2019, 7(12), 1195; https://doi.org/10.3390/math7121195 - 05 Dec 2019
Abstract
In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant (CC-totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen [...] Read more.
In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant ( C C -totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant δ ( 2 , 2 ) for C C -totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of C C -totally real submanifolds of maximum dimension satisfying the equalities. Full article
(This article belongs to the Special Issue Complex and Contact Manifolds)
Open AccessArticle
Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cn
Mathematics 2019, 7(11), 1035; https://doi.org/10.3390/math7111035 - 03 Nov 2019
Abstract
The purpose of this paper is to study the Mergelyan approximation property in Lp and Ck-scales on certain weakly pseudoconvex domains of finite/infinite type in Cn. At the heart of our results lies the solvability of the additive [...] Read more.
The purpose of this paper is to study the Mergelyan approximation property in L p and C k -scales on certain weakly pseudoconvex domains of finite/infinite type in C n . At the heart of our results lies the solvability of the additive Cousin problem with bounds as well as estimates of the ¯ -equation in the corresponding topologies. Full article
(This article belongs to the Special Issue Complex and Contact Manifolds)
Open AccessFeature PaperArticle
On the Sign of the Curvature of a Contact Metric Manifold
Mathematics 2019, 7(10), 892; https://doi.org/10.3390/math7100892 - 24 Sep 2019
Abstract
In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat [...] Read more.
In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field. Full article
(This article belongs to the Special Issue Complex and Contact Manifolds)
Open AccessFeature PaperArticle
Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector
Mathematics 2019, 7(8), 710; https://doi.org/10.3390/math7080710 - 06 Aug 2019
Abstract
The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E3 [...] Read more.
The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors. Full article
(This article belongs to the Special Issue Complex and Contact Manifolds)
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