Geometry Of Foliations Research Articles

Transversal Lightlike Submersions

In this paper, we introduce the concept of transversal lightlike submersions from semi-Riemannian manifolds onto semi-Riemannian manifolds. Specifically, we present the concepts of transversal r-lightlike and isotropic transversal lightlike submersions and examine the geometry of foliations formed by these submersions through various examples. In this way, we demonstrate certain points where transversal r-lightlike submersions differ from semi-Riemannian submersions. Furthermore, we investigate O’Neill’s tensors for transversal r-lightlike submersions and examine the integrability of certain distributions by employing these tensor fields. Thus, valuable information regarding such submersions’ geometric structures and properties can be provided, paving the way for new research avenues. We finally discuss the need for further research.

On the almost h-conformal semi-slant Riemannian maps

As a generalization of conformal semi-slant submersions, semi-slant Riemannian maps, almost h-semi-slant submersions and almost h-semi-slant Riemannian maps, we introduce almost h-conformal semi-slant submersions and almost h-conformal semi-slant Riemannian maps. We give some examples of such maps and also introduce some types of pluriharmonic maps, invariant maps and geodesic maps. We study the geometry of foliations, the integrability of distributions, the properties of pluriharmonic maps, invariant maps and geodesic maps. We also investigate the condition for such maps to be totally geodesic and the harmonicity of such maps.

On slant Riemannian submersions from conformal Sasakian manifolds

In this paper, we study slant Riemannian submersions from conformal Sasakian manifolds onto Riemannian manifolds. We investigate the geometry of foliations associated with the submersion and give sufficient conditions for the submersion to be harmonic.

Problems of Extrinsic Geometry of Foliations

Problems of Extrinsic Geometry of Foliations

Aspects of Submanifolds on (α, β)-Type Almost Contact Manifolds with Quasi-Hemi-Slant Factor

In this study, the authors focus on quasi-hemi-slant submanifolds (qhs-submanifolds) of (α,β)-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the concept of quasi-hemi-slant submanifolds of trans-Sasakian manifolds. We also consider the geometry of foliations dictated by the distribution and the requirements for submanifolds of trans-Sasakian manifolds with quasi-hemi-slant factors to be totally geodesic. Lastly, we give an illustration of a submanifold with a quasi-hemi-slant factor and discuss its application to number theory.

Geometry of foliations of Minkowski spaces

As is well known, foliations of constant curvature and foliations generated by the orbits of Killing vector fields are important classes of foliations from a geometric point of view. The paper studies the geometry of some foliations of the Minkowski space, which arise in a natural way. It is shown that these foliations are foliations of constant curvature. The paper also studies the geometry of some singular foliations generated by the orbits of Killing vector fields.

On h-Quasi-Hemi-Slant Riemannian Maps

In the present article, we indroduce and study h-quasi-hemi-slant (in short, h-qhs) Riemannian maps and almost h-qhs Riemannian maps from almost quaternionic Hermitian manifolds to Riemannian manifolds. We investigate some fundamental results mainly on h-qhs Riemannian maps: the integrability of distributions, geometry of foliations, the condition for such maps to be totally geodesic, etc. At the end of this article, we give two non-trivial examples of this notion.

Pointwise slant Riemannian maps from Kaehler manifolds

Pointwise slant submanifolds were introduced by Chen and Garay (2012) [16] as a natural generalization of slant submanifolds. On the other hand, pointwise slant submersions were defined by Lee and Şahin (2014) [29] as a natural generalization of slant submersions. In this paper, as a generalization of pointwise slant submanifolds and pointwise slant submersions, we introduce pointwise slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, present examples and characterizations. We investigate the geometry of foliations which are arisen from the definition of a pointwise slant Riemannian map and obtain decomposition theorems by using the existence of pointwise slant Riemannian maps. We also investigate the harmonicity of such maps and find necessary and sufficient conditions for pointwise slant Riemannian maps to be totally geodesic. Finally, we study some curvature relations in complex space forms, involving Casorati curvatures for pointwise slant Riemannian maps.

Generic $\xi^{\perp}$-Riemannian Submersions

As a generalization of semi-invariant $\xi ^{\perp }$-Riemannian submersions, we introduce the generic $\xi ^{\perp }$- Riemannian submersions. We focus on the generic $\xi ^{\perp }$-Riemannian submersions for the Sasakian manifolds with examples and investigate the geometry of foliations. Also, necessary and sufficient conditions for the base manifold to be a local product manifold are obtained and new conditions for totally geodesicity are established. Furthermore, curvature properties of distributions for a generic $\xi ^{\perp }$-Riemannian submersion from Sasakian space forms are obtained and we prove that if the distributions, which define a generic $\xi ^{\perp }$-Riemannian submersion are totally geodesic, then they are Einstein.

Bi-slant $\xi^{\perp}$-Riemannian submersions

We introduce bi-slant $\xi^{\perp}$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of slant and semi-slant $\xi^{\perp}$-Riemannian submersion and present some examples. We give the necessary and sufficient conditions for the integration of the distributions used to define the bi-slant $\xi^{\perp}$-Riemannian submersions and examine the geometry of foliations. After we obtain necessary and sufficient conditions related to totally geodesicness of such submersion. Finally we give some decomposition theorems for total manifold.

On the almost h-conformal semi-invariant Riemannian maps

In this paper, we introduce hsR maps and ahsR maps from almost quaternionic Hermitian manifolds into Riemannian manifolds as a generalization of almost h-conformal semi-invariant submersions and conformal semi-invariant Riemannian maps. We give some examples of these maps and study their properties: the geometry of foliations, the integrability of distributions, the conditions for these maps to be harmonic or totally geodesic.

Anti-invariant ξ⊥-cosymplectic-like statistical submersions

Our purpose in this article is to study anti-invariant ??-cosymplectic-like statistical submersions from cosymplectic-like statistical manifolds and an example. Also, we investigate the integrability and the totally geodesicness of the distributions and the geometry of foliations.

A note on quasi-bi-slant submanifolds of Sasakian manifolds

The object of the present paper is to study the notion of quasi-bi-slant submanifolds of almost contact metric manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant, and quasi-hemi-slant submanifolds. We study and characterize quasi-bi-slant submanifolds of Sasakian manifolds and provide non-trivial examples to signify that the structure presented in this paper is valid. Furthermore, the integrability of distributions and geometry of foliations are researched. Moreover, we characterize quasi-bi-slant submanifolds with parallel canonical structures.

Remarks on conformal anti-invariant Riemannian maps to cosymplectic manifolds

M.A. Akyol and B. Şahin [Conformal anti-invariant Riemannian maps to Kaehler manifolds, U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 4, 2018] defined and studied the notion of conformal anti-invariant Riemannian maps to Kaehler manifolds. In this paper, as a generalization of totally real submanifolds and anti-invariant Riemannian maps, we extend this notion to almost contact metric manifolds. In this manner, we introduce conformal anti-invariant Riemannian maps from Riemannian manifolds to cosymplectic manifolds. In order to guarantee the existence of this notion, we give a non-trivial example, investigate the geometry of foliations which are arisen from the definition of a conformal Riemannian map and obtain decomposition theorems by using the existence of conformal Riemannian maps. Moreover, we investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic. Finally, we study weakly umbilical conformal Riemannian maps and obtain a classification theorem for conformal anti-invariant Riemannian maps.

Semi-Invariant Riemannian Submersions with Semi-Symmetric Non-Metric Connection

In this paper, we investigate semi-invariant Riemannian submersion from a Kaehler manifold with semi-symmetric non-metric connection to a Riemannian manifold. We study the geometry of foliations with semi-symmetric non-metric connection. Later, we introduce base manifold to be a local product manifold with semi-symmetric non-metric connection.

ANTI-INVARIANT RIEMANNIAN SUBMERSIONS FROM LOCALLY CONFORMAL KAEHLER MANIFOLDS

Recently, Sahin [10] studied the anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In present work, these notions of anti-invariant and Lagrangian Riemannian submersions have been extended to locally conformal Kaehler manifolds. Certain decomposition results and the geometry of foliation have also been investigated.

On the Geometry of Submersions

Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation defined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.

Almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds

As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic. Finally, we give some examples of such maps.

Semi-slant Riemannian maps from cosymplectic manifolds into Riemannian manifolds

In this paper we study semi-slant Riemannian maps from Cosymplectic manifolds into Riemannian manifolds. Several fundamental results on integrability of distributions and geometry of foliations are proved for such maps. Also we find the conditions for Riemannian maps to be totally geodesic and investigate some decomposition theorems. Finally, we give some examples of semi-slant Riemannian maps such that the characteristic vector field ξ is either vertical or horizontal.

English

We introduce the notions of h-conformal anti-invariant submersions and h-conformal Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, anti-invariant submersions, h-anti-invariant submersions, h-Lagrangian submersion, conformal anti-invariant submersions. We investigate their properties: the integrability of distributions, the geometry of foliations, the conditions for such maps to be totally geodesic, etc. Finally, we give some examples of such maps.