The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an…
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A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Study on gradient solitons in warped product manifolds.
problem Characterizing gradient solitons in warped product manifolds.
method Derive a Bochner-type formula and prove conditions for solitons.
result Warped product manifolds with gradient solitons have constant scalar curvature.
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…
The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.
problem Integral and variation formulas for mixed scalar curvature in multi-product manifolds.
method Generalizes results from pseudo-Riemannian almost product manifolds to multi-product structures.
result Generalizes formulas for mixed scalar curvature in multi-product manifolds.
Study new warped product metric in contact geometry.
problem Define a new type warped product metric.
method Introduced a new warped product metric on product manifolds.
result Showed existence of -Sasakian structure.
Study on Riemannian manifolds with circulant structures and their properties.
problem Characterizing Riemannian manifolds with specific symmetries.
method Analysis of the covariant derivatives and properties of the circulant structure.
result Conditions for an almost product manifold to belong to specific classes.
Almost product structures on statistical manifolds and para-Kähler-like statistical submersionsmath.DG
Study on statistical manifolds with product structures and their properties.
problem Investigating statistical manifolds with almost product structures.
method Proving properties of para-Kähler-like statistical manifolds and deriving properties of statistical submersions compatible with almost product structures.
result The statistical structure of a para-Kähler-like statistical manifold of constant curvature is a Hessian structure.
The paper studies affine connections on singular warped products and their curvature.
problem Analyzing affine connections on singular warped products.
method Introducing semi-symmetric metric and non-metric Koszul forms, and expressing their curvature in terms of factor manifolds.
result Generalized results for singular multiply warped products.
We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for …
Twistorial examples of Riemannian almost product manifolds and their Gil-Medrano and Naveira typesmath.DG
Constructs non-trivial examples of Riemannian almost product structures.
problem Identifying Riemannian almost product structures on specific bundles.
method Uses twistor spaces of oriented Riemannian four-manifolds to construct examples.
result Determines Gil-Medrano and Naveira types of constructed structures.
The paper proves non-existence theorems for certain Riemannian manifolds.
problem Non-existence theorems for specific Riemannian manifolds.
method Proves Liouville-type theorems for complete Riemannian almost product manifolds and special mappings.
result Generalizes similar results for compact manifolds to complete manifolds.
Investigates conditions for product and golden structures to be integrable and parallel, and discusses harmonicity of maps.
problem Conditions for product and golden structures to be integrable and parallel.
method Investigates conditions for integrability and parallelism of product and golden structures.
result Conditions for product and golden structures to be integrable and parallel are established.
In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-He…
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…
Study on properties of tangential hypersurfaces in product-like manifolds.
problem Investigating properties of tangential hypersurfaces in product-like manifolds.
method Analyzing basic properties and computing curvature tensor relations.
result Computed relations involving the Riemannian curvature tensor of tangential hypersurfaces.
Study structures on manifolds to find specific almost contact metric structures.
problem Understanding structures and their induced almost contact metric structures.
method Analyzing 2-fold vector cross products and their effects on manifolds.
result Identified possible classes of induced almost contact metric structures.
A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case - a pseudo-Riemannian one. Each belo…
The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.
problem Investigating the geometry of gradient almost Yamabe solitons in warped product manifolds.
method Presenting geometric rigidity results, investigating existence conditions, and classifying specific solitons.
result Classification of rotational gradient almost Yamabe solitons in .
In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold , where is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.
Study of complex structures on product twistor spaces for 4D manifolds.
problem Understanding complex structures on product twistor spaces.
method Analyzing the product bundle of twistor spaces with Riemannian metrics and almost complex structures.
result Determined Gray-Hervella classes for 4D manifolds.
Study properties of bi-warped product submanifolds in specific geometric spaces.
problem Characterize geometric properties of bi-warped product submanifolds.
method Analyze squared norm of second fundamental form and warping functions.
result Relationship between squared norm and warping functions for proper slant submanifolds.
The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.
The paper classifies warped product almost Ricci solitons.
problem Understanding warped product almost Ricci solitons.
method Analyzing Ricci-Hessian type manifolds and considering two complementary cases.
result The vector field \(
abla\lambda\) belongs to the \(C^\infty(\Bbb{M})\)-module generated by \(
abla f\) and \(
abla\varphi\) in the first case.
Paper finds gradient estimates for warped product gradient almost Ricci solitons.
problem Nonexistence of certain gradient almost Ricci solitons.
method Modified Li-Yau technique to estimate warping function gradients.
result Nonexistence theorem for specific gradient almost Ricci solitons.
Riemannian foliations on quaternion CR-submanifolds of an almost quaternion Kaehler product manifoldmath.DG
The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of an almost quaternion Kaehler product manifold.
The study characterizes almost Kenmotsu manifolds with specific vector fields.
problem Characterizing almost Kenmotsu manifolds with holomorphically planar conformal vector fields.
method Analyzing properties of vector fields and curvature conditions.
result Classification of almost Kenmotsu manifolds as Kenmotsu or having specific geometric properties.
Study gradient Ricci almost solitons as warped products.
problem Characterize gradient Ricci almost solitons as warped products.
method Use Bishop-O'Neill's formulas and Ricci-Hessian equations.
result Prove existence and rigidity results for gradient Ricci almost solitons.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
The notion of relativistic observer is confronted with Naveira's classification of (pseudo-)Riemannian almost-product structures on space-time manifolds. Some physical properties and their geometrical counterparts are shortly discussed.
Study on warped product pointwise semi-slant submanifolds in Sasakian and cosymplectic manifolds.
problem Investigate the geometry of pointwise semi-slant submanifolds and their warped products.
method Using the notion of pointwise slant submanifolds, investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds.
result Prove the existence of no proper pointwise semi-slant warped product submanifolds other than contact CR-warped products in Sasakian manifolds.
We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for t…
The paper studies Einstein-Hilbert action on complex manifolds.
problem Deriving equations for the Einstein-Hilbert action on almost -product manifolds.
method Adapted variations of metric, deriving Euler-Lagrange equations.
result Presented a nice form of Einstein equation.
Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifoldsmath.DG
On a Riemannian almost product manifold we consider a linear connection preserving the almost product structure and the Riemannian metric and having a totally skew-symmetric torsion. We determine the class of the manifolds admitting such a connection and prove that this connection is unique …
Develops a theorem for a 6D manifold with boundary.
problem No specific problem stated; focuses on extending a theorem.
method Extends a Kastler-Kalau-Walze type theorem to 6D almost product Riemannian spin manifolds with boundary.
result Proves a Kastler-Kalau-Walze type theorem for 6D manifolds.
Characterizes Ricci almost solitons on semi-Riemannian warped products.
problem Characterizing Ricci almost solitons on semi-Riemannian warped products.
method Characterization using potential function dependence and fiber properties.
result Fiber is necessarily an Einstein manifold; base and warped product are also Einstein manifolds under certain conditions.
The paper explores properties of conformal vector fields on almost Kenmotsu manifolds.
problem Characterizing properties of conformal vector fields on almost Kenmotsu manifolds.
method Analyzing conformal vector fields as Reeb vector fields and pointwise collinear, proving manifold properties and existence of warped products.
result Conformal vector fields on almost Kenmotsu manifolds lead to specific manifold structures and properties.
The paper studies critical metrics on a specific type of manifold.
problem Investigating critical metrics on almost Kenmotsu manifolds.
method Introducing and studying the -Miao-Tam critical equation on -dimensional -almost Kenmotsu manifolds.
result If a -dimensional -almost Kenmotsu manifold satisfies the -Miao-Tam critical equation, it is -Ricci flat and locally isometric to a specific product of manifolds.
Paper explores existence and nonexistence of submanifolds in contact geometry.
problem Existence and nonexistence of warped product submanifolds in almost contact manifolds.
method Generalizes results for existence and nonexistence, constructs examples, addresses open problems.
result Offers decisive results for existence and nonexistence of submanifolds.
Optimization method yields geometric inequalities for submanifolds in statistical warped product manifolds.
problem Optimizing geometric inequalities for submanifolds in statistical warped product manifolds.
method Optimization techniques applied to statistical submanifolds in statistical warped product manifolds.
result Optimal Casorati inequalities and Chen-Ricci inequality derived for statistical submanifolds.
The study explores dimensions for connected sums of almost complex manifolds and extends results to rational homology spheres.
problem Understanding dimensions for connected sums of almost complex manifolds and extending results to rational homology spheres.
method Obstruction theory and Yang's results on almost complex structures were used to answer questions about dimensions. The index of the twisted spin^c Dirac operator was applied to extend results to rational homology spheres.
result The study partially extends Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres.
Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.
problem Proving a theorem for a specific type of Dirac operator on various manifolds.
method Extending previous results to even-dimensional almost product Riemannian spin manifolds.
result Established the general Kastler-Kalau-Walze type theorem for even-dimensional manifolds.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
Study of warped product pointwise bi-slant submanifolds in Kaehler manifolds.
problem Characterize warped product pointwise bi-slant submanifolds in Kaehler manifolds.
method Extends previous studies on warped product slant submanifolds.
result Extends several important results on warped product slant submanifolds.
Study on new types of manifolds derived from 2D space-forms.
problem Characterizing new types of almost contact B-metric manifolds.
method Used two methods for generating B-metric on product manifolds, characterized with curvature properties.
result New manifolds are classified and their curvature properties are studied.
The paper characterizes specific almost Kenmotsu manifolds with Ricci-Yamabe solitons.
problem Characterizing almost Kenmotsu manifolds with Ricci-Yamabe solitons.
method Analyzing curvature properties and potential vector fields.
result Locally isometric structures of specific almost Kenmotsu manifolds.