The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.
problem Characterizing Einstein doubly warped product manifolds with a semi-symmetric metric connection.
method Deriving curvature formulas and proving necessary and sufficient conditions for a manifold to be a warped product.
result Obtained results for Einstein doubly warped product manifolds and Einstein-like doubly warped product manifolds.
Paper defines new submanifolds in Kaehler manifolds.
problem Understanding submanifolds in Kaehler manifolds.
method Introduced and analyzed warped product quasi bi-slant submanifolds.
result Every warped product quasi bi-slant submanifold is either a Riemannian product or a warped product quasi hemi slant submanifold.
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.
Paper defines and studies Clairaut warped product Riemannian maps.
problem Understanding the geometry of specific Riemannian maps.
method Identify geodesic conditions, derive conditions for Clairaut maps, and calculate curvature.
result Found conditions for a warped product Riemannian map to be Clairaut.
The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.
problem Analyzing the geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds.
method The study extends Chen's inequality for CR-warped product submanifolds and investigates equality cases.
result Several results extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds.
The paper explores geometric properties of Riemannian warped product maps and their curvature.
problem Investigating the geometric properties of Riemannian warped product maps.
method The approach involves establishing conditions for geodesics, deriving curvature tensors, and examining various types of maps.
result Derivation of integral formula for scalar curvature of conformal Riemannian warped product maps.
The paper proves rigidity for submanifolds in warped product manifolds.
problem Rigidity of submanifolds in warped product manifolds.
method Dihedral extremality and rigidity theorem for submanifolds with polyhedral boundary.
result Dihedral rigidity results for hyperbolic polyhedra in flat warped product spaces.
In this paper, we completely classify homogeneous production functions with an arbitrary number of inputs whose production hypersurfaces are flat. As an immediate consequence, we obtain a complete classification of homogeneous production functions with two inputs whose production surfaces are developable.
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.
Generalizes warped product submersion to conformal case.
problem Understanding angles preservation in submersions.
method Introduces conformal warped product submersion.
result Fundamental tensors derived for conformal submersion.
The study examines Einstein-Finsler spaces using Minkowskian products.
problem Characterizing Einstein-Finsler spaces through Minkowskian products.
method Proving conditions for Einstein-Finsler spaces in terms of Minkowskian products.
result If a Minkowskian product Finsler manifold is Einstein, then either the product manifold is Ricci flat or both quotient manifolds are Einstein with same scalar functions.
The paper modifies a warped product space to find conditions for constant height functions.
problem Finding sufficient conditions for the height function to be constant in a modified warped product space.
method The paper modifies the warped product space by adding a warping function and discusses the sufficient condition for the height of immersed surfaces.
result The paper establishes a sufficient condition for the height function to be constant in the modified warped product space.
Paper studies pseudo-projective tensors on warped products.
problem Characterizing pseudo-projectively flat warped products.
method Analyzes sequential warped products and pseudo-projective tensors.
result Necessary and sufficient conditions for pseudo-projectively flat sequential warped products.
In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds.
The paper studies conformal Ricci solitons in warped product spaces.
problem Characterizing conformal Ricci solitons in warped product manifolds.
method Analyzes properties of conformal Ricci solitons in warped product spaces, proving conditions for solitons and characterizing them in terms of vector fields.
result A warped product manifold admitting a conformal Ricci soliton with a concurrent potential vector field is Ricci flat.
Paper reinterprets marginal productivity theory using vectorial products, challenging traditional ethical interpretations.
problem Challenges traditional ethical interpretations of marginal productivity theory.
method Formulates marginal productivity theory using vectorial marginal products, contrasting with traditional scalar approach.
result Vectorial marginal products conflict with traditional distributive shares picture of property.
The study characterizes submanifolds in product spaces.
problem Limited studies on pseudo-umbilical submanifolds.
method Using projections from product structure, conditions for submanifolds to be invariant, anti-invariant, or semi-invariant are derived.
result Necessary and sufficient conditions for pseudo-umbilical submanifolds in locally product Riemannian manifolds.
Estimates heights of special surfaces in warped products.
problem Estimating heights of special surfaces in warped products.
method Provided a vertical height estimate.
result Vertical height estimates for special Weingarten surfaces of elliptic type in warped products.
The paper characterizes biharmonic submersions from product manifolds.
problem Characterizing biharmonic Riemannian submersions from M2imesR. method Local characterizations and by-products of biharmonic submersions.
result Local characterizations of biharmonic Riemannian submersions and uniqueness of a specific submersion.
The paper classifies isoparametric hypersurfaces in product spaces with different curvatures.
problem Characterizing isoparametric hypersurfaces in product spaces with varying curvatures.
method Analyzing hypersurfaces in product spaces with constant sectional curvatures.
result Classification of isoparametric hypersurfaces in product spaces with different curvatures.
B.-Y. Chen initiated the study of warped product submanifolds in his fundamental seminal papers \cite{C1,C2,C2.1}. In this paper, we study contact CR-warped product submanifolds of cosymplectic space forms and prove an optimal inequality by using Gauss and Codazzi equations. In addition, we obtain two geometric inequal…
In this paper, we study the Einstein warped products and multiply warped products with a quarter-symmetric connection. We also study warped products and multiply warped products with a quarter-symmetric connection with constant scalar curvature. Then apply our results to generalized Robertson-Walker spacetimes with a q…
The paper studies Blaschke products, proving uniformization and non-degeneracy of pressure metrics.
problem Analytic aspects of Blaschke products and their moduli space.
method Definition of complex structure and proof of uniformization theorem.
result Pressure semi-norms are non-degenerate outside the super-attracting locus.
The paper proves rigidity for warped product spaces with degenerate ends.
problem Proving rigidity for warped product spaces with degenerate ends.
method Analyzing scalar curvature for specific classes of warped product spaces.
result Proves scalar curvature extremality and rigidity for certain degenerate spaces.
A production function is a mathematical formalization in economics which denotes the relations between the output generated by a firm, an industry or an economy and the inputs that have been used in obtaining it. In this paper, we study the product production functions of 2 variables in terms of the geometry of their a…
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.
The paper explores warped-like product metrics with exceptional holonomy groups.
problem Exploring new metrics with exceptional holonomy groups.
method Presented a general ansatz of warped-like product metric and studied specific examples.
result Explicit examples of (3+3+2) and (3+3+1) warped-like product manifolds with Spin(7) and G2 holonomy, respectively. Paper studies Landsberg curvature of a specific Finsler metric.
problem Analyzing Landsberg curvature of a particular Finsler metric.
method Derived Landsberg curvature and mean Landsberg curvature of the twisted product Finsler metric.
result Necessary and sufficient conditions for the metric to be Landsberg or weakly Landsberg are established.
The paper studies special warped products with a specific connection on super Riemannian manifolds.
problem Investigating curvature and Ricci tensors on super warped product spaces with a semi-symmetric non-metric connection.
method Defined a semi-symmetric non-metric connection, computed curvature and Ricci tensors, and introduced and analyzed two types of super warped product spaces.
result Conditions for two super warped product spaces with a semi-symmetric non-metric connection to be Einstein spaces are provided.
The paper computes non-trivial triple Massey products on specific non-Kähler solvmanifolds.
problem Computing non-trivial triple Massey products on non-Kähler solvmanifolds.
method Explicit computations on specific non-Kähler solvmanifolds.
result Non-Kähler solvmanifolds have non-vanishing triple ABC-Massey products. In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. We obtain a uniqueness result for prescribing the Ricci curvatur…
The paper studies Einstein-type structures in warped product manifolds.
problem Characterizing Einstein-type structures in warped product manifolds.
method Analyzing conditions for minimal, totally umbilical, and geodesic immersions.
result Characterization of rotational hypersurfaces in RimesfRn. In this paper, we study the existence of proper warped product submanifolds in metallic (or Golden) Riemannian manifolds and we discuss about semi-invariant, semi-slant and, respectively, hemi-slant warped product submanifolds in metallic and Golden Riemannian manifolds. Also, we provide some examples of warped product…
The paper examines conditions for Einstein multiply warped products and estimates their parameters.
problem Existence and non-existence of non-trivial Einstein multiply warped products.
method Analyzes conditions for the existence or non-existence of Einstein multiply warped products, especially generalized Kasner type.
result Estimates the Einstein parameter that conditions the existence of such metrics.
The paper examines torsions in Minkowskian product of Finsler metrics.
problem Investigating Cartan torsion and mean Cartan torsion in Minkowskian product of Finsler metrics.
method Deriving explicit formulas for Cartan torsion and mean Cartan torsion, analyzing their geometric behavior, and providing conditions for bounded norm.
result Both Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean, and a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case is provided.
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.
Paper shows isotropic E- and S-curvatures are equivalent in warped Finsler metrics.
problem Equivalence of non-Riemannian curvatures in warped product Finsler metrics.
method Study of warped product Finsler metrics.
result Isotropic E- and S-curvatures are equivalent for these metrics. Paper proves inequalities in sub-static warped product manifolds.
problem Proving inequalities in sub-static warped product manifolds.
method Proved Heintze-Karcher type inequalities involving shifted mean curvature.
result Uniqueness results for hypersurfaces satisfying curvature equations.
Paper studies Hessian quotient equations in warped product manifolds.
problem Analyzing Hessian quotient equations in warped product manifolds.
method Using standard degree theory and a priori estimates.
result Existence of star-shaped compact hypersurface solutions.
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…
This paper compiles formulas involving differential operators and interior products.
problem Scattered identities in differential geometry involving various operators.
method Compilation and extension of formulas using the Schouten-Nijenhuis bracket and interior product.
result New formulas involving the de Rham codifferential and interior product.
In the present paper, we study warped product semi-slant submanifolds of Kenmotsu manifolds. We have obtained results on the existence of warped product semi-slant submanifolds of Kenmotsu manifolds in term of the canonical structure .
The purpose of this paper is to give an effective construction for some induced structures on spheres or product of spheres of codimension 1, 2 or 3, respectively, in Euclidean space endowed with an almost product structure.
The paper examines gradient ρ-Einstein solitons on specific manifolds and spacetimes.
problem Characterizing gradient ρ-Einstein solitons on doubly warped product manifolds.
method Analyzing necessary and sufficient conditions for doubly warped product manifolds to be gradient ρ-Einstein solitons, applying results to specific spacetime models.
result No 3-dimensional essentially conformally symmetric gradient ρ-Einstein soliton exists.
The paper proves rigidity for hypersurfaces with constant shifted curvature functions in warped product manifolds.
problem Characterizing and proving rigidity for hypersurfaces with constant shifted curvature functions.
method Using integral inequalities and Minkowski-type formulas, the paper derives rigidity theorems in sub-static warped product manifolds.
result The paper provides new characterizations and rigidity results for hypersurfaces with constant shifted curvature functions in warped product manifolds.