The paper examines conditions for Einstein-like classes in doubly warped product manifolds.
problem Conditions for a factor manifold to be in the same Einstein-like class as the entire doubly warped product manifold.
method Analysis of the Ricci tensor and warping functions.
result Sufficient and necessary conditions for a factor manifold to inherit the Einstein-like class of the entire manifold.
Study on metrics with harmonic curvature and maximum warped-product decompositions.
problem Understanding metrics with harmonic curvature and their decompositions.
method Analyzing Riemannian manifolds with specific curvature properties and decompositions.
result Local-isometry types form a finite-dimensional moduli space, and specific metrics are realized.
We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We…
We investigate manifolds obtained as a quotient of a doubly warped product. We show that they are always covered by the product of two suitable leaves. This allows us to prove, under regularity hypothesis, that these manifolds are a doubly warped product up to a zero measure subset formed by an union of leaves. We also…
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our …
We will obtain the warped product decompositions of spaces of constant curvature (with arbitrary signature) in their natural models as subsets of pseudo-Euclidean space. This generalizes the corresponding result by S. Nolker to arbitrary signatures, and has a similar level of detail. Although our derivation is complete…
In the present paper, we give a necessary and sufficient condition for a Riemannian manifold (M,g) to have a reducible action of a hyperbolic analogue of the holonomy group. This condition amounts to a decomposition of (M,g) as a warped product of a special form, in analogy to the classical de Rham decomposition th…
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.
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 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 article surveys recent results on warped and CR-warped product submanifolds in Kaehler manifolds.
problem Characterizing submanifolds in Kaehler manifolds using warped and CR-warped products.
method Analyzing properties of warped and CR-warped products in the context of Kaehler manifolds.
result Presented several results on the geometry of warped and CR-warped product submanifolds.
Study on warped product Yamabe solitons with constant fiber curvature.
problem Characterizing nontrivial warped product Yamabe gradient solitons.
method Investigation of warped product manifolds, derivation of scalar curvature estimates.
result Nontrivial warped product Yamabe gradient solitons have constant scalar curvature in the fiber.
Warped product affects divergences in information geometry.
problem Warped product's impact on divergences in information geometry.
method Study of warped product on information geometry.
result Warped product does not preserve canonical divergences.
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 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.
Characterizes and examines gradient solitons on doubly warped product manifolds.
problem Understanding gradient solitons on specific manifold structures.
method Characterizations and examinations of various types of gradient solitons on doubly warped product manifolds.
result Effects of gradient solitons on factor manifolds and specific curvature properties of doubly 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.
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…
Study investigates concircular curvature on warped products.
problem Effects of concircular flatness and symmetry on fibre and base manifolds.
method Investigated concircular flat and symmetric warped product manifolds, considered divergence free curvature tensor.
result Results applied to generalized Robertson-Walker and static space-times.
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…
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.
In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {M=MT×fM⊥} of Lorentzian paracosymplectic manifold such that the characteristic vector field is n…
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.
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.
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 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.
Study on submanifolds in metallic Riemannian manifolds.
problem Existence of specific submanifolds in metallic Riemannian manifolds.
method Analysis of warped product submanifolds in metallic and Golden Riemannian manifolds.
result Existence and properties of various types of warped product submanifolds.
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.
The purpose of this article is to study implications of a Ricci soliton warped product manifold to its base and fiber manifolds. First, it is proved that if a warped product manifold is Ricci soliton then its factors are Ricci soliton. Then we study Ricci soliton on warped product manifolds admitting either a conformal…
Study Einstein warped products with Einstein base and fiber.
problem Characterize Einstein warped products with Einstein base and fiber.
method Investigate necessary and sufficient conditions for a warped product to be Einstein.
result Explicitly determine the warping function when the base is hyperbolic space.
Study information geometry of warped product spaces, finding special connections.
problem Understanding information geometry in warped product spaces.
method Examined warped products with dually flat connections, characterized connections on base space.
result Characterized connections on base space R>0 as α-connections with α=±1. 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.
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.
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.
Defines semi-symmetric metric connection on super warped products.
problem Computing curvature and Ricci tensors on super warped products.
method Introduced semi-symmetric metric connection and conditions for Einstein spaces.
result Conditions for super warped product spaces to be Einstein with semi-symmetric metric connection.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
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.
No proper biharmonic CMC compact hypersurface in a specific warped product space.
problem Existence of proper biharmonic CMC hypersurfaces in a warped product space.
method Finding necessary and sufficient conditions for proper biharmonic CMC hypersurfaces in a special warped product space.
result No proper biharmonic CMC compact hypersurface exists in the specified space.
Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds from extrinsic point of view was initiated by the first author around the beginning of this century in [7, 8]. Since then the study of warped product submanifolds has been investigated by many geo…
The study introduces a new curvature concept for weighted graphs and applies it to warped products.
problem Establishing curvature bounds for doubly warped product graphs.
method Developed a new notion of curvature for weighted graphs and applied it to warped products, establishing bounds in terms of constituent graph curvatures.
result Established curvature bounds for $\left(R_1,R_2
ight)$-doubly warped products of smooth measure spaces.
The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein metrics are unstable. By exploiting the relationship between warped product Einstei…
We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
New rigidity results for warped product domains.
problem Scalar curvature rigidity of domains in warped products.
method Developed a new connection on a twisted spinor bundle and associated Dirac operator.
result Obtained Llarull type scalar curvature rigidity for a general class of domains in a warped product.
Warped product construction for Finsler manifolds with curvature conditions.
problem Constructing new Finslerian manifolds with specific curvature properties.
method Extending the warped product concept to Finsler metrics, particularly (α,β)-metrics. result Demonstrated the feasibility of constructing new Finslerian manifolds with prescribed curvature conditions.
Backward propagation rules for warped products under Ricci flow.
problem Understanding how warped product structures behave under Ricci flow.
method Establishing sufficient conditions for backward propagation of warped product structures.
result Asymptotically conical shrinkers are multiply-warped products over Einstein manifolds.
Recently Hui et al. (\cite{HAP}, \cite{HAN}) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a (LCS)n-manifold Mˉ. In this paper we have studied the characterization for both these classes of warped product submanifolds. It is also shown that there do not ex…
The warped product N1×fN2 of two Riemannian manifolds (N1,g1) and (N2,g2) is the product manifold N1×N2 equipped with the warped product metric g=g1+f2g2, where f is a positive function on N1. The notion of warped product manifolds is one of the most fruitful generalizations of R…
Study shows how certain metrics can be split into warped products.
problem Understanding conditions under which metrics can be split as warped products.
method Investigating warped area-minimizing hypersurfaces and spectral Ricci/scalar curvature bounds.
result Metrics can be locally split as warped products under specific curvature conditions.