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.
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.
Study new warped metrics with vanishing Douglas curvature.
problem Warped product metrics and their curvature properties.
method Solve differential equation for vanishing Douglas curvature.
result Classify Douglas warped product metrics.
Einstein metrics on products are shown to be warped.
problem Characterizing Einstein metrics on conformal products.
method Proving Einstein metrics on conformal products are warped products under natural geometric conditions.
result Einstein metrics on conformal products are proven to be warped products.
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. 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.
Conditions for flat 3-manifolds with diagonal metrics are identified.
problem Characterizing flat 3-manifolds with diagonal metrics.
method Provided necessary and sufficient conditions for flatness.
result Characterized flat manifolds of warped product-type.
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 α-Sasakian structure. 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.
Study constructs non-Riemannian Finsler metrics using warped product.
problem Finding Ricci-flat Finsler metrics.
method Used warped product and PDE characterization for static spacetimes.
result Explicitly constructed two non-Riemannian examples.
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…
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.
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 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.
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. In this paper, we define a semi-symmetric metric Killing vector field, then study semi-symmetric metric Killing vector fields on warped and multiply warped products with a semi-symmetric metric connection. We also study Killing and 2-Killing vector fields on multiply warped products.
Study characterizes Einstein metrics in warped product spaces.
problem Characterizing Einstein metrics in warped product spaces.
method Local characterizations and global restatements of known results.
result Restated global characterizations of Einstein manifolds.
We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be the same up to scaling, while in the non-compact case there are simple examples …
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.
Survey on warped products and their curvature properties.
problem Understanding warped products and their curvature bounds.
method Construction and analysis of warped products between manifolds and metric spaces.
result Warp products have nice curvature properties, especially sectional and Ricci bounds.
We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and hyperbolic geometric flow (HGF) and evolution equations on warped product manifol…
Study warped product metrics on hyperbolic and complex hyperbolic manifolds.
problem Understanding curvature of warped product metrics.
method Prove curvature formulas for warped product metrics on hyperbolic and complex hyperbolic manifolds.
result Curvature formulas expressed in spherical coordinates about totally geodesic submanifolds.
In this paper, we study the Einstein multiply warped products with a semi-symmetric non-metric connection and the multiply warped products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connecti…
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…
In this article we study homogeneous warped product Einstein metrics and its connections with homogeneous Ricci solitons. We show that homogeneous (λ,n+m)-Einstein manifolds (which are the bases of homogeneous warped product Einstein metrics) are one-dimensional extensions of algebraic solitons. This answers a questi…
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.
Study classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.
Classifies smooth metric measure spaces with two weighted Einstein representatives.
problem Classifying smooth metric measure spaces with specific weighted Einstein properties.
method Local and global classification using Einstein and quasi-Einstein warped products.
result Global classification result for complete manifolds, showing specific types of manifolds.
The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.
problem Characterizing the geometry of hypersurfaces in warped product and conformal manifolds.
method Using higher fundamental forms and conformal metrics, the study characterizes the geometries of hypersurfaces in warped product and conformal manifolds.
result Higher conformal fundamental forms play a critical role in the characterization of the geometry of hypersurfaces in conformal manifolds.
Proves triviality and nonexistence of gradient Ricci solitons as warped metrics.
problem Proving triviality and nonexistence of gradient Ricci solitons as warped metrics.
method Proved through the construction of gradient Ricci solitons as warped products and studying Ricci-Hessian type manifolds.
result Gradient Ricci solitons are trivial and non-existent as warped metrics.
Study on special warped product manifolds with Einstein metrics.
problem Characterizing Einstein metrics in specific warped product manifolds.
method Derive general formulas and prove existence of solutions for invariant families.
result Existence of families of solutions invariant under transformations for positive constant Ricci curvature.
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. Warped products play very important roles in differential geometry as well as in physic…
Warped product metrics are a class of Riemannian metrics on cross products B×F which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base B …
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.
Characterizes certain Kähler manifolds with doubly-warped product structures.
problem Understanding Kähler manifolds with specific geometric properties.
method Obata-type characterization for Kähler manifolds with doubly-warped product structures.
result Complete description of Kähler doubly-warped product structures with Einstein metrics.
We investigate Kähler metrics conformal to gradient Ricci solitons, and base metrics of warped product gradient Ricci solitons. The latter we name quasi-solitons. A main assumption that is employed is functional dependence of the soliton potential, with the conformal factor in the first case, and with the warping funct…
Study shows curvature rigidity of specific metric types.
problem Curvature rigidity of specific metric types.
method Spin geometry based arguments.
result Scalar curvature rigidity of specific metric types.
Sharp upper bounds found for Steklov eigenvalues of warped products.
problem Finding bounds for Steklov eigenvalues of specific metric configurations.
method Investigation of Steklov spectrum for warped products with a fiber of dimension 2.
result Sharp upper bounds for Steklov eigenvalues in terms of the eigenvalues of the Laplacian on the fiber.
New findings extend rigidity results to broader classes of manifolds.
problem Extending rigidity results to non-warped product spaces.
method Establishing rigidity theorems for manifolds conformal to those with nonnegative curvature.
result New families of manifolds exhibit scalar-mean rigidity.
Study Finsler metrics with vanishing Landsberg curvature.
problem Characterize Finsler metrics with specific curvature properties.
method Derive expressions for curvatures, solve PDEs, construct examples.
result Construct non-regular Landsberg metrics not of Berwald type.
In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consi…
In this short note, we show that the negative curvature is preserved in the deformation of hyperbolic warped product metrics under Ricci flow. It is also showed that the flow converges to a flat metric as time going to infinity.
Study on four-dimensional Ricci solitons and multiply warped Ricci flow solutions.
problem Characterizing singularities in Ricci flow solutions.
method Classification and rigorous examples of multiply warped Ricci flow solutions.
result Formation of generalized cylinder singularity models.
Study on special symmetries in biwarped product 3-manifolds.
problem Characterizing Killing vector fields on biwarped product-type 3-manifolds.
method Derived system of equations for Killing fields and described their structure.
result Families of solutions found, including explicit examples.
In this paper, we study the quasi-Einstein and generalized quasi-Einstein warped products with a semi-symmetric non-metric connection. We give the expressions of the Ricci tensors and scalar curvatures for the bases and fibres. In some cases we give some obstructions to the existence of the quasi-Einstein and generaliz…
The study constructs gradient Einstein-type warped metrics and proves nonexistence and rigidity results.
problem Proving nonexistence and rigidity for gradient Einstein-type warped metrics.
method Establishing necessary and sufficient conditions for constructing these metrics, leading to a Lichnerowicz equation.
result Nonexistence and rigidity results for a class of gradient Einstein-type warped metrics.
Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…
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…