Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Groups with Property (T) have fiber products with Property (T).
problem When does the fiber product of groups with Property (T) have Property (T)?
method Analyzing fiber products of groups with Property (T).
result Fiber products of groups with Property (T) also have Property (T).
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.
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.
Paper defines conditions for warped product gradient Yamabe solitons.
problem Conditions for warped product gradient Yamabe solitons.
method Investigated necessary and sufficient conditions for warped product M=BimesfF to be a gradient Yamabe soliton. result Obtained solutions in steady case with fiber scalar-flat.
Local Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds.
problem Characterizing local immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds.
method Analyzing local Sasakian immersions of Sasaki-Ricci solitons into fiber products of homogeneous Sasakian manifolds.
result Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds. We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.
This paper computes the number of fiberings for non-product manifolds.
problem Computing the number of fiberings for non-product manifolds.
method Analyzes specific manifolds and uses bundle properties.
result Fib(M)=2 for the Atiyah-Kodaira manifold and its finite covers, and Fib(M)=4 for an example. In this paper we show that a compact warped product Einstein manifold with vanishing Bach tensor of dimension n≥4 is a finite quotient of a warped product with (n−1)-dimensional Einstein fiber. The fiber has constant curvature if n=4.
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.
The study proves conditions for quasi-Einstein manifolds with specific structures to be Einstein.
problem Conditions for quasi-Einstein manifolds to have Einstein structures.
method Proved conditions for quasi-Einstein semi-Riemannian warped products to have Einstein fibers.
result Found conditions for quasi-Einstein manifolds with specific structures to be Einstein.
In this paper, we give some necessary and sufficient conditions for a normal subgroup of an amalgamated product of groups to be finitely generated. We apply these conditions together with Stallings' fibering theorem to prove that an irreducible multilink in a homology 3-sphere fibers if and only if each of its multilin…
The volume spectrum of fiber bundles is bounded by the product of the base's volume spectrum and the fiber's volume.
problem Bounding the volume spectrum of fiber bundles and understanding its relationship with the base and fibers.
method Established an inequality relating the volume spectrum of a fiber bundle to the volume spectrum of its base and the volume of the largest fiber.
result The volume spectrum of a fiber bundle is bounded by the product of the volume spectrum of the base and the volume of the largest fiber.
I construct an algebraic model for a typical fiber on a 1+1 dimensional spacetime. The vector space comprising the fiber is composed of elements formed from the direct product of two copies of an element x in the D2=C2xC2 finite group algebra over the real numbers. The fiber contains subspaces whose elements are associ…
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
Maps are shown to be Riemannian products with Ricci-flat fibers.
problem Understanding maps between manifolds and their geometric properties.
method Spin geometry and representation theory of curvature operators.
result Scalar-rigid maps are essentially Riemannian products of base and Ricci-flat fibers.
Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.
problem Characterizing and understanding spacelike hypersurfaces in twisted product spacetimes.
method Analysis of conditions ensuring global hyperbolicity, use of conformal geometry results.
result Non-existence results for constant mean curvature hypersurfaces under certain conditions.
Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypothes…
New singularity theorems for warped products help analyze extra dimensions stability.
problem Stability of extra dimensions in warped product spacetimes.
method Derived new singularity theorems for warped-product spacetimes, analyzed conditions for geodesic incompleteness, and solved conditions for parallel transportation.
result Explicit conditions on the warping function that lead to geodesic incompleteness, providing insights into the intrinsic geometry of extra dimensions.
Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of `generalized boundary blow-up' in which a new manifold and blow-down…
Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.
problem Maintain geodesic fibers and positive sectional curvatures in Riemannian submersions.
method Vary Riemannian metrics while keeping fibers totally geodesic and horizontal distribution fixed.
result Conditions for making sectional curvatures positive and existence of fat submersions.
Study shows hyperbolic subgroups can be free products of surface and free groups.
problem Characterizing hyperbolic subgroups within larger groups.
method Analyzing fiber bundles and using properties of hyperbolic groups.
result Non-elementary hyperbolic commensurated subgroups are virtually free products of surface and free groups.
This paper extends curvature results to higher fiber dimensions.
problem Investigating curvature constraints on base for (2+m)-Einstein warped product manifolds. method Extending previous work to higher fiber dimensions, considering (2,m)-PNDP manifolds. result The dimension of the fiber does not affect curvature results.
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.
In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spa…
This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Krö16]. Up to a small gap in the case of sin-c…
New bases found for Kauffman bracket skein module of fibered torus.
problem Computing Kauffman bracket skein module of fibered 3-manifolds.
method Constructed bases for Kauffman bracket skein module of product annulus and circle, then applied to (β,2)-fibered torus. result Found a new basis for KBSM of (β,2)-fibered torus. 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 examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such…
Paper solves the Hurwitz existence problem using fiber products.
problem Determining when a combinatorial map datum corresponds to a holomorphic map.
method Using fiber products of holomorphic maps between Riemann surfaces.
result Proves non-realizability of many branch data and constructs new data.
Generalized complex structures on certain torus bundles are explored.
problem Exploring generalized complex structures on specific torus bundles.
method Analyzing principal torus bundles over complex manifolds with even dimensional fibers and characteristic class of type (1,1).
result Generalized complex structures on these bundles are equivalent to products of complex and symplectic structures in tubular neighborhoods of fibers.
Given a closed connected Riemannian manifold M and a connected Riemannian manifold N, we study fiberwise volume decreasing diffeomorphisms on the product M x N. Our main theorem shows that in the presence of certain cohomological condition on M and N such diffeomorphisms must map a fiber diffeomorphically onto another …
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free G-spaces. method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
Upper bounds for Steklov eigenvalues of warped products are derived.
problem Finding upper limits for Steklov eigenvalues of warped product manifolds.
method Using volume, boundary volume, fiber Laplace eigenvalues, and warping function norms.
result Optimal upper bounds and stability estimates for eigenvalues are obtained.
Bi-orderable groups from left-orderable ones, showing non-profinite properties.
problem Non-profinite properties in bi-orderability and generalized torsion elements.
method Using fiber products to construct bi-orderable groups, showing non-profinite properties.
result Bi-orderability and generalized torsion elements are not profinite properties.
In this paper we consider M=B×fF warped product gradient Ricci solitons. We proved that the potential function depends only on the base and the fiber F is necessarily Einstein manifold. We provide all such solutions in the case of steady gradient Ricci solitons when the base is conformal to an n-dimensi…
In this paper, we generalize the dualistic structures on warped product manifolds to the dualistic structures on generalized warped product manifolds. we develop an expression of curvature for the connection of the generalized warped product in relation to those corresponding analogues of its base and fiber and warping…
We show that on a closed smooth manifold M equipped with k fiber bundle structures whose vertical distributions span the tangent bundle, every smooth diffeomorphism f of M sufficiently close to the identity can be written as a product f=f1...fk, where fi preserves the ith-fiber. The factors …
The paper studies biharmonic submersions in Riemannian geometry.
problem Characterizing biharmonic Riemannian submersions.
method Deriving bitension field equations and using them to construct biharmonic examples and characterize submersions.
result Examples of proper biharmonic Riemannian submersions with 1-dimensional fibers and characterizations of warped products.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
New method constructs degenerate Sasakian manifolds from hyperkähler bundles.
problem Constructing degenerate 3-(α,δ)-Sasakian manifolds. method Using fiber products of Boothby-Wang bundles over hyperkähler manifolds.
result No non-trivial compact examples exist, and one family of nilpotent Lie groups with this geometry is identified.
Constructs moduli space for discs mapping to homogeneous spaces.
problem Stable holomorphic discs mapping to homogeneous spaces.
method Orbifold with corners, fibered products, pushforward, pullback of differential forms.
result Moduli space of stable holomorphic discs as an orbifold with corners.
For a Poincare duality space X and a map X -> B, consider the homotopy fiber product X x^B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X x^B X has the structure of a graded associative algebra. When X -> B is the diagonal…
Extends width estimates to family case using index theory.
problem Sharp width estimates for Riemannian bands with positive scalar curvature.
method Dirac operators and family index theory.
result Proves width estimate for fiber bundles with infinite A-hat area.
Study monodromy relations in Seifert fibered spaces for 3-manifold fillings.
problem Understanding symplectic fillings of Seifert fibered 3-manifolds.
method Analyzes monodromy factorizations and their impact on fillings.
result Computes and bounds Euler characteristics of different fillings.
In this paper, we show that the projection of a dualistic structure defined on a twisted product manifold induces dualistic structures on the base and the fiber manifolds, and conversely. Then under some conditions on the Ricci curvature and the Weyl conformal tensor we characterize dually flat structures on twisted pr…
There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of tw…
We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension …