Geometric cohomology model uses co-oriented maps to define a product structure.
problem Constructing a geometric model for cohomology of smooth manifolds.
method Develops a cochain complex model based on co-oriented smooth maps, focusing on their pull-back product structure.
result Geometric cochains with a partially defined product structure induce the cup product in cohomology.
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.
Geometric structure reveals optimal investment and hedging products.
problem Optimal design of investment and hedging products.
method Investigation of geometric structure in risks and returns using a simple formula.
result Duality between hedging and investment with geometric interpretation of rationality.
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 paper examines metrics on foliated manifolds that have special geometric properties.
problem Characterizing metrics on foliated manifolds with specific harmonic properties.
method Examining the properties of bundle-like metrics on foliated manifolds.
result The interior product of basic harmonic forms is basic harmonic under certain conditions.
Geometrically describes surfaces with parallel mean curvature in warped product spaces.
problem Understanding surfaces with parallel mean curvature in warped product spaces.
method Obtained a geometric description using the normal connection.
result Extended a result by Alencar-do Carmo-Tribuzy on surfaces with parallel mean curvature.
The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpre…
Study geometric properties and spectral estimates on warped products.
problem Investigate Ricci curvature and spectral estimates in warped products.
method Establish integral inequalities and sufficient conditions for geometric properties.
result Sufficient conditions for intersection of warped products with totally geodesic hypersurfaces.
In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.
New method compares geometric and standard cup products.
problem Reconciling partially defined and fully defined cochain products.
method Vector field flow through cubulation to compare products.
result Explicit cochain level comparison between intersection and cup products.
The paper establishes sharp geometric inequalities for hypersurfaces in warped product manifolds.
problem Geometric inequalities involving three distinct quantities in warped product manifolds.
method Two families of inequalities comparing three geometric quantities in space forms or warped product manifolds.
result Generalizes and extends previous results on Weinstock-type inequalities and Steklov/Wentzell eigenvalues.
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.
Proves small cancellation free products have geometric actions on CAT(0) cube complexes.
problem Proving small cancellation free products have geometric actions on CAT(0) cube complexes.
method Using a blown-up complex of groups and a boundary separation criterion, proving wall stabilizers form a rich family of subgroups.
result Proves $C'( rac16)$--small cancellation free products of residually finite groups are residually finite.
The study classifies geometrically finite polynomials on the boundary of Blaschke products.
problem Understanding the boundaries of hyperbolic components of Blaschke products.
method Combinatorial classification and construction of self-bumps.
result The closure of the main hyperbolic component is not a topological manifold with boundary for d≥4. Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
Study geometric formal metrics and Massey products on Kähler manifolds with torsion.
problem Interplay between geometrically-Bott-Chern-formal metrics and SKT metrics on Kähler manifolds.
method Analyzing nilmanifolds and Kähler solvmanifolds, proving conditions for existence of SKT metrics and Massey products.
result Any Kähler solvmanifold is geometrically formal, and explicit constructions of lattices with non-vanishing Massey products.
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.
Some basic geometric properties of doubly twisted product immersions are established.
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 RimesfRn. Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
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.
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.
Researchers find a way to bound the complexity of certain subgroup geometric invariants.
problem Understanding the geometric invariants of subgroups of direct products of free groups.
method Generalizing techniques for 'pushing fillings' into normal subgroups.
result Finitely presented subgroups of direct products of three free groups and subgroups of finiteness type Fn−1 in a direct product of n free groups have Dehn functions bounded above by N9. Quasi-isometries in horospherical products are close to product maps.
problem Understanding the rigidity of quasi-isometries in specific geometric spaces.
method Proving quasi-isometries are uniformly close to product maps in horospherical products of hyperbolic spaces.
result Quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid.
Study geometric actions of groups on horocyclic products.
problem Understanding geometric actions of groups on horocyclic products.
method Analyzing geometric actions of groups on horocyclic products of CAT(-κ) spaces.
result Groups acting on horocyclic products are either ascending HNN extensions of finitely-generated virtually nilpotent groups or not finitely presented.
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.
Paper unifies and simplifies proof of free product conditions.
problem Conditions for normal closure in free product.
method Geometric approaches to normal closure in ambient groups.
result Unified and simplified proof of Dahmani-Guirardel-Osin theorem.
In this article we obtain classification results on the quasi-product production functions in terms of the geometry of their associated graph hypersurfaces, generalizing in a new setting some recent results concerning basic production models. In particular, we obtain several results on the geometry of Spillman-Mitscher…
Study describes conformal product structures on compact Kähler manifolds.
problem Characterizing compact Kähler manifolds with conformal product structures.
method Geometric description of conformal product structures on compact Kähler manifolds.
result Geometric characterization of compact Kähler manifolds with conformal product structures.
Geometric theory explains substitutability in market outcomes based on production constraints.
problem Understanding substitutability in markets with structured feasible products.
method Modeling the set of feasible products as a compact Riemannian manifold to study intrinsic geometry and its effects on substitutability.
result Intrinsic geometry of the feasible set governs substitutability and market outcomes, with curvature controlling technological substitution elasticity.
The study classifies cellular pseudomanifolds and their properties.
problem Understanding the structure of cellular pseudomanifolds.
method Analyzing the combinatorial and geometric properties of cellular pseudomanifolds.
result Complete classification of cellular pseudomanifolds with excess < 2, and progress towards excess 2.
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 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.
In this note I use cup-products and higher Massey products to find topological lower bounds on the number of geometrically distinct critical points of any closed 1-form in a given cohomology class.
Study quantifies geometric complexity of connections on product surfaces.
problem Understanding geometric complexity of connections on product manifolds.
method Establishes a topological lower bound on the holonomy of cohomologically calibrated connections.
result Proves a bound on the dimension of the holonomy that is a topological invariant.
The paper introduces a statistical version of contact CR-product for Sasakian statistical manifolds.
problem Characterizing geometric properties of contact CR-submanifolds in Sasakian statistical manifolds.
method Characterization of integrability of invariant and anti-invariant distributions, development of results on specific types of contact CR submanifolds, introduction of statistical contact CR-product.
result Introduction of a statistical version of contact CR-product for Sasakian statistical manifolds.
The study examines minimal surfaces in Riemannian products of surfaces.
problem Exploring geometric and topological restrictions on minimal surfaces in Riemannian products of surfaces.
method Analyzes totally geodesic surfaces and minimal 2-spheres, 2-tori, and 2-spheres in Riemannian products of surfaces with constant curvature.
result Generically, a totally geodesic surface in a Riemannian product is either a slice or a product of geodesics. Minimal 2-spheres and 2-tori have specific properties under certain curvature conditions.
Study shows observability for Schrödinger equations on product manifolds with specific conditions.
problem Observability of Schrödinger equations on product manifolds with product metrics.
method Proof of observability in finite time on open subsets satisfying Vertical Geometric Control Condition, under gap condition on spectrum of F(g).
result Observability on ω for the Schrödinger equation is strictly weaker than Geometric Control Condition on product of spheres.
Abstract: Study gyrations of sphere products and connected sums, generalizing Fico's Lemmata.
problem Understanding the homotopy type of gyrations of sphere products and connected sums.
method Recasting Fico's Lemmata into modern homotopy theoretic setting.
result Generalization of Fico's Lemmata to sphere products and connected sums.
Introduces halo products and studies their geometric properties.
problem Understanding the large-scale geometry of halo groups.
method Introduces halo products and builds a geometric framework.
result Provides refined invariants distinguishing halo groups up to quasi-isometry.
Geometrically interprets cup products and defines combinatorial Pin structures.
problem Understanding Steenrod's cup products and their geometric interpretation.
method Constructs vector fields and combinatorial frames to interpret cochain-level formulas.
result Geometrically interprets cup products and defines Pin structures combinatorially.
We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstru…
We prove that for geometrically finite groups cohomological dimension of the direct product of a group with itself equals 2 times the cohomological dimension dimension of the group.
Study on geometric properties of second Ricci solitons.
problem Understanding the geometry of second Ricci solitons.
method Investigation of closed and compact second Ricci soliton manifolds, and immersed submanifolds as well as warped product manifolds.
result Investigation of geometric properties of second Ricci solitons.
Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riem…
We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities i…
For product manifolds, cohomologically calibrated affine connections are geometrically irreducible.
problem Establishing geometric irreducibility of cohomologically calibrated affine connections on product manifolds.
method Proof relies on Hodge theory and integral arguments showing non-cancellation of off-diagonal components in the Riemann curvature tensor.
result Cohomologically calibrated affine connections on product manifolds are holonomically irreducible.
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.