Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.
problem Identifying Clairaut constants from Fermat constants for specific geodesics.
method Analytical proof for a specific class of geodesics on a surface of revolution.
result Fermat constants do not fully determine Clairaut constants for some geodesics, except for a standard sphere.
The paper defines and explores Clairaut conformal submersions in Riemannian geometry.
problem Defining and characterizing Clairaut conformal submersions.
method Analyzing necessary and sufficient conditions, deriving geometric properties, and providing examples.
result Clairaut conformal submersions have constant dilation along fibers and are harmonic.
Defines and studies Clairaut Riemannian maps between manifolds and Ricci solitons.
problem Characterizing and analyzing Clairaut Riemannian maps.
method Using geodesic curves, necessary and sufficient conditions for harmonicity and Clairaut maps are derived.
result Necessary conditions for various properties of Clairaut Riemannian maps are established.
Study Clairaut anti-invariant submersions on nearly Kaehler manifolds.
problem Characterize geometric properties of Clairaut anti-invariant submersions.
method Investigate conditions for total geodesic maps and totally umbilical fibers.
result Conditions for Clairaut anti-invariant submersions to be totally geodesic.
The paper characterizes Clairaut conformal submersions on Ricci solitons.
problem Characterizing Clairaut conformal submersions on Ricci solitons.
method Calculating scalar and Ricci tensors, providing necessary conditions for fibres and base manifolds to be Ricci solitons and Einstein, and solving Poisson equations.
result Necessary and sufficient conditions for Clairaut conformal submersions to be harmonic.
Study Clairaut semi-slant/hemi-slant maps to Kähler manifolds.
problem Understanding Riemannian maps between specific types of manifolds.
method Analyzing Clairaut semi-slant/hemi-slant maps to Kähler manifolds.
result New insights into the properties of these maps.
Study on Clairaut maps from nearly Kahler to Riemannian manifolds.
problem Characterizing Clairaut maps from nearly Kahler manifolds.
method Analyzing conditions for Clairaut maps to be totally geodesic foliations.
result Non-trivial examples of Clairaut maps are provided.
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.
This study examines Clairaut slant Riemannian maps from Riemannian to Kähler manifolds.
problem Characterizing Clairaut slant Riemannian maps between Riemannian and Kähler manifolds.
method Analyzing necessary and sufficient conditions for geodesics, Clairaut slant maps, total geodesy, integrability, and harmonicity.
result Obtained inequalities involving second fundamental forms of Clairaut slant Riemannian maps.
The paper studies Clairaut maps on Sasakian manifolds.
problem Investigating Clairaut maps on Sasakian manifolds.
method Analyzing necessary and sufficient conditions for geodesics and biharmonicity.
result Conditions for Clairaut anti-invariant Riemannian maps on Sasakian manifolds.
The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.
problem Exploring geodesic curves on rotational surfaces in pseudo-Euclidean 4-space.
method Expressing Clairaut's theorem and deriving equations for geodesic curves.
result Characterization of geodesic curves on hyperbolic and elliptic surfaces of rotation.
The article defines and analyzes Clairaut anti-invariant maps between Riemannian and trans-Sasakian manifolds.
problem Characterizing Clairaut anti-invariant Riemannian maps between specific types of manifolds.
method Deriving conditions for Clairaut maps, discussing integrability, and establishing harmonicity.
result Necessary and sufficient conditions for Clairaut anti-invariant maps are derived.
The article generalizes Clairaut's formula for geodesics on submanifolds.
problem Conditions for geodesics on specific submanifolds.
method Study of geodesics on submanifolds involving Euclidean distance.
result Generalization of Clairaut's formula for higher dimensions.
We investigate new Clairaut conditions for anti-invariant submersions from normal almost contact metric manifolds onto Riemannian manifolds. We prove that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field when the total manifold is Sasakian. Several illustrative examples are also inclu…
The paper introduces and studies a new type of submersion in Riemannian geometry.
problem Exploring new submersions in Riemannian geometry.
method Defining and studying Clairaut Riemannian warped product submersions.
result Established conditions for a Riemannian warped product submersion to satisfy the Clairaut condition.
We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three dimensional spaces; the local description of the geodesics; the explicit description o…
Investigates physical properties on surfaces of rotation using Clairaut's theorem.
problem Understanding specific energy and angular momentum on surfaces of rotation.
method Used Clairaut's theorem with geodesic conditions to derive specific energy and angular momentum.
result Physical expressions for specific energy and angular momentum on surfaces of rotation were derived.
Study Clairaut maps on Kähler manifolds with Ricci solitons, finding curvature and scalar relations.
problem Exploring Clairaut maps on Kähler manifolds with Ricci solitons.
method Analyzing curvature relations, calculating Ricci tensor, and finding conditions for Einstein spaces.
result Conditions for range and kernel spaces to be Einstein and finding scalar curvature for range space.
The paper studies maps between Riemannian and Kähler manifolds, focusing on Clairaut semi-invariant Riemannian maps.
problem Analyzing maps between Riemannian and Kähler manifolds, particularly Clairaut semi-invariant Riemannian maps.
method Recalled and defined Clairaut semi-invariant Riemannian maps, derived necessary and sufficient conditions for geodesic curves and maps, and explored foliations and product manifolds.
result Necessary and sufficient conditions for various properties of Clairaut semi-invariant Riemannian maps were derived.
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.
Complete left-invariant metrics on Lie groups with specific properties.
problem Completeness of left-invariant semi-Riemannian metrics on Lie groups.
method Introducing bi-Lipschitz Riemannian Clairaut metrics and proving completeness conditions.
result All left-invariant metrics are complete for certain Lie groups.
We study the behavior of geodesics on a Randers surface of revolution. The main tool is the extension of Clairaut relation from Riemannian case to the Randers case. Moreover, we show that our Randers surface of revolution can be embedded in a Minkowski space as hypersurface.
Catenaries defined on any Riemannian surface using intrinsic distance.
problem Defining catenaries on Riemannian surfaces.
method Defining catenaries as critical points of a potential functional, calculating potential with intrinsic distance, and characterizing using curvature.
result Characterization of catenaries on various Riemannian surfaces.
This is a postprint of our paper "Force free Moebius motions of the circle" (J. Geom. Symmetry Phys. 27 (2012) 59-65), which we hadn't uploaded to arXiv previously. We would like to draw attention to the relationship with the article "A geometry where everything is better than nice", by Larry Bates and Peter Gibson (to…
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
The paper classifies hypersurfaces in H2imesH2 with constant curvature.
problem Classifying hypersurfaces in H2imesH2 with constant sectional curvature. method Analyzing the geometry of H2imesH2 and constructing specific examples. result Examples of hypersurfaces in H2imesH2 with non-constant product angle function. Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms Nn+1(c), proving properties and finding specific examples. result Proves that biconservative hypersurfaces with constant scalar curvature in N4(c) have constant mean curvature, and in N5(c), they are either rotational or constant mean curvature. We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…
Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
The paper studies curves of constant-ratio in pseudo-Galilean space.
problem Characterizing curves of constant-ratio in pseudo-Galilean space.
method Analyzing spacelike curves with constant-ratio in terms of curvature functions.
result Characterization of special curves of constant-ratio in pseudo-Galilean space.
The Cheeger constant increases under Ricci flow on spheres.
problem Behavior of the Cheeger constant under Ricci flow.
method Evolution identities for parallel curves and viscosity formulation of logh. result The Cheeger constant is non-decreasing under Ricci flow on surfaces diffeomorphic to S2. We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…
The paper defines new constants for p-Laplacian on manifolds.
problem Bounding eigenvalues of the p-Laplacian on compact manifolds. method Introducing Steklov and Neumann isocapacitary constants.
result Two-sided bounds for (p,α)-Sobolev constants and eigenvalues. Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in R3 with constant width, constant brightness, and boundary of class C2 is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.
Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
Curves with constant torsion can be deformed arbitrarily.
problem Deforming curves of constant torsion in Euclidean space.
method Convex integration and degree theory.
result Existence of knots with constant torsion in each isotopy class.
New upper bound for Cheeger constant of hyperbolic surfaces.
problem Bounding the Cheeger constant of hyperbolic surfaces.
method Random construction based on Poisson--Voronoi tessellation.
result The Cheeger constant of closed hyperbolic surfaces is less than that of the hyperbolic plane.
Simplified proof for Cheeger's isoperimetric constant.
problem Cheeger's isoperimetric constant
method Simplified proof of Buser's result
result Simplified proof for Cheeger's isoperimetric constant
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
This paper mixes constant sum and constant product market makers to improve their features.
problem Improving the balance between stable exchange rates and liquidity in automated market makers.
method Mixing and designing new methods for AMMs with specific features.
result Demonstrates new tools for creating markets with desired characteristics.
Ruled surfaces with Ricci metrics use curves of constant torsion.
problem Characterizing ruled surfaces with Ricci metrics.
method Using curves of constant torsion to construct ruled surfaces.
result Helicoid is the only surface with constant mean curvature.
Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.
problem Estimating eigenvalues and spectrum for graph substructures.
method Introducing Cheeger type constants via isocapacitary constants to estimate eigenvalues and spectrum.
result Estimates for first Dirichlet, Neumann, and Steklov eigenvalues, as well as the bottom of the spectrum of the Laplace operator and Dirichlet-to-Neumann operator.
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
A number of results for C2-smooth surfaces of constant width in Euclidean 3-space E3 are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in the Heisenberg group.
method Proves surfaces in a neighborhood of non-umbilic points are solutions to a sinh-Gordon equation with a differential constraint.
result Surfaces in Heisenberg group with constant mean curvature described by solutions to sinh-Gordon equation.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.