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.
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.
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.
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.
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.
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.
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 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.
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.
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…
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…
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.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.
problem Evaluating theorem proving agents' ability to generalize to unseen theorems.
method INT benchmark based on a theorem generation and proof procedure with adjustable knobs for measuring 6 types of generalization.
result MCTS can help agents prove new theorems.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.