The study characterizes Euclidean submanifolds with a conformal canonical vector field.
problem Characterizing Euclidean submanifolds with a specific vector field property.
method Investigating properties of the canonical vector field and its conformal nature.
result Characterization of Euclidean submanifolds with conformal canonical vector fields.
A Ricci soliton (M,g,v,λ) on a Riemannian manifold (M,g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is …
The paper classifies almost Yamabe solitons on hypersurfaces and submanifolds in Euclidean spaces.
problem Classifying almost Yamabe solitons on various geometric structures.
method Analyzing hypersurfaces and submanifolds with position and concurrent vector fields.
result Complete classification of almost Yamabe solitons on hypersurfaces and submanifolds.
The study proves that conformal Killing vector fields on manifolds with positive Ricci curvature are non-trivial.
problem Investigating conformal Killing vector fields on manifolds under curvature pinching conditions.
method Establishing a new Bochner-type identity and using Moser iteration for gradient estimates.
result Conformal Killing vector fields are non-trivial on manifolds with positive Ricci curvature.
Characterizes affine vector fields on Finsler manifolds with rigidity results.
problem Understanding affine vector fields on Finsler manifolds.
method Utilizing the Jacobi type equation and spray characterization, proving rigidity theorems.
result Rigidity theorems for affine vector fields on Finsler manifolds with non-positive total Ricci curvature.
Study proves conformal vector fields on certain Finsler manifolds are Killing fields.
problem Characterizing conformal vector fields on compact homogeneous Finsler manifolds.
method Analyzes properties of conformal vector fields and homogeneous Finsler metrics.
result Conformal vector fields on compact homogeneous Finsler manifolds are Killing fields.
The presence of a conformal gradient vector field on a Riemannian manifold implies it is isometric to a hemisphere.
problem Characterizing Riemannian manifolds with boundary using conformal gradient vector fields.
method Analyzing the properties of conformal gradient vector fields on Riemannian manifolds with boundary and applying Ricci curvature and scalar curvature conditions.
result Riemannian manifolds with boundary are isometric to a hemisphere under certain conditions on conformal gradient vector fields and curvature.
The study characterizes quasi Yamabe solitons with potential vector fields.
problem Characterizing quasi Yamabe solitons with specific properties.
method Analyzing potential vector fields and their norms in quasi Yamabe solitons.
result If the potential vector field has a finite global norm in a complete non-trivial, non-compact quasi Yamabe soliton with finite volume, the scalar curvature becomes constant and the soliton reduces to a Yamabe soliton.
Kahler-Einstein metrics linked to eigenvalue gaps on Fano manifolds.
problem Existence of Kahler-Einstein metrics on Fano manifolds.
method Characterization via eigenvalue gaps of Cauchy-Riemann and Hamiltonian vector fields.
result Existence of Kahler-Einstein metrics linked to eigenvalue gaps.
The study of Euclidean submanifolds with incompressible canonical vector fields.
problem Characterizing Euclidean submanifolds with incompressible canonical vector fields.
method Analyzing the canonical vector field properties and conditions for incompressibility.
result Necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible.
Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano…
Study describes lcK structures on Vaisman-type manifolds with holomorphic Lee vector field.
problem Characterizing lcK structures with holomorphic Lee vector field on Vaisman-type manifolds.
method Complete description through potential analysis and vector field properties.
result Examples of lcK structures with non-homothetic Lee vector field.
Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.
problem Understanding the derivations of Tanaka prolongations of transitive nilpotent Lie algebras.
method Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.
result Derivations of degree 0 are given by vector fields of degree 0, and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation.
Study on Yamabe and quasi-Yamabe solitons on Euclidean submanifolds.
problem Characterizing solitons on Euclidean submanifolds.
method Investigation of solitons with tangential components of position vector fields.
result Classification of Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.
In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the …
A space curve in a Euclidean 3-space E3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…
On a Hermitian manifold we construct a symmetric (1,1)- tensor H using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor H for a harmonic 1-form to be analytic and for an analytic 1-form to be harm…
Study of IMCF on warped product manifolds for long time existence.
problem Analyzing IMCF on conformal warped product manifolds.
method Relating gradient conformal vector field to conformal vector field, using control of the flow.
result Showed long time existence of IMCF on conformal warped product manifolds.
Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare co…
The paper classifies conformal solitons in pseudo-Euclidean spaces.
problem Classifying conformal solitons in pseudo-Euclidean spaces.
method Classification through pseudo-Riemannian hypersurfaces and position vector fields.
result Complete classification of conformal solitons in pseudo-Euclidean spaces.
In this paper we address the following questions: (i) Let C⊂C2 be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is C contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positi…
The paper classifies a type of solitons in Euclidean spaces.
problem Classifying generalized Yamabe solitons on hypersurfaces.
method Completely classified solitons arising from the position vector field.
result Classification of generalized Yamabe solitons on hypersurfaces in Euclidean spaces.
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…
Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
Study vector fields with complex singularities, proving bounds and formulas.
problem Understanding the Milnor number of vector fields with specific singularities.
method Global and local formulas expressing Milnor/Poincare-Hopf contributions, sharp lower bounds under perturbations.
result Sharp lower bounds for Milnor number contributions under holomorphic perturbations.
The paper explores properties of conformal vector fields on almost Kenmotsu manifolds.
problem Characterizing properties of conformal vector fields on almost Kenmotsu manifolds.
method Analyzing conformal vector fields as Reeb vector fields and pointwise collinear, proving manifold properties and existence of warped products.
result Conformal vector fields on almost Kenmotsu manifolds lead to specific manifold structures and properties.
Study lightlike hypersurfaces in a specific type of manifold.
problem Characterize lightlike hypersurfaces in indefinite almost contact metric-manifolds.
method Analyze the position of the structure vector field and classify hypersurfaces into two types.
result Prove there are only two types of lightlike hypersurfaces: ascreen and inascreen.
Develops Morse theory for commuting gradient-like vector fields.
problem Understanding the algebraic structure of the infrared data.
method Formal analogue of Morse theory for commuting vector fields, constructing L-infinity algebra and Maurer-Cartan elements.
result The algebraic formalism is similar to the algebra of the infrared of Gaiotto, Moore, and Witten.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
Study stabilizes second-order systems to first-order dynamics.
problem Stabilizing second-order systems to first-order dynamics.
method Feedback control of second-order systems on manifolds.
result Second-order systems can globally exponentially stabilize first-order dynamics for fully actuated systems.
The study characterizes almost Kenmotsu manifolds with specific vector fields.
problem Characterizing almost Kenmotsu manifolds with holomorphically planar conformal vector fields.
method Analyzing properties of vector fields and curvature conditions.
result Classification of almost Kenmotsu manifolds as Kenmotsu or having specific geometric properties.
We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth …
Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
The paper introduces new metrics on Finsler manifolds and characterizes associated vector fields.
problem Characterizing vector fields on Finsler manifolds with new metrics.
method Introducing F-natural metrics and characterizing conformal, homothetic, and Killing vector fields. result Characterization of vector fields on slit tangent bundles of Finsler manifolds.
Efficiently visualizes uncertainty in local divergence of 2D vector fields.
problem Uncertainty in vector field data leads to inaccurate divergence computations.
method Closed-form approach for highly efficient and accurate uncertainty visualization of local divergence, assuming independently Gaussian-distributed vector uncertainties.
result Significantly enhanced efficiency and accuracy of our algorithms over classical MC approach.
We study the convergence of the Kähler-Ricci flow on a compact Kähler manifold (M,J) with positive first Chern class c1(M;J) and vanished Futaki invariant on πc1(M;J). As the application we establish a criterion for the stability of the Kähler-Ricci flow (with perturbed complex structure) around a Kähler-Einste…
Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.
problem Understanding properties of holomorphic tensor fields on Kähler manifolds.
method Established vanishing theorems for uniformly rational connected (RC) k-positive Hermitian holomorphic vector bundles. result Holomorphic tangent bundles of Kähler manifolds with positive k-Ricci curvature are uniformly RC k-positive. Study on mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.
problem Characterizing mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.
method Analyzing the condition LVLVg=fLVg and using rigidity phenomena. result Dimension of complete mixed Killing fields is 5 and a basis is explicitly determined.
The limiting behavior of the normalized Kähler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabu…
New definitions and properties of harmonic vector fields on Finsler manifolds.
problem Defining and understanding harmonic vector fields in Finsler geometry.
method Natural definitions of differential, divergence, and p-harmonic form; proving Hodge theorem; Bochner-Yano classification theorem. result A closed orientable Finsler manifold with a positive harmonic Ricci scalar has a zero Betti number.
Local gradient estimates for eigenfunctions on conformal solitons improve Liouville theorems.
problem Estimating eigenfunctions on conformal solitons.
method Proving local gradient estimates for positive eigenfunctions of L-operator. result Improved Liouville theorems for Lu=0 on conformal solitons. The study extends and generalizes a result about quasi Einstein manifolds, proving conditions for Killing vector fields.
problem Characterizing conditions for Killing vector fields in quasi Einstein manifolds.
method Extending and generalizing Cochran's result, proving conditions for Killing vector fields under specific integrals and conformal conditions.
result Conditions for Killing vector fields in quasi Einstein manifolds, including integral identities and global isometry to spheres.
We discuss the solution theory of operators of the form ∇X+A, acting on smooth sections of a vector bundle with connection ∇ over a manifold M, where X is a vector field having a critical point with positive linearization at some point p∈M. As an operator on a suitable space of smooth section…
Geometric proof shows complex endomorphisms have invariant axes.
problem Proving complex endomorphisms have invariant axes.
method Dual geometric proof using vector fields and bordism.
result Arbitrary complex endomorphisms have invariant axes.
RVGP learns vector fields over unknown manifolds, preserving singularities.
problem Learning vector fields over unknown non-Euclidean manifolds.
method RVGP uses positional encoding with eigenfunctions of the connection Laplacian.
result RVGP preserves singularities in vector fields over unknown manifolds.
The paper studies conformally Einstein-Maxwell Kähler metrics and automorphism group structure.
problem Analyzing conformally Einstein-Maxwell Kähler metrics and their automorphism groups.
method Using a Hessian formula for the Calabi functional and extending the Lichnerowicz-Matsushima Theorem.
result Proves a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds.
Discover new identities linking hypersurface mean curvatures.
problem Understanding mean curvatures of hypersurfaces in Riemannian manifolds.
method Developed three most general Minkowski or Hsiung-Minkowski identities.
result Classical Minkowski identity is natural to all Riemannian manifolds.
3D Lorentzian manifolds can't be closed Einstein with positive constant.
problem Prohibiting closed Lorentzian 3-manifolds with positive Einstein constant.
method Analyzing Ricci tensor properties and unit timelike vector fields.
result No closed Lorentzian 3-manifolds with positive Einstein constant exist.