The paper extends and generalizes nullity distributions on real hypersurfaces in complex space forms.
problem Analyzing nullity distributions on real hypersurfaces in non-flat complex space forms.
method Extending and generalizing the concept of nullity distributions to three-dimensional cases and introducing new distributions.
result Results on real hypersurfaces in non-flat complex space forms, including new nullity distributions.
A computational technique for calculating nullity vectors and kernel vectors, using the new Finsler package, is introduced. As an application, three interesting counterexamples are given. The first counterexample shows that the two distributions K e r R \mathrm{Ker}_R Ker R and N R \N_R N R do not coincide. The second shows that the nul…
The Klein-Grifone approach to global Finsler geometry is adopted. The nullity distributions of the three curvature tensors of Cartan connection are investigated. Nullity distributions concerning certain relevant special Finsler spaces are considered. Concrete examples are given whenever the situation needs.
The nullity distributions of the two curvature tensors \, $\overast{R}$ and $\overast{P}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution N R ∗ \N_{R^\ast} N R ∗ is proved. Two counterexamples are given: the first shows that $\N_{R…
We give a conceptual proof of the fact that if M is a complete submanifold of a space form, then the maximal integral manifolds of the nullity distribution of its second fundamental form through points of minimal index of nullity are complete.
Develops theory for homogeneous spaces with non-trivial nullity.
problem Understanding homogeneous spaces with non-trivial nullity.
method Constructs invariant distributions and uses transvections to find examples.
result First homogeneous examples with non-trivial nullity found.
The study classifies Riemannian manifolds with curvature nullity.
problem Classifying Riemannian manifolds with nontrivial curvature nullity.
method Classification theorems based on curvature nullity, scalar curvature, and quotient existence.
result New classification theorems and revisited previous results.
If M is a submanifold of a space form, the nullity distribution N of its second fundamental form is (when defined) the common kernel of its shape operators. In this paper we will give a local description of any submanifold of the Euclidean space by means of its nullity distribution. We will also show the following glob…
Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dim…
Study submanifolds with relative nullity in space forms using splitting tensor.
problem Characterize submanifolds with relative nullity in space forms.
method Use splitting tensor and Codazzi equation to express second fundamental form.
result Derive new strong consequences in hyperbolic and Euclidean spaces.
The paper explores conditions for certain submanifolds to be cylinders.
problem Conditions for isometric immersions with positive index of relative nullity to be cylinders.
method Analyzes geometric conditions and properties of submanifolds with nullity.
result Nonminimal n n n -dimensional submanifolds in space forms of any codimension are locally cylinders under specific conditions. Survey shows deformations of homogeneous metrics in curvature homogeneous manifolds.
problem Understanding curvature homogeneous manifolds with nontrivial κ-nullity.
method Analyzes deformations of homogeneous metrics along submersions.
result Identifies two new examples of curvature homogeneous manifolds.
Complete Euclidean submanifolds with high nullity must be cylinders.
problem Characterizing submanifolds with high relative nullity.
method Analyzing submanifolds with controlled Ricci curvature decay.
result Hartman cylindricity theorem extended to higher nullity.
Study on submanifolds with specific nullity ranks, proving inequalities between nullity ranks.
problem Characterizing submanifolds with specific nullity ranks.
method Analyzing submanifolds with given nullity ranks and proving inequalities.
result Characterization of submanifolds with specific nullity differences.
The study examines complete conformally flat submanifolds with nullity in Euclidean space.
problem Investigating properties of conformally flat submanifolds with nullity.
method Analyzing the index of relative nullity and scalar curvature to deduce manifold properties.
result Conditions for the manifold to be flat and the immersion to be a cylinder over a submanifold.
New conditions on Riemannian manifolds' curvature tensor.
problem Conditions for nullity in homogeneous Riemannian manifolds.
method Analyzing Lie algebras and transvections.
result Transvections generate abelian ideals in the isometry group.
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
problem Stability of Yang-Mills connections' Morse indices and nullity in 4D under weak convergence.
method Proves stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence.
result Stability of the sum of Morse indices and nullity of a sequence of Yang-Mills connections.
Study links' concordance using signatures and nullity.
problem Link concordance invariance of Levine-Tristram signatures.
method Determine invariance for complex numbers on the unit circle.
result Signature and nullity give link concordance invariants.
The study classifies quasi-minimal surfaces in 4D pseudo-Riemannian space-forms with positive nullity.
problem Characterizing surfaces in pseudo-Riemannian space forms with positive nullity.
method Analyzing the relative null space and classifying quasi-minimal surfaces.
result Classifications of quasi-minimal surfaces with positive relative nullity.
For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form η η η is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity c…
The paper examines minimal submanifolds with a specific nullity in Euclidean space.
problem Investigating minimal submanifolds with a positive index of relative nullity.
method Analyzing complete Riemannian manifolds and minimal isometric immersions with specific conditions.
result If certain conditions are met, the submanifold must be a cylinder over a minimal surface.
Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.
problem Characterize space-like and time-like surfaces in Robertson-Walker space-times with positive relative nullity.
method Provide necessary and sufficient conditions, local classification theorems, and analyze special spaces.
result Local classification theorems for space-like and time-like surfaces in L 1 4 ( f , 0 ) L^4_1(f,0) L 1 4 ( f , 0 ) with positive relative nullity. The Lawson surface ξ_{g,1} has index 2g+3 and nullity 6.
problem Characterizing the geometric properties of Lawson surfaces.
method Analyzing the linearized stability of the Lawson surface.
result The Lawson surface ξ_{g,1} has no exceptional Jacobi fields.
New bounds refine how multivariable signature and nullity change under link cobordisms.
problem Understanding how multivariable signature and nullity change under link cobordisms.
method Refined bounds on multivariable signature and nullity using link cobordisms and 0.5 0.5 0.5 -solvable cobordism. result Multivariable signature and nullity are invariant under 0.5 0.5 0.5 -solvable cobordism. Study on minimal submanifolds with nullity in hyperbolic space.
problem Characterize complete minimal submanifolds with nullity in hyperbolic space.
method Investigate properties of submanifolds with index of relative nullity at least one in hyperbolic space, considering scalar curvature bounds.
result If the scalar curvature is bounded from below, submanifolds are either totally geodesic or generalized cones over complete minimal surfaces.
Minimal surface doublings have specific index and nullity values.
problem Analyzing the properties of minimal surface doublings.
method Analytical proof of index and nullity values.
result Minimal surface doublings have index 2 γ + 5 = 2 m + 7 2γ+5=2m+7 2 γ + 5 = 2 m + 7 and nullity 6 6 6 . Researchers compute index and nullity of biharmonic maps in spheres.
problem Computing the exact index and nullity of proper biharmonic maps.
method Detailed analysis of specific examples, including non-compact domains.
result A large family of proper biharmonic maps in \(\mathbb{R}\) to \(\mathbb{S}^2\) is strictly stable.
Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
problem Proving upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
method Analyzing a weighted eigenvalue problem and using a Lorentz-Sobolev inequality to study eigenfunctions and index/nullity in neck regions.
result Upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces proved.
In this note we investigate the ( E ) (E) ( E ) -index and the ( E ) (E) ( E ) -nullity of vacuum solutions from the two-torus into the two sphere.
Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.
problem Prove nullity of biharmonic maps from flat 2-torus to round 2-sphere.
method Use spectral geometry, construct polynomial isomorphism with elliptic curve, determine rational points on spectral curve.
result Nullity of every map in the family is 5, confirming conjecture.
The nullity of a minimal submanifold M ⊂ S n M\subset S^{n} M ⊂ S n is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions S O ( n + 1 ) SO(n+1) S O ( n + 1 ) of the ambient space, modulo those motions which preserve M M M , whose dimension is the Killing nulli…
The paper calculates Morse indices and nullities for embedded networks on spheres.
problem Computing Morse indices and nullities for embedded networks on spheres.
method Using the Dirichlet-to-Neumann map and properties of eigenvalues and eigenfunctions.
result For all stationary triple junction networks in S 2 \mathbb{S}^2 S 2 , there is only one eigenvalue -1. We investigate 3-dimensional almost Kenmotsu manifolds satisfying special types of nullity conditions depending on two smooth functions κ , μ κ,μ κ , μ . When either κ < − 1 κ<-1 κ < − 1 and μ = 0 μ=0 μ = 0 or h = 0 h=0 h = 0 , such conditions coincide with the κ κ κ -nullity condition which we show to be equivalent to the η η η -Einstein one. As an application of this …
The paper calculates Morse indices and nullities for triply periodic minimal surfaces.
problem Computing Morse indices and nullities for triply periodic minimal surfaces.
method Developed an algorithm to compute Morse index and nullity using matrix properties of abelian differentials.
result Explicitly determined key matrices for five families of triply periodic minimal surfaces.
Study on CR structures of hypersurface type, focusing on Levi-form nullity.
problem Understanding nullity of Levi-form in CR structures.
method Analyzing sequence of CR invariant subsets and characteristic polynomial of Levi-form.
result Local existence of complex submanifolds determined by coefficients of characteristic polynomial.
New insights into projective structures with nullity conditions.
problem Analyzing projective structures with specific curvature nullity conditions.
method Discovering a fundamental 2-tensor invariant and a new canonical tractor connection.
result Strong local and global results on the nature of solutions and geometries.
Study calculates indices and nullities of focal manifolds in spheres.
problem Calculating indices and nullities of focal manifolds in spheres.
method Analyzes isoparametric hypersurfaces in spheres with three distinct principal curvatures.
result Index equals the ambient space dimension, nullity determined by Killing vector fields.
We regard a contact metric manifold whose Reeb vector field belongs to the ( κ , μ ) (κ,μ) ( κ , μ ) -nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric ( κ , μ ) (κ,μ) ( κ , μ ) -spaces in terms of a canonical connection which can be naturally defined on them.
Minimal submanifolds in spheres can be produced via Clifford type minimal products, and their Morse indices and nullities are calculated.
problem Understanding the properties of minimal submanifolds in spheres via Clifford products.
method Analyzing the first eigenfunctions and Morse indices of minimal products of minimal submanifolds.
result The Morse index and nullity of the minimal product are calculated and shown for specific cases.
Paper derives second variational formula for statistical manifold mappings.
problem Variational formulas for mappings between statistical manifolds.
method Develops second variational formula for harmonic mappings, defines stability, index, and nullity.
result Shows weakly stability for harmonic mappings into statistical manifolds of non-positive curvature.
Study introduces a new Allen-Cahn energy on hypersurfaces and analyzes its properties.
problem Analyzing geometric variations of the Allen-Cahn energy on hypersurfaces.
method Establishes Γ-convergence, computes variations, and analyzes the linearized equation.
result Shows that the index and nullity of the energy are related to the Allen-Cahn index and nullity.
Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.
problem Calculating Morse index and nullity for homogeneous minimal hypersurfaces with g = 4 g = 4 g = 4 or 6 6 6 . method Analyzing two specific homogeneous minimal hypersurfaces in S n S^n S n with g = 4 g = 4 g = 4 . result Obtained irrational eigenvalues in Laplace spectra for the two hypersurfaces.
The paper bounds the Morse index and nullity of bipolar surfaces related to Otsuki tori.
problem Bounding the Morse index and nullity of bipolar surfaces.
method Lawson's bipolar surface construction applied to Otsuki tori.
result Upper and lower bounds on the Morse index and nullity for p / q p/q p / q close to 2 / 2 \sqrt 2/2 2 /2 . The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
Introduces Finsler entropy for smooth distributions and foliations.
problem Entropy of smooth distributions and foliations.
method Definition of Finsler entropy based on distances on compact metric sets.
result Nullity of Finsler entropy for controllable distributions and singular Riemannian foliations.
The aim of this work is to extend the results of S. Nayatani about the index and the nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces for values of the genus bigger than 37. That allows us to state that these minimal surfaces are non degenerate for all the values of the genus in the sense of the definition …
Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
The paper studies minimal submanifolds in spheres with specific nullity properties.
problem Investigating minimal submanifolds with nullity properties in Euclidean spheres.
method Analyzing submanifolds with index of relative nullity at least \(m-2\) and providing a complete local parametric description using 1-isotropic surfaces.
result Any complete submanifold is either totally geodesic or has dimension three.