arXiv research

The study classifies Riemannian manifolds with curvature nullity.

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…

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $M^…

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.

New conditions on Riemannian manifolds' curvature tensor.

Survey shows deformations of homogeneous metrics in curvature homogeneous manifolds.

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 $\N_{R^\ast}$ is proved. Two counterexamples are given: the first shows that $\N_{R…

Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.

Study calculates indices and nullities of focal manifolds in spheres.

We show that a complete Euclidean submanifold with minimal index of relative nullity $ν_0>0$ and Ricci curvature with a certain controlled decay must be a $ν_0$-cylinder. This is an extension of the classical Hartman cylindricity theorem.

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on comple…

In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$, fully described by Dajczer and Florit \cite{DF2} in terms of a certain c…

The study examines properties of $f$-contact manifolds with constant curvature.

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $ν$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two tr…

The paper studies contact pseudo-metric manifolds with a specific curvature condition.

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let $M^m$ be a complete Riemannian manifold and let $f\colon M^m\to\R^n$ be a minimal isometric immersion with index of relative nullity at least $m-2$ at any point. We show that if the Omori-Yau maximum princ…

Paper derives second variational formula for statistical manifold mappings.

Study curvature properties of w.a. S-manifolds with new conditions.

Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.

The mobility of a Kaehler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kaehler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kaehler metric to have m…

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in \cite{dksv1} and \cite{dksv2}, respectively. If the scalar curvature i…

Found a minimal surface in curved space.

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $$ and $\tildeμ$). This class of pseudo-Riemannian manifolds, which includes para-Sasak…

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 $\mathrm{Ker}_R$ and $\N_R$ do not coincide. The second shows that the nul…

This paper is a complete study of almost α-paracosmplectic manifolds. We characterize almost α-paracosmplectic manifolds which have para Kaehler leaves. Main curvature identities which are fulfilled by any almost α-paracosmplectic manifold are found. We also proved that ξ is a harmonic vector field if and only if it is…

Study on submanifolds with specific nullity ranks, proving inequalities between nullity ranks.

We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the…

Study on sphere immersions and their stability indices.

Stability of Yang-Mills connections' Morse indices and nullity in 4D.

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.

For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form $η$ 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…

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invaria…

In this paper, we consider immersed two-sided minimal hypersurfaces in $\mathbb{R}^n$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $n=4$, …

We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be ar…

We study almost Kaehler manifolds whose curvature tensor satisfies the third curvature condition of Gray. We show that the study of manifolds within this class reduces to the study of a subclass having the property that the torsion of the first canonical Hermitian connection has the simplest possible algebraic form. Th…

Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.

In this paper the result of real hypersurfaces in non-flat complex space forms, whose structure vector field $ξ$ belongs to the $κ$-nullity distribution is extended in case of three dimensional real hypersurfaces in non-flat complex space forms. Furthermore, generalization of notion ($κ$,$μ$)-nullity distribution defin…

Minimal surface doublings have specific index and nullity values.

In this note we investigate the $(E)$-index and the $(E)$-nullity of vacuum solutions from the two-torus into the two sphere.

Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.

We find necessary and sufficient conditions for a complete $n$-dimensional Riemannian manifold of finite volume, whose curvature tensor has nullity at least $n-2$, to be a geometric graph manifold. In the process, we show that Nomizu's conjecture, well known to be false in general, is true for manifolds with finite vol…

We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition o…

Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.

The nullity of a minimal submanifold $M\subset S^{n}$ 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 $SO(n+1)$ of the ambient space, modulo those motions which preserve $M$, whose dimension is the Killing nulli…

The paper calculates Morse indices and nullities for embedded networks on spheres.

We investigate 3-dimensional almost Kenmotsu manifolds satisfying special types of nullity conditions depending on two smooth functions $κ,μ$. When either $κ<-1$ and $μ=0$ or $h=0$, such conditions coincide with the $κ$-nullity condition which we show to be equivalent to the $η$-Einstein one. As an application of this …

In this paper, we consider surfaces in 4--dimensional pseudo--Riemannian space--forms with index 2. First, we obtain some of geometrical properties of such surfaces considering their relative null space. Then, we get classifications of quasi--minimal surfaces with positive relative nullity.

In this paper we investigate $m$-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least $m-2$ at any point. These are austere submanifolds in the sense of Harvey and Lawson \cite{harvey} and were initially studied by Bryant \cite{br}. For any dimension and codimension the…