arXiv research

Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.

Introduces a variational framework for indefinite Lagrangians with specific symmetries.

We study the indefinite metric $G$ in the contact phase space $(P,θ)$ of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of $P$ - constitutive surfaces of different homogeneous thermodynamical syste…

The study extends Hano's theorem to semi-Riemannian product manifolds with specific conditions.

Study on hypersurfaces in Einstein manifolds using Killing spinors.

The set E of Levi-Civita connections of left-invariant pseudo-Riemannian Einstein metrics on a given semisimple Lie group always includes D, the Levi-Civita connection of the Killing form. For the groups SU(l,j) (or SL(n,R), or SL(n,C) or, if n is even, SL(n/2,IH)), with 0<=j<=l and j+l>2 (or, n>2), we explicitly descr…

The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we cons…

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are know…

Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.

We classify all spacetimes with a closed rank-2 conformal Killing-Yano tensor. They give a generalization of Kerr-NUT-de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness condition…

The paper bounds the genus of surfaces in four-manifolds with indefinite forms.

We present a compared analysis of some properties of indefinite almost $\mathcal{S}$-manifolds and indefinite $\mathcal{S}$-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and $φ$-sectional curvature of indefinite almost $\…

We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overlineφ,ζ, η)$, tangent to the characteristic vector field $ζ$, called; {\it contact screen conformal} and {\it contact screen umbilic} null hypersurfaces. These hypersurfaces come in to fill the existing gap in screen…

Developed new Crofton formulas for pseudo-Riemannian spaces.

We construct invariant complex product (hyperparacomplex, indefinite quaternion) structures on the manifolds underlying the real noncompact simple Lie groups $SL(2m-1,\RR)$, $SU(m,m-1)$ and $SL(2m-1,\CC)^\RR$. We show that on the last two series of groups some of these structures are compatible with the biinvariant Kil…

We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a gr…

Researchers solve conformal Killing forms on Kaehler manifolds.

Study on Riemannian Poisson warped product spaces and their properties.

The study explores indefinite nilsolitons and Einstein solvmanifolds, revealing new geometric properties.

We show that an indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form. We further study whether two compact Fubini-Study spaces are relatives or not.

We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of $d ω$ and its Hodge dual $* dω$ where $ω$ is the fundamental 2-form of the nearly Kähler stru…

Study of Killing spinor-valued forms and their integrability conditions.

In the present note, first we derive an intrinsic inequality for Pseudo-umbilical spacelike submanifold in an indefinite space form. We use this inequality to show that such submanifold is totally geodesic. In the rest part of this paper, using a result of Aiyama [1], we prove that Pseudo-umbilical spacelike subamnifol…

Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a…

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-ze…

First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characteriz…

Survey on invariant conformal Killing forms on Lie groups.

Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.

Classifies Killing forms of arbitrary degree on specific nilpotent Lie groups.

In a metric $g.f.f$-manifold we study lightlike hypersurfaces $M$ tangent to the characteristic vector fields, and owing to the presence of the $f$-structure, we determine some decompositions of $TM$ and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the …

Study Minkowski formula for conformal Killing-Yano 2-forms in constant curvature spacetimes.

Study conformal Killing forms on specific nilpotent Lie groups.

Study of spheres and circles on a manifold with a specific metric structure.

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of…

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equatio…

In a noncompact harmonic manifold $M$ we establish finite dimensionality of the eigenspaces $V_λ$ generated by radial eigenfunctions of the form $\cosh r + c$. As a consequence, for such harmonic manifolds, we give an isometric imbedding of $M$ into $(V_λ,B)$, where $B$ is a nondegenerate symmetric bilinear indefinite …

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing $p$--form ($p\ge2$) if and only if it isometric to a Riemannian product $S^k\times N$, where $S^k$ is a round sphere…

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and releva…

On a closed, connected Riemannian manifold with a Kähler foliation of codimension $q=2m$, any transverse Killing $r\ (\geq 2)$-form is parallel (S. D. Jung and M. J. Jung [\ref{JJ2}], Bull. Korean Math. Soc. 49 (2012)). In this paper, we study transverse conformal Killing forms on Kähler foliations and prove that if th…

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$…

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of …

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that on a compact manifold with holonomy G2 or Spin7 any Killing form has to be parallel. The main tool is a universal Weitzenboeck formula. We show how such a formula can be obtained for any given…

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemanni…

On a pseudo-Riemannian manifold $\mathcal{M}$ we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on $\mathcal{M}$ and parallel fields on the metric cone over $\mat…

We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we c…

It is well known that if the dimension of the Sasaki cone is greater than one, then all Sasakian structures are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that the dimension of the Sasaki cone is greater than one, there are three possibiliti…

In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing-Yano 2-forms is one-dimensiona…