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Piotr Dacko

Publications and source records attributed to Piotr Dacko.

13 recordsLinked to original sources
arXiv2023

Sasakian lift of Kaehler manifold and $α$-Sasakian Ricci solitons

In this paper we provide a local construction of a Sasakian manifold given a Kähler manifold. Obatined in this way manifold we call Sasakian lift of Kähler base. Almost contact metric structure is determined by the operation of the lift of vector fields - idea similar to lifts in Ehresmann connections. We show that Sasakian lift inherits geometry very close to its Kähler base. In some sense geometry of the lift is in analogy with geometry of hypersurface in Kähler manifold. There are obtained structure equations between corresponding Levi-Civita connections, curvatures and Ricci tensors of the lift and its base. We study lifts of symmetries different kind: of complex structure, of Khler metric, and Kähler structure automorphisms. In connection with $η$-Ricci solitons we introduce more general class of manifolds called twisted $η$-Ricci solitons. As we show class of $α$-Sasakian twisted $η$-Ricci solitons is invariant under naturally defined group of structure deformations. As corollary it is proved that orbit of Sasakian lift of steady or shrinking Ricci-Kähler soliton contains $α$-Sasakian Ricci soliton. In case of expanding Ricci-Kähler soliton existence of $α$-Sasakina Ricci solition is assured provided expansion coefficient is small enough.

math.DG↗
arXiv2022

Remarks on anti-quasi-Sasakian manifolds

In this short note we present some remarks concerning anti-quasi-Sasakian manifolds. Some proofs of their basic properties are simplified. We also discuss some canonical invariant distributions which exist on every anti-quasi-Sasakian manifold.

math.DG↗
arXiv2020

$η$-Normality, CR-structures, para-CR structures on almost contact metric and almost paracontact metric manifolds

For almost contact metric or almost paracontact metric manifolds there is natural notion of $η$-normality. Manifold is called $η$-normal if is normal along kernel distribution of characteristic form. In the paper it is proved that $η$-normal manifolds are in one-one correspondence with Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case of almost paracontact metric manifolds. There is provided characterization of $η$-normal manifolds in terms of Levi-Civita covariant derivative of structure tensor. It is established existence a Tanaka-like connection on $η$-normal manifold with autoparallel Reeb vector field. In particular case contact metric CR-manifold it is usual Tanaka connection. Similar results are obtained for almost paracontact metric manifolds. For manifold with closed fundamental form we shall state uniqueness of this connection. In the last part is studied bi-Legendrian structure of almost paracontact metric manifold with contact characteristic form. It is established that such manifold is bi-Legendrian flat if and only if is normal. There are characterized semi-flat bi-Legendrian manifolds.

math.DG↗
arXiv2020

Classification results for three-dimensional (para)contact metric and almost (para)cosymplectic $(κ,μ)$-spaces

It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu $(κ,μ)$-manifolds. In the case of dimension three it is described in full details structure of contact metric or almost cosymplectic $(κ,μ)$-spaces. The second part of the paper addresses three-dimensional paracontact metric and almost paracosymplectic $(κ,μ)$-spaces. There is obtained local classification of paracontact metric $(κ,μ)$-spaces, and almost paracosymplectic $(κ,μ)$-spaces, for every possible value of $κ$.

math.DG↗
arXiv2020

Almost (para-) contact metric $(κ,μ)$-manifolds. Part 1: Riemannian

The author is planning if possible classify all three-dimensional $(κ,μ)$-manifolds wether contact metric, almost cosymplectic, para-contact metric, almost para-cosymplectic. Of course classification in contact or almost cosymplectic cases already is provdied. Up to authors knowledge there is no classification for para-contact or almost para-cosymplectic $(κ,μ)$. Conjecture is described by the author in coming paper structures provide classification. The main goal however is to show that these three dimensional manifolds are essentially building blocks of higher-dimensional manifolds. The other possiibilty is to introduce class of manifolds which contain both almost contact metric and almost para-contact metric manifolds as proper subclasses.

math.DG↗
arXiv2018

Rank of Jacobi operator and existence of quadratic parallel differential form, with applications to geometry of almost para-contact metric manifolds

It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible. In particular result can be applied directly to known classes of almost (para-) contact metric manifolds when considered Jacobi operator of characteristic vector field has maximal rank. There is effective algorithmic procedure which resolves the problem of existence of such vector field in pure algebraic way - there is canonically defined homogeneous differential form, with coefficients determined purely by the coeffcients of curvature operator - such that non-isotropic vector field has non-degenerate Jacobi operator if and only if it is non-zero of this form.

math.DG↗
arXiv2017

On a class of immersions between almost para-Hermitian manifolds

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an analogue of the idea of slant submanifold in almost Hermitian geometry. There are classified pointwise slant surfaces of four dimensional almost para-Hermitian manifold.

math.DG↗
arXiv2015

On the existence of proper Nearly Kenmotsu manifolds

This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly Kähler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly Kähler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.

math.DG↗
arXiv2014

Note on classical notion of Lee form

This note is devoted to partial study of recurrent equation $dω=β\wedge ω$, based on linear algebra of exterior forms. Such equation was considered by Lee, for non-degenerate 2-form. In this note we approach general case, when $ω$ is arbitrary. Particularly, we extend results obtained by Lee, on odd-forms.

math.DG↗
arXiv2013

Five dimensional almost para-cosymplectic manifolds with contact Ricci potential

There are studied in details 5-dimensional pseudo-Riemannian manifolds equipped with the structure analogous to the almost cosymplectic (almost coKaehler) structure. The curvature by assumption commutes with the structure affinor and all these manifolds are Walker spaces. There are obtained classifications for manifolds for contact Ricci potential, locally flat manifolds and described all connected, simply connected Lie groups admiting left-invariant structure. There are some other more general results.

math.DG↗
arXiv2012

On CR (Cauchy-Riemann) almost cosymplectic manifolds

In the paper we develop a framework for the alternative way of the study of a local geometry of almost cosymplectic manifolds with Kahlerian leaves. The main idea is to apply the concept of a geometry and analysis of CR manifolds. Locally the almost cosymplectic manifold is modeled on the 'mixed' space RxCn. There is given a complete local description of the underlying almost contact metric structure in the system of local, mixed - real, complex- coordinates. We also introduce a notion of a canonical Hermitian complex connection in the CR structure of a CR almost cosymplectic manifold. As an example we provide detailed descritpion of almost cosymplectic $(-1,μ,0)$-spaces.

math.DG↗
arXiv2012

Sewing cells in almost cosymplectic and almost Kenmotsu geometry

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 conditions. If cells satisfy nullity conditions - then - in the case of almost cosymplectic or almost $α$-Kenmotsu manifolds - "sewed cells" also satisfies nullity condition - but generally with different constants. It is important that even in the case of the generalized nullity conditions - "sewed cells" are the manifolds which satisfy such conditions provided the cells satisfy the generalized nullity conditions.

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