The present paper aims to study the higher-order complete and vertical lifts of the extended almost complex structures on an extended complex manifold kM. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure Jk on the extended complex manifold kM are proved. Also, a tensor field J~k of type (1,1) is introduced and shows that it is an extended almost complex structure. Finally, the Lie derivative concerning higher-order lifts is studied and basic results on the almost analytic complex vector concerning an extended almost complex structure on kM are investigated.
The tangent bundle of an almost Norden manifold and the complete lift of the Norden metric is considered as a 4n-manifold. It is equipped with an almost hypercomplex Hermitian-Norden structure. It is characterized geometrically. The case when the base manifold is an h-sphere is considered.
In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metric on the base manifold M. This metric is in some senses more general than the other lift metrics defined previously on TM and then we study the complete lift vector fields on TM. More precisely we prove; Let (M,g) be a Finsler manifold, TM its tangent bundle and G a Riemannian me…
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…
The goal of this paper, using lifting theory it is to produce almost paracomplex struc- tures on the tangent bundle of almost Lorentzian r-paracontact manifold endowed with almost Lorentzian r-paracontact structure. Finally, we discuss the effect over dynamics systems of the produced geometrical structures.
In this article, we introduce some metallic structures on the tangent bundle of a P-Sasakian manifold by complete lift, horizontal lift and vertical lift of a P-Sasakian structure $(ϕ, η,ξ)$ on tangent bundle. Then we investigate the integrability and parallelity of these metallic structures.
In this study, taking into considering lifting theory, we shall obtain both almost complex and paracomplex structures on the tangent bun- dle, based on almost Lorentzian r-contact and r-paracontact manifold.
We show how to lift positive Ricci and almost non-negative curvatures from an orbit space $M/G$ to the corresponding $G$-manifold, $M$. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously.
This article presents the further steps of the previously done studies taking into consideration the k-th order extensions of a complex manifold. In the previous studies higher order vertical and complete lifts of structures on the complex manifold were introduced. Presently, k-th extended spaces of a product manifold have been set and the higher order vertical, complete, complete- vertical and horizontal lifts of geometric structures on the product manifold have been presented.
In this article we present a study of the subspaces of the manifold OscM, the total space of the osculator bundle of a real manifold M. We obtain the induced connections of the canonical metrical N-linear connection determined by the homogeneous prolongation of a Finsler metric to the manifold OscM. We present the relation between the induced and intrinsic geometric objects of the associated osculator submanifold.
In this paper for a given Banach, possibly infinite dimensional, manifold $M$ we focus on the geometry of its iterated tangent bundle $T^rM$, $r\in {\N}\cup\{\infty\}$. First we endow $T^rM$ with a canonical atlas using that of $M$. Then the concepts of vertical and complete lifts for functions and vector fields on $T^rM$ are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply $T^\infty M$ with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on $M$, can be lifted to a vector field…
Let (M,g) be a pseudo-Riemannian manifold and $T^2M$ be its the second-order tangent bundle equipped with the deformed 2-nd lift metric g which obtained from the 2-nd lift metric by deforming the horizontal part with a symmetric (0,2)-tensor field c. In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of $(T^2M,g)$. We give necessary and sufficient conditions for $(T^2M,g)$ to be semi-symmetric. Secondly, we show that $(T^2M,g)$ is a plural-holomorphic B-manifold with the natural integrable nilpotent structure. Finally, we get the conditi…
In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^A M$ and establish their properties. We prove that the complete and the vertical lifts induce homomorphisms of the Poisson cohomology spaces. We compute the modular classes of the lifts of Poisson structures.
In this paper we construct a functor from the category of two-dimensional Riemannian manifolds to the category of three-dimensional manifolds with generalized metric tensors. For each two-dimensional oriented Riemannian manifold $(M,g)$ we construct a metric tensor $\hat g$ (in general, with singularities) on the total space $SO(M,g)$ of the principal bundle of the positively oriented orthonormal frames on $M$. We call the metric $\hat g$ the Wagner lift of $g$. We study the relation between the isometry groups of $(M,g)$ and $(SO(M,g),\hat g)$. We prove that the projections of the geodesics o…
In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold.
We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this setting and hence can be considered as the special Lagrangian lift of $M$. For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by Bryant and Salamon, minimal surfaces with vanishing complex quartic form (…
We construct some lift of an almost complex structure to the cotangent bundle, using a connection on the base manifold. This generalizes the complete lift defined by I.Sato and the horizontal lift introduced by K.Yano and S.Ishihara. We study some geometric properties of this lift and its compatibility with symplectic forms on the cotangent bundle.
The aim of this article is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also some suitable examples of Kenmotsu and $(κ,μ)'$-almost Kenmotsu manifolds are constructed to justify our results.
In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain a sharp eigenvalue estimate on quasi Kaehler manifolds and a sharp Hessian comparison on nearly Kaehler manifolds.
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of obtained results.
We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a Kählerian structure $(G,J)$ of general natural lift type (see \cite{Druta1}), is Einstein. We first obtain a general natural Kähler-Einstein structure on the cotangent bundle $T^*M$. In this case, a certain parameter, $λ$ involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expressed as a rational function of the other two, the value of the constant sectional curvature, $c$, of the base manifold $(M,g)$ and the constant $ρ$ involved in the condition for the str…
This paper is a study of almost contact statistical manifolds. Especially this study is focused on almost cosymplectic statistical manifolds. We obtained basic properties of such manifolds. It is proved a characterization theorem and a corollary for the almost cosymplectic statistical manifold with Kaehler leaves. We also study curvature properties of an almost cosymplectic statistical manifold. Moreover, examples are constructed.
We give a condition for an almost constant-type manifold to be a constant-type manifold, and holomorphic and $R$-invariant submanifolds of almost Hermitian manifolds are studied. Generalizations of some results in [5] are given.
In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.
We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $ω$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if a…
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 man…
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension are studied. Such structures are constructed on hyperspheres in 4-dimensional spaces, Euclidean and pseudo-Euclidean, respectively. The obtained manifolds are studied and characterised in terms of the classification used and their geometric properties.
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.
As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic, etc. Finally, we give some example…
In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.
Lie groups considered as three-dimensional almost paracontact almost paracomplex Riemannian manifolds are investigated. In each basic class of the classification used for the manifolds under consideration, a correspondence is established between the Lie algebra and the explicit matrix representation of its Lie group.
Relations between conjugate connections with respect to the pair of Norden metrics and to the almost complex structure on almost Norden manifolds are studied. Conjugate connections of the Levi-Civita connections induced by the Norden metrics are obtained. Statistical structures on almost Norden manifolds are considered.
Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the…
The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an example by a Lie group.
A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian manifold from a holomorphic complex Riemannian manifold is presented and called an $S^1$-solvable extension.
The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.
This paper is devoted to the study of curvature and torsion of almost contact curves in trans-Sasakian 3-Manifolds. The conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained.
Almost contact manifolds with B-metric are considered. Of special interest are the so-called vertical classes of the almost contact B-metric manifolds. Curvature properties of these manifolds are studied. An example of 5-dimensional manifolds is constructed and characterized.
In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(κ,μ)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimesional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $η$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton.
It is introduced a differentiable manifold with almost contact 3-structure which consists of an almost contact metric structure and two almost contact B-metric structures. The product of this manifold and a real line is an almost hypercomplex manifold with Hermitian-Norden metrics. It is proven that the introduced manifold of cosymplectic type is at. Some examples of the studied manifolds are given.
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 an eigen vector field of the Ricci operator. We locally classify three dimensional almost α-para-Kenmotsu manifolds satisfying a certain nullity condition. We show that this condition is invariant under D_{γ,β}-homothetic deformation. Furthermore, we construct examples of almost…
In this paper, by introducing a notion of local quasi holomorphic frame, we obtain a curvature formula for almost Hermitian manifolds which is similar to that of Hermitian manifolds. Moreover, as applications of the curvature formula, we extend a result of H.S. Wu and a result of F. Zheng to almost Hermitian manifolds.
Almost contact manifolds with B-metric are considered. A special linear connection is introduced, which preserves the almost contact B-metric structure on these manifolds. This connection is investigated on some classes of the considered manifolds.
The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kähler manifold, and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost Kähler metric vanishes for all …
We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.
Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and suffcient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that the normal almost paracontact metric manifolds are para-CR. We establish necessary and suffcient conditions for paracontact metric manifolds as well as for almost paracosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their conseque…
A four-parametric family of linear connections preserving the almost complex structure is defined on an almost complex manifold with Norden metric. Necessary and sufficient conditions for these connections to be natural are obtained. A two-parametric family of complex connections is studied on a conformal Kähler manifold with Norden metric. The curvature tensors of these connections are proved to coincide.
In the present paper, we introduce the notion of $\ast$-Miao-Tam critical equation on almost contact metric manifolds and studied on a class of almost Kenmotsu manifold. It is shown that if the metric of a $(2n + 1)$-dimensional $(k,μ)'$-almost Kenmotsu manifold $(M,g)$ satisfies the $\ast$-Miao-Tam critical equation, then the manifold $(M,g)$ is $\ast$-Ricci flat and locally isometric to the Riemannian product of a $(n + 1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold. Finally, an illustrative example is presented to support the main theorem.
In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spaces, is introduced.
The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold.
