Liana David
We introduce and study the notion of Sasaki--Weyl manifold, which is a natural generalization of the notion of Sasaki manifold. We construct a reduction of Sasaki--Weyl manifolds and we show that it commutes with several reductions already existing in the literature.
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Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor
本文研究完备梯度收缩Sasaki-Ricci孤立子，建立其积分曲率估计，并证明具有调和Weyl张量的此类孤立子必为球面的有限商，推广了Munteanu和Sesum关于Ricci孤立子的结果。
Para-Sasaki-like Manifolds with Generalized Symmetric Metric Connection
本文研究了类抛物Sasaki流形上的广义对称度量联络，推导了其与Levi-Civita联络的关系，探究了相关曲率性质与孤立子，并通过两个实例验证了结果。
The object of the present paper is to study locally $ϕ$-symmetric LP-Sasakian manifolds admitting semi-symmetric metric connection and obtain a necessary and sufficient condition for a locally $ϕ$-symmetric LP-Sasakian manifold with respect to semi-symmetric metric connection to be locally $ϕ$-symmetric LP-Sasakian manifold with respect to Levi-Civita connection.
We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.
Generalized $η-$Ricci solitons on LP-Sasakian manifolds admitting the general connection
本文研究了在允许一般连接的LP-Sasakian流形上广义η-Ricci溶剂的性质，并通过构造例子证明了其存在性。
On a sub-Riemannian manifold, a connection with skew-symmetric torsion is defined as the unique connection from the class of $N$-connections that has this property. Two cases are considered separately: sub-Riemannian structure of even rank, and sub-Riemannian structure of odd rank. The resulting connection, called the canonical connection, is not a metric connection in the case when the sub-Riemannian structure is of even rank. The structure of an almost quasi-Sasakian manifold is defined as an almost contact metric structure of odd rank that satisfies additional requirements. Namely, it is re…
The present study initially identify the generalized symmetric connections of type $(α,β)$, which can be regarded as more generalized forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained respectively when $(α,β)=(1,0)$ and $(α,β)=(0,1)$. Taking that into account, a new generalized symmetric metric connection is attained on Lorentzian para-Sasakian manifolds. In compliance with this connection, some results are obtained through calculation of tensors belonging to Lorentzian para-Sasakian manifold involving curvature tensor, Ricci tensor and Ri…
We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condition is equivalent to the existence of a Sasaki-Einstein 5-manifold and vice versa, any Sasaki-Einstein 5-manifold generates a two parametric family of connections with skew torsion satisfying the Einstein metricity condition. Formulas for the curvature and the Ricci tensors of t…
The present paper deals with the study of Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds of SQ-Sasakian manifolds with respect to Levi-Civita connection and semisymmetric metric connection and obtain that these two classes are equivalent with a certain condition. Also the invariant and anti-invariant submanifolds of SQ-Sasakian manifolds with respect to Levi-Civita connection as well as semisymmetric metric connection whose metrics are Ricci solitons are studied.
The geometric characteristics of SGL submanifolds in an indefinite Sasakian statistical manifold equipped with a quarter symmetric metric connection
本文研究了在特定几何结构下SGL子流形的几何特性，分析了其分布的可积性和平行性，并通过示例展示了其应用。
The space of invariant affine connections on every $3$-Sasakian homogeneous manifold of dimension at least $7$ is described. In particular, the remarkable subspaces of invariant affine metric connections, and the subclass with skew-torsion, are also determined. To this aim, an explicit construction of all $3$-Sasakian homogeneous manifolds is exhibited. The unique $3$-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew-torsion invariant affine connections are those of dimension $7$, that is, $\mathbb{S}^7=\mathrm{Sp} (2)/ \mathrm{Sp(1)}$, $\mathbb{R} P^7=\mathrm{Sp}(2)/ \m…
We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the $5$-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures. By deforming such a $SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Fu…
We study compact Sasakian manifolds whose Tondeur connection has holonomy group either trivial or contained in Sp(n). We show that the first condition forces the manifold to be a compact quotient of the Heisenberg Lie group, while in the simply-connected case a Sasakian structure has the holonomy of the Tondeur connection contained in Sp(n) if and only if there exists a transverse hypercomplex structure. This latter result is the "Sasakian version" of a theorem of Verbitsky.
We show that a compact K-contact manifold $(M,g,ξ)$ has a closed Weyl-Einstein connection compatible with the conformal structure $[g]$ if and only if it is Sasaki-Einstein.
Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the special orthogonal group, of dimension considerably smaller. We realize the computations of the holonomies in a unified way, by using as a main algebraic tool a nonassociative structure, that one of symplectic triple system.
In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the newly defined classes should admit 'good' metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-$(α,δ)$-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heis…
In this study, we introduce indefinite sasakian statistical manifolds and lightlike hypersurfaces of an indefinite sasakian statistical manifold. Some relations among induced geometrical objects with respect to dual connections in a lightlike hypersurface of an indefinite sasakian manifold are obtained. Some examples related to these concepts are also presented. Finally, we prove that an invariant lightlike submanifold of indefinite sasakian statistical manifold is an indefinite sasakian statistical manifold.
We define a class of metrics that extend the Sasaki metric of a tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback of the big tangent bundle of a manifold to the tangent manifold. We obtain the expression of the transferred metric and we define a canonical, metric connection with torsion. We calculate the torsion, curvature and Ricci curvature of this connection and give a few applications of the results. We also discuss the transfer of generalized complex and generalized Kaehler structures.
We investigate curvature properties of 3-$(α,δ)$-Sasaki manifolds, a special class of almost 3-contact metric manifolds generalizing 3-Sasaki manifolds (corresponding to $α= δ= 1$) that admit a canonical metric connection with skew torsion and define a Riemannian submersion over a quaternionic Kähler manifold with vanishing, positive or negative scalar curvature, according to $δ= 0$, $αδ> 0$ or $αδ< 0$. We shall investigate both the Riemannian curvature and the curvature of the canonical connection, with particular focus on their curvature operators, regarded as symmetric endomorphisms of the …
We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. In particular, we generalize the class of quasi-Sasaki manifolds and characterize these structures by their intrinsic torsion. Among other things, we see that these manifolds admit a unique metric connection that is compatible with the underlying almost contact metric structure. Finally, we construct a family of examples that are not quasi-Sasaki.
Note on geodesics of cotangent bundle with Berger-type deformed Sasaki metric over Kählerian manifold
本文在 Kählerian 流形的余切丛及单位余切丛上引入了 Berger 型变形 Sasaki 度量，研究了其 Levi-Civita 联络，并探讨了相应的测地线。
Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that a…
Connection Towers and Sasaki Metrics on Higher-Order Tangent Bundles
本文引入高阶切丛上的连接塔概念，研究其诱导的几何结构，并构造由Levi-Civita连接诱导的具体连接塔，推广了Dombrowski连接映射，导出一族高阶Sasaki度量及其Levi-Civita连接与测地方程。
For the Riemannian manifold $M^{n}$ two special connections on the sum of the tangent bundle $TM^{n}$ and the trivial one-dimensional bundle are constructed. These connections are flat if and only if the space $M^{n}$ has a constant sectional curvature $\pm 1$. The geometric explanation of this property is given. This construction gives a coordinate free many-dimensional generalization of the connection from the paper: R. Sasaki 1979 Soliton equations and pseudospherical surfaces, Nuclear Phys., {\bf 154 B}, pp. 343-357. It is shown that these connections are in close relation with the imbeddi…
It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along with its parallel spinor field can be used for a thorough investigation of the geometric properties of 7-dimensional 3-Sasakian manifolds. Many known and some new properties can be easily derived from the properties of this connection and the spinor field, yielding thus an appropria…
We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only it is flat.
Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles
本文在拟正则紧致Sasakian流形上引入基本向量丛的拟正则性与正则性概念，建立轨形基本群半单表示与Higgs轨形丛之间的非阿贝尔Hodge对应，并将其推广到平坦丛与基本Higgs丛，同时证明了数值平坦丛刻画的Sasakian类比。
Sasaki with torsion manifolds and string backgrounds
本文受埃尔米特几何中Bismut的启发，研究带挠率的Sasaki流形，引入∇-爱因斯坦流形概念，推导带挠率Sasaki结构的流并证明其与广义里奇流规范等价。
We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate ap…
Holonomy algebras of Weyl connections in Lorentzian signature are classified. In particular, examples of Weyl connections with all possible holonomy algebras are constructed.
We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product $\mathbb S^1 \times M$ carries canonical families of Weyl connections with such a property, for any Riemmanian manifold $M$. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, is non-positive in a stronger sense (streched non-positive), then it must be locally of the product type.
Constructions of Frenet Curves with respect to Semi-Symmetric Metric Connection
本文基于半对称度量连接，研究了三维流形中Frenet曲线的构造及其在不同流形中的示例。
The object of this paper is to obtain the concircular curvature tensor of the semi symmetric non-metric connection on the Weyl manifold and to give a necessary and sufficient condition for a semi symmetric non-metric connection to be S-concircular.
This article examines the coincidence of the projective and conformal Weyl tensors associated to a given connection D. The connection may be a general Weyl connection associated to a conformal class of metrics [g]. The main result for n>3 is that the Weyl tensors coincide iff D is the Levi-Civita connection of an Einstein metric.
Stretched non-positive Weyl connections on solvable Lie groups
本文研究了可解李群上的不变拉伸非正Weyl联络，确定了其结构并分类了允许或不允许此类联络的可解李群类，同时完成了4维情形下可解李群与紧可解流形的全面分类。
On non-positive Weyl connection on Lie groups
研究李群上非正不变外尔联络问题，通过证明完全可解三维李群中仅SOL群有此类联络证实猜想，猜想方法可扩展到完全可解李群一般情形。
Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}^{1}(M)$ be its $(1,1)-$tensor bundle equipped with the rescaled Sasaki type metric $% ^{S}g_{f}$ which rescale the horizontal part by a nonzero differentiable function $f$. In the present paper, we discuss curvature properties of the Levi-Civita connection and another metric connection of $T_{1}^{1}(M)$. We construct almost paracomplex Norden structures on $T_{1}^{1}(M)$ and investigate conditions for these structures to be para-Kähler (paraholomorphic) and quasi-Kähler. Also, some properties of almost paracomplex Norden structur…
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natural way. For the case of the odd dimensional spheres $\mathbb{S}^{2n+1}$ viewed as homogeneous spaces of the special unitary groups, the classical Nomizu's Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the sphe…
Starting from a real analytic surface $\mathcal{M}$ with a real analytic conformal Cartan connection A. Borówka constructed a minitwistor space of an asymptotically hyperbolic Einstein-Weyl manifold with $\mathcal{M}$ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on $\mathcal{M}$ can be extended to symmetries of the obtained Einstein-Weyl manifold.
In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection $(M^4,c,D)$. We show that there is an Eells-Salamon type correspondence between nonvertical $\mathcal{J}$-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When $(M,c,J)$ is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality \begin{equation}\label{adj} χ(T_fΣ)+χ(N_fΣ) \le \pm c_1(f^*T^{(1,0)}M). \end{equ…
We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable $L^2$-energy identity known as…
Weyl forms in parabolic contact geometries
本文在无挠正规抛物接触几何中推导了Rho张量与Weyl联络的关系，给出了Weyl联络关于Tanaka--Webster联络的公式，并应用该结果得到可积Legendrian接触几何的局部同构完全障碍。
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds c…
In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection $\bar{\nabla}$ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler contact distribution is of constant holomorphic sectional curvatures if and only if its 3-Sasakian manifold is of constant $φ_α$-sectional curvatures. Moreover, it is shown that there is an interesting relation between the sectional curvatures of $φ_α$-planes on $TM$ of metric connect…
Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces $G/P$ with $G$ semisimple and $P$ parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, closed and exact Weyl connections, and Rho--tensors, we characterize the classes of such objects, and we use the results to give a new description of the Cartan bundles and connections for all parabolic geometries.
The Levi-Civita connections of manifolds with prescribed optical geometries
本文针对具有Kähler-Sasaki型光学几何的Lorentzian流形，用适配标架显式推导了Levi-Civita联络的Christoffel符号，并指出其在几何不变量确定和曲率约束求解中的应用。
We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor $ψ$ of algebraic type $F \cdot ψ= 0$ with respect to the unique connection $\nabla$ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of almost contact metric manifolds and discuss a link between them and the dilation function in 5-dimensional string theory. We find natural conditions implying conformal invariances of parallel spinors. We present topological obstructions to the existence of parallel spinors in the…
We find necessary and sufficient conditions for the bi-Legendrian connection $\nabla$ associated to a bi-Legendrian structure $(\cal F,\cal G)$ on a contact metric manifold $(M,ϕ,ξ,η,g)$ being a metric connection and then we give conditions ensuring that $\nabla$ coincides with the (generalized) Tanaka-Webster connection of $(M,ϕ,ξ,η,g)$. Using these results, we give some interpretations of the Tanaka-Webster connection and we study the interactions between the Tanaka-Webster, the bi-Legendrian and the Levi Civita connection in a Sasakian manifold.
We apply the theory of Weyl structures for parabolic geometries developed by A. Cap and J. Slovak in to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot-Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold. In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern-Moser tensor in CR geometry. This ten…
The heterotic G$_2$ system with reducible characteristic holonomy
本文提出了一种在退化3-(α,δ)-Sasaki流形上构造异质G₂系统精确解的方法，通过分析G₂连接获得新解。
The invariants of the Thomas and the Weyl type for a mapping between non-symmetric affine connection spaces are obtained with respect to the factored deformation tensor in this paper. Motivated by two invariants of the Weyl type obtained in (N. O. Vesić, Basic Invariants of Geometric Mappings, [17]), we founded novel invariants of the Weyl type. Invariants for almost geodesic mappings of the third type are searched at the end of this paper.
In this paper, we introduce the notions of ALF Weyl connection and of associated mass, and we prouve the positive mass theorem for the ALF Weyl structures.
We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the `twistor' bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadr…
We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magnetic relaxation and the hydrodynamic stability of steady Euler flows. We characterize the Sasakian manifolds where the Reeb field is a minimizer in terms of the first positive eigenvalue of the curl operator and show that for $a>a_0$ (a constant that depends on the Sasakian struc…
Generalized quasi-statistical structures
该研究在微分几何框架下引入广义准统计结构概念，刻画了广义切丛上广义复/积结构的∇可积性与准统计流形的等价关系，构造了准统计结构到切/余切丛的延拓并定义了相关几何结构。
For a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl's result that the Weyl curvature vanishes if and only if the manifold is (locally) diffeomorphic to a real projective space with the connection, when transported to the projective space, in the projective class of the Levi-Civita connection of the Fubini-Study metric. Associated to a connection on an even-dimensional M is an almost complex structure on J(M) the bundle of all complex structures on the tan…
Generalized Lagrange-Weyl structures and compatible connections are introduced as a natural generalization of similar notions from Riemannian geometry. Exactly as in Riemannian case, the compatible connection is unique if certain symmetry conditions with respect to vertical and horizontal Christoffel symbols are imposed.
We classify the possible local holonomy groups of Weyl connections. The Berger-Simons theorem and the Merkulov-Schwachhöfer classification of holonomy groups of irreducible torsion-free connections leaves us with the remaining case, where the Weyl connection $D$ is reducible and non-closed. In this case, it was shown by F. Belgun and A. Moroianu that the Weyl structure is an adapted Weyl structure of a non-closed conformal product. Furthermore we prove that non-closed Einstein-Weyl product structures only exist in dimension $4$.
In the present paper, the $(HM',S,T)$-Cartan connections on pseudo-Finsler manifolds, introduced by A. Bejancu and H.R. Farran, are obtained by the natural almost complex structure arising from the nonlinear connection $HM'$. We prove that the natural almost complex linear connection associated to a $(HM',S,T)$-Cartan connection is a metric linear connection with respect to the Sasaki metric $G$. Finally we give some conditions for $(M', J, G)$ to be a Kähler manifold.
The aim of this paper is to study from the point of view of linear connections the data $(M,\mathcal{D},g,W),$ with $M$ a smooth $(n+p)$ dimensional real manifold, $(\mathcal{D},g)$ a \textit{$n$}\textit{\emph{dimensional semi-Riemannian distribution}}\emph{}on $M,$ $\mathcal{G}$ the conformal structure generated by $g$ and $W$ a Weyl substructure: a map $W:$ $\mathcal{G}\to$ $Ω^{1}(M)$ such that $W(\overline{g})=W(g)-du,$ $\overline{g}=e^{u}g;u\in C^{\infty}(M)$. Compatible linear connections are introduced as a natural extension of similar notions from Riemannian geometry and such a connecti…
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