Ercument H. Ortacgil
We show that the theory of geometric structures proposed in the recent book "An Alternative Approach to Lie Groups and Geometric Structures" can be developed independently of connections.
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In this paper we calculate the curvature of the Hitchin connection. We further show that a slight (possibly trivial) modification of the Hitchin connection has curvature equal to an explict given multiple of the Weil-Petersen symplectic form on Teichmüller space.
This paper presents a brief study on connections on fiber, principal and vector smooth bundles as well as some relations with their curvatures.
We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a linear symmetric connection on the base manifold. We prove the generalized Bianchi identity for linear connections and we prove that the antisymmetrization of the second order covariant differential is expressed via the curvature tensors of both connections.
Torsion-free Connections with prescribed Curvature
本文利用幂级数方法给出了曲率映射由无挠联络曲率张量产生的充要条件，唯一确定了该联络，并利用曲率映射有效计算了其和乐群。
The Chern Sectional Curvature of a Hermitian Manifold
本文定义了厄米流形的陈截面曲率，推导了其局部表达式，并证明当它与黎曼截面曲率一致时流形为凯勒流形，同时给出了里奇曲率和标量曲率。
We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian Geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections, and we find a family of anisotropic connections which are well suited to study the geometry of Finsler metrics.
There is considered a connection with skew symmetric torsion on a quasi-Kähler manifold with Norden metric. Some necessary and sufficient conditions are derived for the corresponding curvature tensor to be Kählerian. In the case when this tensor is Kählerian, some relations are obtained between its scalar curvature and the scalar curvature of other curvature tensors. Conditions are given for the considered manifolds to be isotropic-Kähler.
We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.
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.
A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Backlund transformations can be formulated for such equations. It is discussed how Lax pairs and Backlund transformations can be formulated for …
In this paper, we study the Einstein multiply warped products with a semi-symmetric non-metric connection and the multiply warped products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connection and generalized Kasner spacetimes with a semi-symmetric non-metric connection and find some new examples of Einstein affine manifolds and affine manifolds with constant scalar curvature.ß We also consider the multiply warped products with an affine connection with a zero torsi…
In this paper, authors have established Chen's inequalities for the submanifolds of quaternionic Kaehler manifolds characterized by Ricci quarter-symmetric metric connection. Other than these inequalities, we have also derived generalized normalized Casorati curvature inequalities for the same submanifolds.
A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived.
Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.
We introduce a new approach for computing curvature of sub-Riemannian manifolds. Curvature is here meant as symplectic invariants of Jacobi curves of geodesics, as introduced by Zelenko and Li. We describe how they can be expressed using a compatible affine connection and induced tensors, without any restriction on our sub-Riemannian manifold or the choice of connection. In particular, we obtain a universal Bonnet-Myers theorem of Riemannian type. We also give universal formulas for the canonical horizontal frame along each geodesic and an algorithm for computing the curvature in general. Seve…
Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie bracket of vectorfields on the manifold. The motion of a ball rolling on an oriented surface is the parallel transport of a similar connection on the trivial SO(3)-bundle over the surface. If the surface is a plane or a sphere, then the curvature of the connection is a scalar multi…
Curvatures of curves in $\mathbb{R}^{2}$ endowed with a canonical linear connection
研究配备典范线性联络的 $\mathbb{R}^2$ 中平面曲线的几何，引入切向和测地曲率，证明给定曲率曲线的存在唯一性定理，并分类常曲率曲线及推广曲率为弧长线性函数的曲线。
In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors …
Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We give expressions of $m$'s associated connection, and Riemann curvature tensor $R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of $m$ (with respect to the Levi-Civita connection associated with $g$). The formulas turn out to be generalizations of the coordinate expressions. Coordinate expression formulas can be recovered from ours by setting $g$ as the Euclidean metric induced by a given coordinate chart. As the covariant derivative induced by $g$ becomes the ordinary partial derivative and th…
For a subRiemannian manifold and a given Riemannian extension of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizing Tanaka's notion for CR manifolds to the subRiemannian case. Under the assumption of normality, we construct local frames that simplify computations in a manner analogous to Riemannian normal coordinates. We then use these frames to establish Bianchi Identities and symmetries f…
Canonical metric connections with constant holomorphic sectional curvature
本文研究了紧致厄米流形典范度量联络空间形式的Chen-Nie猜想，并证明了该猜想在具有幂零$J$的复零矩阵流形和非平衡Bismut挠平行流形这两类特殊厄米流形上成立。
Let $(M,g,J)$ be a Riemannian manifold with a compatible integrable complex structure $J\in\mathrm{End}(T_\mathbb{R} M)$ and $\mathcal{A}_{g,J}$ be the space of real connections on $T_\mathbb{R} M$ preserving both $g$ and $J$. In this paper, we investigate the relationship between the geometry of real connections in $\mathcal{A}_{g,J}$ and that of Hermitian connections on $T^{1,0}M$. In particular, we study the geometry of the real Chern connection $\nabla^{\mathrm{Ch,\mathrm{R}}}$ on $(M,g,J)$, and obtain Kähler-Einstein metrics by using real Chern-Einstein metrics.
In this paper, we discuss the invariant properties of curvatures effected by the Matsumoto's $C$ or $L$-process. We find equivalent conditions on curvatures by comparing the difference between the corresponding curvatures of closely related connections. As an application, Matsumoto's $L$-process on Randers manifold is studied. Shen connection can not be obtained by using Matsumoto's processes from other well-known connections. This leads us to two new processes which we call Shen's $C$ and $L$-processes. We study the invariant properties of curvatures under the Shen's processes.
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 …
The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The main aim of this work is to obtain properties of manifolds of this class with connections, whose curvature tensors have similar properties as the Kähler tensors in Hermitian geometry.
In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the symplectic connection and its covariant derivatives. Some relations between various more general conditions on the symplectic sectional curvature and the geometry of the symplectic connection or that induced on a symplectic submanifold are explored as well.
In this paper, we systematically compute the Bianchi identities for the canonical connection on an almost Hermitian manifold. Moreover, we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection. As applications of the curvature identities, we obtain some results about the integrability of quasi Kaehler manifolds and some properties of nearly Kaehler manifolds.
Constant sectional curvature surfaces with a semi-symmetric non-metric connection
本文研究赋予典范半对称非度量联络的三维欧氏空间中常截面曲率曲面，分类了母线与联络向量场正交/平行的柱面、轴平行于该场的锥形旋转曲面，并证明截面曲率为1/2时存在与轴正交相交的旋转曲面。
A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvature tensor for the Levi-Civita connection and the torsion tensor is parallel, are obtained.
Hermitian connections with parallel torsion and constant holomorphic sectional curvature
本文研究具有平行挠率和常全纯截面曲率的Hermitian联络，证明非零常数迫使挠率为零，从而度量是Kähler的且局部为复空间形式，并应用于Chern和Bismut联络。
Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold
本文研究几乎厄米特流形上四分之一对称联络的性质，证明保持广义黎曼度量的情形等价于凯勒流形，并构造了与联络生成元无关的曲率张量，揭示其与Weyl射影曲率张量及全纯射影曲率张量的关系。
Classification of translation surfaces in $\mathbb{R}^3$ with constant sectional curvature
本文研究赋予典范半对称非度量联络的欧氏空间中的平移曲面，完成常截面曲率平移曲面的完全分类，证明其为广义柱面，同时指出该框架下存在截面曲率可常可非常的广义柱面，与Levi-Civita联络情形存在差异。
The aim of this paper is to introduce a cosymplectic analouge of conformal connection in a cosymplectic manifold and proved that if cosymplectic manifold M admits a cosymplectic conformal connection which is of zero curvature, then the Bochner curvature tensor of M vanishes.
We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder.
The twistor space \Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h_t compatible with the almost complex structures J_1 and J_2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (\Z,h_t,J_n), n=1,2 and find the geometric conditions on M under which the curvature of its Chern connection D^n is of type (1,1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D^n and show that the Nijenhuis te…
The present study initially identified the generalized symmetric connections $(α,β)$ typed, which can be regarded as more generalised forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained respectively particularly when $(α,β)=(1,0)$ and $(α,β)=(0,1)$ are taken into consideration. Taking that into account, a new generalized symmetric metric connection was attained upon Kenmotsu manifolds. In compliance with the new connection, some results were provided through calculation of tensors belonging to Kenmotsu manifold involving curvature tensor, ri…
A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kaehler manifold of constant holomorphic sectional curvatures.
In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result is based on a tensoriality argument and extends to nonlinear connections on fibre bundles a well-known formula for linear connections on vector bundles.
Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to relate geometric properties of the classifying map to geometric properties of the pulled-back connection. More specifically, we describe conditions on the classifying map under which the pulled-back connection: (1) is fat (in the sphere bundle), (2) has a parallel curvature tenso…
We investigate L-sectional curvature of S-manifolds with respect to the Rieman- nian connection and to certain semi-symmetric metric and non-metric connections naturally related with the structure, obtaining conditions for them to be constant and giving examples of S-manifolds in such conditions. Moreover, we calculate the scalar curvature in all the cases.
Uniqueness of the torsion-curvature pair
本文证明了在光滑流形上，torsion和curvature在满足线性和微分Bianchi恒等式的情况下是唯一的，扩展了对称连接曲率张量的特征化结果。
In this note, we give the correct statements of [2,Proposition 3.3 and Theorem 3.4] and a formula of the Chern curvature in terms of the curvature tensor $R^V$ of the affine connection $\nabla^V$ and the Chern tensor $P$.
For a manifold with an affine connection, we prove formulas which infinitesimally quantify the gap in a certain naturally defined open geodesic quadrilateral associated to a pair of tangent vectors $u$, $v$ at a point of the manifold. We show that the 1st order infinitesimal obstruction to the quadrilateral to close is always zero, the 2nd order infinitesimal obstruction to the quadrilateral to close is $-T(u,v)$ where $T$ is the torsion tensor of the connection, and if $T = 0$ then the 3rd order infinitesimal obstruction to the quadrilateral to close is $(1/2)R(u,v)(u+v)$ in terms of the curv…
It is developed the considerations from (S. M. Minčić, [14, 15]) about curvature tensors and pseudotensors for a non-symmetric affine connection space in this paper. How many kinds of covariant derivatives are enough to be defined for complete researching in the field of non-symmetric affine connection spaces is examined here. This result is interpreted. After that, we searched how many curvature tensors and pseudotensors of a non-symmetric affine connection space are linearly independent. In the next, it is examined is it possible to express all curvature tensors as the linear combinations of…
In the present paper it is considered a class V of 3-dimensional Riemannian manifolds M with a metric g and two affinor tensors q and S. It is defined another metric \bar{g} in M. The local coordinates of all these tensors are circulant matrices. It is found: 1)\ a relation between curvature tensors R and \bar{R} of g and \bar{g}, respectively; 2)\ an identity of the curvature tensor R of g in the case when the curvature tensor \bar{R} vanishes; 3)\ a relation between the sectional curvature of a 2-section of the type \{x, qx\} and the scalar curvature of M.
In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometric inequalities and eigenvalue estimates under a much more general Ricci curvature conditions. The new Ricci curvature condition interpolates between static Ricci tensor and $1$-Bakry-Emery Ricci, and also includes the conformal Ricci as an intermediate case.
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifol…
Discrete Bakry-Émery curvature tensors and matrices of connection graphs
本文提出连接图顶点处Bakry-Émery曲率的重新表述，通过单位正交基的曲率张量表示，并探讨了其在Cartesian积中的扩展应用。
All the connections, pure toward the nilpotent structure, are found. Examples of manifolds, for which the curvature tensor is pure or hybrid, are given. For a manifold of B-type a necessary and sufficient condition for purity of the curvature tensor is proved. It is verified that the conformal change of the metric of a B-manifold does not retain its purity.
Conformal product structures on compact manifolds with constant sectional curvature
本文证明了常数截面曲率且非平坦的紧致流形不具有共形积结构，并进一步展示了这些方法可以自然推广到非正曲率的不可约紧致局部对称空间。
We study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.
Post-Lie Algebra Structure of Manifolds with Constant Curvature and Torsion
本文研究具平行挠率与曲率的仿射联络，证明通过加入完整群无穷小作用对应的自同态，可在向量场空间上构造 post-Lie 代数，并说明其泛李代数及微分方程应用。
Choi-Wang inequality for affine connections
本文扩展了 Choi-Wang 不等式到 Li-Xia 类型仿射连接的正曲率环境，研究最小超曲面的拉普拉斯算子特征值下界。
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…
Positive biorthogonal curvature on $S^2 \times T^2$ via affine connection
本文通过引入带反对称挠率且经非平凡上同调类校准的仿射联络，克服 \\(S^2 \times T^2\\) 上 \\(χ=0\\) 的拓扑约束，在不要求度量相容性的条件下构造出严格正双正交曲率。
The $B$-connection on almost complex manifolds with Norden metric is an analogue of the first canonical connection of Lihnerovich in Hermitian geometry. In the present paper it is considered a $B$-connection in the class of the quasi-Kähler manifold with Norden metric. Some necessary and sufficient conditions are derived for the corresponding curvature tensor to be Kählerian. Curvature properties for this connection are obtained. Conditions are given for the considered manifolds to be isotropic-Kähler.
In this paper, we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean $C^2$-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean $C^2$-smooth curves on surfaces. We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.
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 …
A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of $2$ forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that can be used to decide when a matrix of $2$ forms is equivalent to a skew symmetric matrix. We apply this algorithm to verify whether a "{\it full rank}" curvature connection is locally metric.
On 5-dimensional almost contact B-metric manifolds, the form of any Kähler-type tensor (i.e. a tensor satisfying the properties of the curvature tensor of the Levi-Civita connection in the special class of the parallel structures on the manifold) is determined. The associated 1-forms are derived by the scalar curvatures of the Kähler-type tensor for the canonical connection on the manifolds from the main classes with closed 1-forms.
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