Bozhidar Z. Iliev, Sawa S. Manoff
Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base. This is done via imposing certain conditions on the Lie derivatives with respect to the tangential vector of the basic trajectory.
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We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalised connection.
The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO(3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.
A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mechanics are provided.
The classification of general affine connections in Newton--Cartan geometry: Towards metric-affine Newton--Cartan gravity
本文针对伽利略流形上的一般仿射联络，基于可独立指定张量场完成完整分类，推广了已有伽利略联络相关结论，明确了非度量相容联络的自由度来源。
The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimens…
Special Affine Connections on Symmetric Spaces
该研究针对对称空间上的特殊仿射联络，证明其与对应李代数分量上的特殊乘积一一对应，引入强半单对称对并证明其上典范联络是唯一特殊仿射联络，还计算了特殊仿射联络的和乐李代数。
For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant derivatives along MVFs which are not induced by affine connections on $TM$. We call this more general class of covariant derivatives \textit{higher affine connections}. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obta…
We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.
We develop the method of anholonomic frames with associated nonlinear connection (in brief, N--connection) structure and show explicitly how geometries with local anisotropy (various type of Finsler--Lagrange--Cartan--Hamilton geometry) can be modeled in the metric--affine spaces. There are formulated the criteria when such generalized Finsler metrics are effectively induced in the Einstein, teleparallel, Riemann--Cartan and metric--affine gravity. We argue that every generic off--diagonal metric (which can not be diagonalized by coordinate transforms) is related to specific N--connection conf…
On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the α-divergence, for αin (-1,1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the α-divergence belongs to the class of quasi-entropies, defined by Petz.
On the Finsler variational nature of autoparallels in metric-affine geometry
本文研究了无 torsion 仿射连接在向量非度量性下的自平行线变分性质，证明了其可由Finsler拉格朗日量描述，从而解决自平行线的变分性问题。
We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its $Γ$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $Γ$-variation of the functional respects a certain symmetry then the functional is …
We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi-Civita connection of some Riemannian metric. We show for the first time that this result does not extend to Finsler spacetimes. More precisely, we find a large class of Berwald spacet…
The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.
Chern-Simons field theory on the general affine group, $3d$-gravity and the extension of Cartan connections
该研究从几何力学角度探究结构群为一般仿射群时3d引力与Chern-Simons场论的对应，通过推广Cartan联络下的联络延拓约化概念，建立了主纤维丛上Chern-Simons型变分问题与常规理论的关联。
Commutation formulae with respect to a non-symmetric affine connection are obtained in this paper. The components of commutation formulae in this paper are covariant derivatives of tensors with respect to symmetric and non-symmetric affine connection.
In this paper we prove that all manifolds with affine connection are globally projectively equivalent to some space with equiaffine connection (equiaffine manifold). These manifolds are characterised by a symmetric Ricci tensor.
This paper is a review of the twistor theory of irreducible G-structures and affine connections. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections, while the situation with holonomies of non-metric torsion-free affine connections was and remains rather unclear. One of the results discussed in this review asserts that any torsion-free holomorphic affine connection with irreducibly acting holonomy group can, in principle, be constructed by twistor methods. Another…
In this paper, we introduce a new structure, namely, affine Szabó connection. We prove that, on $2$-dimensional affine manifolds, the affine Szabó structure is equivalent to one of the cyclic parallelism of the Ricci tensor. A characterization for locally homogeneous affine Szabó surface is obtained. Examples of two- and three-dimensional affine Szabó manifolds are also given.
We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.
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.
The main purpose of present paper is to study the affine connection induced from the horizontal lift on the cross-section determined by a vector field in Mn with respect to the adapte frame of .
In this paper, we compute the Bott connections and their curvature on three-dimensional Lorentzian Lie groups with three different distributions, and we classify affine Ricci solitons associated to the Bott connection on three-dimensional Lorentzian Lie groups with three different distributions.
We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane.
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 $P$.
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, we investigate the existence of a subclass of quotients of affine connection control systems, which preserve the mechanical structures. Both local and global sufficient and necessary conditions are given for the geodesically accessible affine connection control systems such that they can admit this subclass of quotients. The structural properties of the quotient map and the quotient mechanical control system are discussed.
In this paper, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated to Bott connections, canonical connections and Kobayashi-Nomizu connections are Codazzi tensors associated to these connections. We also classify three-dimensional Lorentzian Lie group with the quasi-statistical structure associated to Bott connections, canonical connections and Kobayashi-Nomizu connections.
In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.
Let $M$ be a complex surface. We show that there is a one-to-one correspondence between torsion-free affine connections on $M$ and Riccati distributions on $\mathbb{P}(TM)$. Furthermore, if $M$ is compact, then this correspondence induces a one-to-one correspondence between affine structures on $M$ and Riccati foliations on $\mathbb{P}(TM)$.
We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.
The goal of this paper is to introduce the lifting theory that has an important role in geometry. Therefore, using the lifts of differential geometric structures we show that tangent bundle TM of paracomplex manifold M admits para-complex torsion-free affine connection.
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen's inequalities for non integrable distributions of real space forms endowed with a semi-symmetric metric connection and a semi-symmetric non-metric connection. We give so…
We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.
We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.
We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of $2$-jets of the geodesic symmetries. Conversely, any connection is described in terms of the geodesic symmetries by a simple formula involving only the Lie bracket of vector fields. We then formulate, in terms of the symmetry jet, several aspects of the theory of affine connections and obtain geometric and intrinsic descript…
We study the Schouten-van Kampen connection associated to an almost contact or paracontact metric structure. With the help of such a connection, some classes of almost (para) contact metric manifolds are characterized. Certain curvature properties of this connection are found.
We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborh…
In this paper, we define deformed Schouten-Van Kampen connections which are metric connections and compute sub-Riemannian limits of Gaussian curvature for a Euclidean C2-smooth surface associated to deformed Schouten-Van Kampen connections with two kinds of distributions in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. According to above results, we get Gauss-Bonnet theorems associated to two kinds of deformed Schouten-Van Kampen connections in the affine group and…
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。
A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.
Affine connections and second-order affine structures
本文为光滑流形对角线的高阶邻域构造了自然的无穷小结构，并证明任意对称仿射联络均可扩展为二阶无穷小仿射结构。
In this paper, we study the quasi-Einstein and generalized quasi-Einstein warped products with a semi-symmetric non-metric connection. We give the expressions of the Ricci tensors and scalar curvatures for the bases and fibres. In some cases we give some obstructions to the existence of the quasi-Einstein and generalized quasi-Einstein warped products with a semi-symmetric non-metric connection.
In this paper, we generalize the results in [Y. Wang: Affine connections on singular warped products. Int. J. Geom. Methods Mod. Phys. 18(5), 2150076, (2021).] to singular multiply warped products and singular twisted products. We study singular multiply warped products and singular twisted products and their curvature with the semi-symmetric metric connection and the semi-symmetric non-metric connection. We also discuss Koszul forms associated with the almost product structure and their curvature of singular multiply warped products and singular twisted products. Finally, several examples are…
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connection on a Lie group. In the last case, we show the existence of a Lie group endowed with a flat affine bi-invariant connection whose Lie algebra contains the Lie algebra of complete infinitesimal affine transformations of the given Lie group. We also prove some results about flat a…
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.
On the classification of Meromorphic Affine Connections on Complex Compact Surfaces
本研究分类了代数维数为1的紧复曲面上的亚纯仿射联络，拓展了前人在全纯情形下的相关工作，旨在探索复紧曲线单值化原理在高维带奇点几何结构中的拓展可能。
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…
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…
Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.
In this paper we consider existence and multiplicity results concerning affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs, following the regularity theory introduced in arXiv:1908.04442. We show that if $M$ admits a $B_{α,β}^{k}$-structure, then the existence of such regular connections can be established in terms of properties of the structural presheaf $B$. In other words, we propose a solution to the existence problem in this setting. With regard to the multiplicity problem, we show that the space of regular affine connections is an affine space of t…
We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin--Novikov type for a given one--dimensional system of hydrodynamic type. We give several examples including Zoll connections, and Hamiltonian systems arising from two--dimensional Frobenius manifolds.
We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open subset $Ω\subset M$ associated to any vector field $V$ on $Ω$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$. Then we show some properties of the curvature tensor $R^V$ associated to $\nabla^V$ and in particular we prove that the Jacobi operator of $R^V$ along a geodesic coincides with the one given by the Chern curvature.
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.
Affine connections on the algebra of differential forms
本文在微分形式代数上定义并研究了半对称度量联络，计算了其曲率与里奇张量，推导了分布的Gauss-Codazzi-Ricci方程，并定义了典范、Schouten和Vrancreanu联络及其性质。
Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces
本文提出用仿射联络和协变导数研究强黏性摩擦力实现非完整约束的奇异摄动问题，建立慢流形不变性条件并给出递归近似方法，推导滑移速度与近似动力学，并以垂直滚动圆盘验证。
We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if $\nabla$ is a generic connection on a principal elliptic bundle over a Riemann surface of genus $g \geq 2$, with odd first Betti number. In the last case, the local Killing Lie algebra is of dimension one, generated by the fundamental vector field of the principal fibration.
We give a combinatorial/geometric argument of the classical result that an affine connection, which is both torsion free and curvature free, is locally an affine space.
We will introduce formal frames of manifolds, which are a generalization of ordinary frames. Their fundamental properties are discussed. In particular, canonical forms are introduced, and torsions are defined in terms of them as a generalization of the structural equations. It will be shown that the vanishing of torsions are equivalent to the realizability of given formal frames as ordinary frames. We will also discuss deformations of linear connections on tangent bundles. An application to deformations of foliations are then given.
Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric Spaces
本文提出一种在仿射连接空间中具有三级精度的平行运输方案，该方案在仿射对称空间中精确，且在黎曼对称流形中几乎全局精确。
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