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 tangent spaces of M, c.f. [O'Brian-Rawnsley]. We show that this structure is a projective invariant, and when integrable can be obtained from a torsionless connection which must then have W=0. We also show that two torsionless connections define the same almost complex structure if and only if they are projectively equivalent.
A dualistic structure on a smooth Riemaniann manifold $M$ is a triple $(M,g,\nabla)$ with $g$ a Riemaniann metric and $\nabla$ an affine connection, generally assumed to be torsionless. From $g$ and $\nabla$, the dual connection $\nabla^*$ can be defined and the triple $(M, \nabla,\nabla^*)$ is called a statistical manifold, a basic object in information geometry. In this work, we give conditions based on this notion for a manifold to admit an almost contact structure and some related structures: almost contact metric,contact, contact metric, cosymplectic, and coKähler in the three-dimensional…
Let $(M,\nabla,\langle\;,\;\rangle)$ be a manifold endowed with a flat torsionless connection $\nabla$ and a Riemannian metric $\langle\;,\;\rangle$ and $(T^kM)_{k\geq1}$ the sequence of tangent bundles given by $T^kM=T(T^{k-1}M)$ and $T^1M=TM$. We show that, for any $k\geq1$, $T^kM$ carries a Hermitian structure $(J_k,g_k)$ and a flat torsionless connection $\nabla^k$ and when $M$ is a Lie group and $(\nabla,\langle\;,\;\rangle)$ are left invariant there is a Lie group structure on each $T^kM$ such that $(J_k,g_k,\nabla^k)$ are left invariant. It is well-known that $(TM,J_1,g_1)$ is Kähler if…
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.
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.
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.
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 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…
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.
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.
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…
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 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.
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.
Affine deformations serve as basic examples in the continuum mechanics of deformable 3-dimensional bodies (referred as homogeneous deformations). They preserve parallelism and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, the parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time - dependent deformations (motions) after establishing formulas for the variation of the connection, but in the more general context of hypersurfac…
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…
That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible ones and some previous (unpublished) works by the author giving the classi cation for the pseudo-riemannian totally reducible case. One describes those Lie subalgebras through a general structure theorem involving two constructions and some lists. These constructions give new examples of non irreducible totally reducible holonomy algebras and also recover som…
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…
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, or, more generally a directed curve or a frontal curve which has well-defined tangent directions along the curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the c…
The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original li…
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of aff…
The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstein with skew-torsion up to $\mathbb{S}^3$. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemaniann Berger spheres, not Einstein, …
The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.
In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated to the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated to the almost product structure and their curvature of semi-regular almost product warped products are expressed in …
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Elie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkows…
The real form Spin(6,H) in End(R^{32}) of Spin(12,C) in End(C^{32}) is absolutely irreducible and thus satisfies the algebraic identities (40) and (41). Therefore, it also occurs as an exotic holonomy and the associated supermanifold M_g admits a SUSY-invariant polynomial. This real form has been erroneously omitted in our paper. Also, the two real four-dimensional exotic holonomies, whose occurrences were un known at the time of writing, have been shown to exist very recently by R. Bryant.
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 $\mathbb{P}(TM)$.
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.
Dual affine connections on Riemannian manifolds have played a central role in the field of information geometry since their introduction by Amari. Here I would like to extend the notion of dual connections to general vector bundles with an inner product, in the same way as a unitary connection generalizes a metric affine connection, using Cartan decompositions of Lie algebras. This gives a natural geometric interpretation for the Amari tensor, as a "connection form term" which generates dilations, and which is reversed for the dual connections.
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 begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
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 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.
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.
