Torsion Free Linear Connection Research Articles

On the cotangent bundle with vertical modified riemannian extensions

Let $M$ be an n-dimensional differentiable manifold with a torsion-free linear connection $\nabla $ which induces on its cotangent bundle ${T^*}M$. The main purpose of the present paper is to study some properties of the vertical modified Riemannian extension on ${T^*}M$ which is given as a new metric in [17]. At first, we investigate a metric connection with torsion on ${T^*}M$. And then, we present the holomorphy properties with respect to a compatible almost complex structure. urthermore, we study locally decomposable Golden pseudo-Riemannian structures on the cotangent bundle endowed with vertical modified Riemannian extension.

Notes on some properties of the natural Riemann extension

Let $(M,\nabla)$ be an $n$-dimensional differentiable manifold with a torsion-free linear connection and $T^{*}M$ its cotangent bundle. In this context we study some properties of the natural Riemann extension (M. Sekizawa (1987), O. Kowalski and M. Sekizawa (2011)) on the cotangent bundle $T^{*}M$. First, we give an alternative definition of the natural Riemann extension with respect to horizontal and vertical lifts. Secondly, we investigate metric connection for the natural Riemann extension. Finally, we present geodesics on the cotangent bundle $T^{*}M$ endowed with the natural Riemann extension.

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Let M be an n-dimensional differentiable manifold equipped with a torsion-free linear connection $$\nabla $$ and $$T^{*}M$$ its cotangent bundle. The present paper aims to study a metric connection $$\widetilde{ \nabla }$$ with nonvanishing torsion on $$T^{*}M$$ with modified Riemannian extension $${}\overline{g}_{\nabla ,c}$$ . First, we give a characterization of fibre-preserving projective vector fields on $$(T^{*}M,{}\overline{g} _{\nabla ,c})$$ with respect to the metric connection $$\widetilde{\nabla }$$ . Secondly, we study conditions for $$(T^{*}M,{}\overline{g}_{\nabla ,c})$$ to be semi-symmetric, Ricci semi-symmetric, $$\widetilde{Z}$$ semi-symmetric or locally conharmonically flat with respect to the metric connection $$ \widetilde{\nabla }$$ . Finally, we present some results concerning the Schouten–Van Kampen connection associated to the Levi-Civita connection $$ \overline{\nabla }$$ of the modified Riemannian extension $$\overline{g} _{\nabla ,c}$$ .

Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M endowed with a torsion free linear connection, ∇ . We introduce the concept of ∇ -integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M induced by complex structures on the generalized tangent bundle of M .

Universal Star Products

One defines the notion of universal deformation quantization: given any manifold M, any Poisson structure Λ on M and any torsionfree linear connection ∇ on M, a universal deformation quantization associates to this data a star product on (M, Λ) given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor Λ, the curvature tensor R and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poissoncohomology.

Relating Thomas–Whitehead Projective Connections by a Gauge Transformation

Thomas-Whitehead projective connections, or TW-connections, are torsionfree linear connections, satisfying certain properties, on a naturally defined principal R-bundle over a manifold. The name credits T. Y. Thomas and J. H. C. Whitehead, who originally studied these connections in the 1920's and 1930's. Three equivalence classes of TW-connections will be considered. This leads to a necessary and sufficient condition for TW-connections to be related by a gauge transformation; namely, they induce the same projective structure on the base manifold, have identical Ricci tensor, and induce the identity element in the one-dimensional de Rham cohomology vector space of the base manifold.

Non-commutative geometry of theh-deformed quantum plane

The $h$-deformed quantum plane is a counterpart of the $q$-deformed one in the set of quantum planes which are covariant under those quantum deformations of GL(2) which admit a central determinant. We have investigated the noncommutative geometry of the $h$-deformed quantum plane. There is a 2-parameter family of torsion-free linear connections, a 1-parameter sub-family of which are compatible with a skew-symmetric non-degenerate bilinear map. The skew-symmetric map resembles a symplectic 2-form and induces a metric. It is also shown that the extended $h$-deformed quantum plane is a noncommutative version of the Poincar\'e half-plane, a surface of constant negative Gaussian curvature.

Examples of Thomas-Whitehead projective connections

Projective connections on a manifold can be represented by Thomas-Whitehead (TW) connections which are just linear connections on a natural R-bundle over the manifold satisfying certain special properties. We show TW-connections with recurrent curvature are flat, and we characterize the TW-connections which are metric connections. Finally, we classify invariant torsion-free linear connections, invariant projective structures, and invariant TW-conections for a homogeneous space.

The projective connections of T.Y. Thomas and J.H.C. Whitehead applied to invariant connections

The purpose of this paper is to define a Thomas-Whitehead projective connection and apply it to the study of invariant connections. If G is a Lie group acting on a manifold M of dimension ⩾ 2 and M admits a G-invariant torsionfree linear connection, it is shown that every G-invariant projective equivalence class on M contains a G-invariant torsionfree linear connection. This generalizes a result obtained by Agaoka and Podestà for the case where M is a reductive homogeneous space.

THE BOCHNER TYPE CURVATURE TENSOR OF PSEUDO CONVEX CR-STRUCTURES

For a pseudo-convex CR-structure on an odd dimensional manifold, we introduce a family of canonical torsion-free linear connections. Every connection in this family is uniquely determined by an almost contact structure associated with the given pseudo convex CR-structure. We study the change of the connections in this family under the gauge transformations and, accordingly, the corresponding change of the gauge tensor fields. The Bochner type curvature tensor field we get is invariant under gauge transformations.

Yano-ledger connection and induced connection on vector bundles

K. Yano and A. J. Ledger [13] const ructed f rom a linear connection V on a manifold B a torsion-free linear connection on T B (called the Yano-Ledger connection). M. Ma t sumoto [7] proved tha t : a) V determines a Finsler connection (75, V) in the space V T B of Finsler vectors; b) the symmetr iza t ion of the extension V ~ of (75, V) to T T B is exactly the Yano-Ledger connect ion on TB; c) the Levi-Civi ta connection of the Riemannian metr ic on T B derived f rom a Riemannian metric g on B coincides with the Yano-Ledger connect ion derived f rom the Levi-Civi ta connection V of g iff the Riemannian curvature tensor of g vanishes (see also [2]). On the other hand R. Miron [8] developed a theory of Finsler connections on vector bundles. The purpose of this paper is to construct a Yano-Ledger connect ion on vector bundles, and then to prove the analogues of Matsumoto ' s theorems a), b), and c) for vector bundles and vector bundle Finsler connections. In our considerations and construct ions we apply pullback of pseudoconnections. They are developed and invest igated in §§1, 2, and 3. §4 yields the Yano-Ledger connect ion for vector bundles, and §§4, 5, and 6 present the ment ioned theorems analogous to those of Matsumoto . Concerning no ta t ion and terminology we refer to the monographs [1],

Conditions on a connection to be a metric connection

It is shown that a torsion free linear connection is determined by a metric of given signature if and only if its holonomy group is a subgroup of the orthogonal group corresponding to the signature.