SUB-WEYL GEOMETRY AND ITS LINEAR CONNECTIONS

Starting from the notion of conformal metrical structure in the tangent bundle, given by R. Miron and M. Anastasiei in [10], [11], we define the notion of conformal metrical d-linear connection with respect to a conformal metrical structure corresponding to the 1-forms ω andω in T M. We determine all conformal metrical d-linear connections in the case when the nonlinear connection is arbitrary and we give important particular cases. Further, we find the transformation group of these connections. We study the role of the torsion tensor fields T and S in this theory, especially the semi-symmetric d-linear connections, and the group of transformations of semi-symmetric conformal metrical d-linear connections, having the same nonlinear connection N and its important invariants.

Facta Universitatis, Series: Mathematics and Informatics, 2021

The differential geometry of the tangent bundle is an effective domain of differential geometry which reveals many new problems in the study of modern differential geometry. The generalization of connection on any manifold to its tangent bundle is an application of differential geometry. Recently a new type of semi-symmetric non-metric connection on a Riemannian manifold has been studied and a relationship between Levi-Civita connection and semi-symmetric non-metric connection has been established. The various properties of a Riemannian manifold with relation to such connection have also been discussed. The present paper aims to study the tangent bundle of a new type of semi-symmetric non-metric connection on a Riemannian manifold. The necessary and sufficient conditions for projectively invariant curvature tensors corresponding to such connection are proved and show many basic results on the Riemannian manifold in the tangent bundle. Furthermore, the properties of group manifolds of the Riemannian manifolds with respect to the semi-symmetric non-metric connection in the tangent bundle have been studied. Moreover, theorems on the symmetry property of Ricci tensor and Ricci soliton in the tangent bundle are established.

arXiv (Cornell University), 2011

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

Cambridge University Press eBooks, 2013

We provide invariant formulas for the Euler-Lagrange equation associated to sub-Riemannian geodesics. They use the concept of curvature and horizontal connection introduced and studied in the paper.

Journal of Mathematical Physics, 2005

On a (pseudo-) Riemannian manifold of dimension n 3, the space of tensors which transform covariantly under Weyl rescalings of the metric is built. This construction is related to a Weyl-covariant operator D whose commutator [D, D] gives the conformally invariant Weyl tensor plus the Cotton tensor. So-called generalized connections and their transformation laws under diffeomorphisms and Weyl rescalings are also derived. These results are obtained by application of BRST techniques.

Balkan J. Geom. Appl, 2008

We give an algebraic characterization of the case when conformal Weyl and conformal Lyra connections have the same curvature tensor. It is determined a (1,3)-tensor field invariant to certain transformation of semi-symmetric connections, compatible with Weyl structures on conformal manifolds. It is studied the case when this tensor is vanishing.

Applied Mathematics Letters, 2011

In the papers [19], [20] several Ricci type identities are obtained by using non-symmetric affine connection. In these identities appear 12 curvature tensors, 5 of which being independent [21], while the rest can be expressed as linear combinations of the others. In the general case of a geodesic mapping f of two non-symmetric affine connection spaces GAN and GAN it is impossible to obtain a generalization of the Weyl projective curvature tensor. In the present paper we study the case when GAN and GAN have the same torsion in corresponding points. Such a mapping we name "equitorsion mapping". With respect to each of mentioned above curvature tensors we have obtained quantities E θ i jmn (θ = 1, • • • , 5), that are generalizations of the Weyl tensor, i.e. they are invariants based on f. Among E θ only E 5 is a tensor. All these quantities are interesting in constructions of new mathematical and physical structures.

2000

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.

2007

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.

Arxiv preprint gr-qc/0508023, 2005

The researches resulting in this massive book have been initiated by S. Vacaru fifteen years ago when he prepared a second Ph. Thesis in Mathematical Physics. Studying Finsler-Lagrange geometries he became aware of the potential applications of these geometries in exploring nonlinear aspects and nontrivial symmetries arising in various models of gravity, classical and quantum field theory and geometric mechanics.

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.

2014

Conformal Riemannian P-manifolds with connections whose curvature tensors are Riemannian P-tensors Dobrinka Gribacheva & Dimitar Mekerov 1 2 3 Your article is protected by copyright and all rights are held exclusively by Springer Basel. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".