Akademik Veri Yönetim Sistemi
Akbulut K., Yıldırım F. (Yürütücü)
TÜBİTAK Projesi, 2022 - 2025
The aim of this study is to investigate the lifts and applications of various geometric objects (complete, vertical, horizontal, etc. lifts of tensor fields) which previously investigated in the tangent bundles, and their applications in the semi-tangent bundles. Tangent bundles theory is a popular subject in Engineering, Physics and especially Differential Geometry and there are many studies on these bundles. Semi-Tangent Bundle which is based on the proposed project is different from Tangent Bundle and defines a pull-back bundles. Within the scope of the project, firstly, almost complex, almost contact and almost para contact structures will be defined in semi-tangent bundles and Lie and covariant derivatives of almost contact and almost para contact structures will be investigated according to the complete, vertical and horizontal lifts of the vector fields. In addition, the application of Tachibana and Vishnevskii operators to the complete, vertical and horizontal lifts of the vector fields according to the almost contact and almost para contact structures in the semi-tangent bundles will be investigated and the Tachibana and Vishnevskii operators for the almost r-contact structure and the Lorentzian almost r-para-contact structure will be examined. Also, using the lifts of projectable linear connections and various geometric objects in semi-tangent bundles and the lifts of Tachibana, Vishnevskii etc., which will make important contributions to the solution of some problems (lift problems of projective linear connections and tensor structures) that have not been solved, Lie and covariant derivatives of almost contact and almost para-contact structures according to the complete, vertical and horizontal lifts of the vector fields in the semi-tangent bundles will be investigated. As theoretical basis, general methods and research techniques in the project, Tangent and semi-tangent bundles geometry, classical tensor analysis (use of indices that is local coordinates) and Covariant differential formalism (global analysis technique) will be used.
The project is planned to be completed in 36 months and it is also planned that the project will consist of four important stages:
(a) Defining the almost complex, almost contact and almost para-contact structures in the semi-tangent bundles,
(b) Investigation of Lie and Covariant derivatives of the almost contact and almost para-contact structures according to the complete, vertical and horizontal lifts of the vector fields in the semi-tangent bundles,
(c) Applying of Tachibana and Vishnevskii Operators in almost contact and almost para contact structures to complete, vertical and horizontal lifts of vector fields in semi-tangent bundles
(d) Examining Tachibana and Vishnevskii operators for almost r-contact and Lorentzian almost r-para-contact structures in semi-tangent bundles.
There are 9 different work packages in the project. In the realization of these work packages, the project coordinator and 4 scholarship students two of whom are Ph.D and two master students will take charge. In addition, at all stages of the work carried out in the project, the knowledge, thought and guidance of a consultant faculty member who is an expert in the subject area will be sought. The consultant faculty member is competent and expert in this field. The project coordinator has successfully completed the TUBITAK-3001 project, which is the basis for the subject.
Upon completion of the project, it is envisaged that 9 articles in the international indexed journals will be published and 9 papers will be presented in international scientific meetings. In addition, it is planned to publish an international book called “Semi-Bundles Theory” for graduate level. Also, 2 doctoral theses and 2 master's theses will be produced from the project. The project will contribute to current issues of semi-cotangent bundles geometry and lift theory. Finally, general results will be obtained for the project outputs and half-bundle theory using the induction method.