TUBITAK Project, 2018 - 2020
i. The aim of this project is to study the complete and horizontal lifts of projectable linear connections in the semi-tangent bundle tM. The properties of complete and horizontal lifts of projectable linear connections for semi-tangent bundle tM will also be examined. In addition, infinitesimal linear transformations, will be discussed. In this project the vertical, complete and horizontal lifts of tensor fields of type (0, 2) to semi-tangent bundle and their properties will be found. We will get a new example for good square in this project.
ii. Yano and Ishihara (1967)  defined and studied prolongations called complete, vertical and horizontal lifts of tensor fields and connections. The geometry of the tangent bundle and lifting of tensor fields and connections to tangent bundle was studied by Morimoto (1970) . The differential geometry of cotangent bundle and lifting of the geometric objects to cotangent bundle was studied by Yano and Petterson in 1967 . Then, the vertical, complete, horizontal and diagonal lifting of tensor fields to tangent and cotangent bundle was developped by Yano and Ishihara (1973) . Semi-tangent bundle geometry for the first time came out by the works of Duc (1979) . Vishnevskii in 2002  studied Semi-tangent bundle and its some properties. The complete lifts of derivations to the semi-tangent bundle was investigated by Salimov and Kadıoğlu (2000) . Projectable linear connections in the semi-tangent bundles and their some properties (curvature, torsion tensors and Lie derivatives) were studied by Vishnevskii (2002) and Vishnevskii, Shirokov, Shurygin (1985) [19,20]. Our primary goal in this project, we will continue to study the complete and horizontal lifts of projectable linear connection from differentiable bundle to the semi-tangent (pull-back) bundle initiated by Vishnevskii (2002) . In addition, we will consider infinitesimal linear transformations and lifting problem of projectable linear connections to the semi-tangent bundle. We will also have a new example for good square in this work. The vertical, complete and horizontal lifts of tensor fields of type (0, 2) to the semi-tangent bundle and properties of these lifts will be examined. Also, in this project, we will define a pullback (semi-tangent) bundle tM of tangent bundle TM by using projection (submersion) of the cotangent bundle T*M. Finally, the complete and horizontal lifts of vector and affinor (tensor of type (1,1)) fields will be constructed for pullback (semi-tangent) bundle tM.
This project will take 24 months to complete. Finally, the project is going to be become true in 4 phases:
1. Complete and horizontal lifts of projectable linear connection from differentiable bundle to the semi-tangent bundle, good square and infinitesimal linear transformations,
2. Metrics in the semi-tangent bundle,
3. Horizontal lifts of projectable linear connection from differentiable bundle to the semi-tangent bundle,
4. Complete and horizontal lifts to pullback (semi-tangent) bundle tM of tangent bundle TM by using projection (submersion) of the cotangent bundle T*M.
Subject selection and original value: Semi-tangent bundles are different from tangent bundles and define a pull-back bundle. In this study, complete and horizontal lift of projectable linear connections in pull-back bundles will be analyzed in detail, the connection coefficients will be calculated and various lift problems of the projectable linear connections in semi-tangent bundles will be dealt with. The complete and horizontal lift of projectable linear connections in the semi-tangent bundles will provide great distance in the solution of some problems that have not been solved so far in the semi-bundle theory which will be studied later. The project will make an important contribution to the study of the relationship between lifts (complete and horizontal) of the projectable linear connections in the semi-tangent bundle and the lifts (complete and horizontal) of the projectable linear connections in the semi-cotangent bundle with the transformation of musical isomorphism (from semi-tangent bundle to semi-cotangent bundle). It will also be seen that semi-tangent bundles are concrete examples of good square transformations. The metrics in the tangent bundle are non-degenerate (regular) metrics. It is predicted that the metrics in the semi-tangent bundle are the degenerate (singular) metric. Degenerate metrics in physics and differential geometry have an important place, and many studies have been carried out with these metrics. Therefore, with the new degenerate metrics to be obtained, many studies will be made in the future.
iii. In relation to the project, the following will be used as theoretical basis, general methods and research techniques:
1. Tangent and semi-tangent bundle geometry (Theoretical basis-Tangent and semi-tangent bundles and lifts in these bundles and various operators)
2. Classical tensor analysis (the use of indices, ie. local coordinates)
3. Covariant differentiation formalism (global review technique).
iv. If the project is successful, it is envisaged to publish at least an article in international indexed journals and 3 articles in international peer-reviewed journals. In addition, at least 4 International presentations are planned. This project will create an important content for my book (in English), which is the semi-bundle theory I would like to publish in the future. Project subjects will be completed with a doctoral thesis and a master's thesis. The project will contribute to the semi-cotangent bundle geometry and lift theory, which are current issues. Also in the future, project outputs will provide general results for the semi-bundle theory by using the method of mathematical induction.