TUBITAK Project, 2018 - 2020
Project Summary
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) [24] 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) [10]. The
differential geometry of cotangent bundle and lifting of the geometric
objects to cotangent bundle was studied by Yano and Petterson in 1967 [27].
Then, the vertical, complete, horizontal and diagonal lifting of tensor fields
to tangent and cotangent bundle was developped by Yano and Ishihara (1973) [23].
Semi-tangent bundle geometry for the first time came out by the works of Duc
(1979) [3].
Vishnevskii in 2002 [20] 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) [15]. 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) [20]. 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. |