The thesis is devoted to the study of metrics of bundles, their properties, and their generalizations to a broader class of metrics that lead to different geometries in the fibers and the entire tangent bundle.
The standard metric on the tangent bundle is the Sasaki metric, whose geometric properties are well known. However, in general, the Sasaki metric possesses only the properties of a general Riemannian metric since, on the fibers, it coincides with the metric of the base manifold. Nevertheless, one can generalize the Sasaki metric definition allowing the fiber metric to be different from the base one. The main question, which has been and continues to be studied by many renowned mathematicians, is: to what extent the new fiber-wise metric changes the geometry of the tangent bundle?
The aim of the study is to identify the dependence of the geometric properties of general vector bundles on the metric of the bundle, as well as to establish the relationship between the properties of the base manifold and the fibers of a general bundle. Additionally, the study explores the geometric properties of sections of a general bundle for different types of generalized metrics.
The object of the study is the geometric properties of tangent bundles of Riemannian manifolds and their sections with various deformations of the Sasaki metric.
The subjects of the study are generalizations of the Sasaki metric, geodesic lines of the tangent bundle, properties of the Sasakian manifold, as well as harmonic, minimal, and totally geodesic vector fields.
The objectives of the study are:
1) generalization of the cigar soliton metric to the tangent bundle;
2) investigation of geodesic lines in the tangent bundle with a fiberwise cigar soliton metric;
3) study of harmonic unit vector fields on the unit tangent bundle with a twisted Sasaki metric;
4) classification of left-invariant harmonic unit vector fields that define harmonic maps in the case of a vertical rescaled metric on the unit tangent bundle;
5) derivation of the expression for the second fundamental form of a submanifold defined by a unit vector field in the unit tangent bundle of a Riemannian manifold with a g-natural metric;
6) determination of the conditions under which a map defined by a unit vector field in the unit tangent bundle of a Riemannian manifold with a g-natural metric can be totally geodesic.
The study employs methods of differential geometry, Riemannian geometry, differential equations, mathematical analysis, linear algebra, group theory, and Lie algebras.
Section 1 of the thesis is devoted to fundamental concepts in the geometry of bundles, as well as a literature review. It provides basic information on the geometry of the tangent bundle and the unit tangent bundle, contact manifolds, the Sasaki metric, and vector fields on a Riemannian manifold as maps and submanifolds. An overview of concepts such as harmonicity, minimality, and totally geodesicity of vector fields and maps is also given.
Section 2 of the thesis is devoted to the generalization of Richard Hamilton's cigar soliton metric as a deformation of the Sasaki metric on the tangent bundle of a Riemannian manifold. In the thesis, for the first time, a generalization of the Hamiltonian cigar soliton metric, which is a two-dimensional manifold, was studied as a metric on the tangent bundle. The concept of "fiberwise cigar soliton deformation of the Sasaki metric" (or "fiberwise Hamiltonian cigar soliton metric") was introduced for the first time, and the geodesic lines of the tangent bundle with this metric were investigated.
Section 3 of the thesis is devoted to the study of harmonic unit vector fields on the unit tangent bundle with the twisted Sasaki metric. In the thesis, for the first time, deformations preserving the existence of harmonic left-invariant unit vector fields on three-dimensional unimodular Lie groups with a left-invariant metric and harmonic maps defined by a unit vector field in the case of the twisted Sasaki metric on the unit tangent bundle were described. Additionally, such vector fields and maps were classified for the first time in the case of the vertical rescaled metric.
Section 4 of the thesis is devoted to studying the case when a non-parallel unit vector field on a Riemannian manifold defines an isometric immersion with the Riemannian g-natural metric on the unit tangent bundle. In the thesis, the total geodesicity of non-parallel unit vector fields defining an isometric embedding on a K-contact metric manifold with a g-natural metric on the unit tangent bundle was studied for the first time. Additionally, for the first time, an expression for the second fundamental form of the map defined by a unit vector field on a Riemannian manifold with a g-natural metric on the unit tangent bundle was obtained.
The practical significance of the obtained results lies in complementing the existing findings in the geometry of bundles.
