Investigation of Certain Curves and Metallic Structures on Manifolds

Abstract

The title of the present thesis is ``Investigation of certain curves and metallic structures on manifolds'. The main objective of this thesis is to investigate the magnetic curves, slant curves, contact CR-submanifold of a Kenmotsu manifold with Killing tensor field, metallic structures, adapted connections, and differential equations for indicatrices, spacelike and timelike curves. newlineThere are two types of renowned structures on manifolds, namely, almost contact metric structures and almost Hermitian structures. The Kenmotsu, Sasakian and trans-Sasakian are the classes of almost contact metric structures. On the other side, the complex space forms, Kaehler, nearly Kaehler manifolds are particular cases of almost Hermitian manifolds. The thesis is divided into seven chapters and each chapter is further divided into various sections and subsections. newline newlineIn Chapter 1, firstly, we give a historical background of differential Geometry. Secondly, we have discussed two important structures on manifolds, i.e., contact structure and complex structure. Moreover, we give some basic definitions and results: manifold, differentiable manifold, tangent space, vector field, Lie bracket, affine connection, torsion tensor, Riemannian manifold, Levi-Civita connection, curvature tensors, complex manifold, Hermitian manifold, contact manifold, almost contact metric manifold, Cosymplectic manifold, Kenmotsu manifold, and Sasakian Manifold. newline newlineChapter 2 deals with the study of magnetic curves and slant curves in Kenmotsu manifolds. In this chapter, the magnetic trajectories associated with contact magnetic fields have been investigated and classification theorem is proved for the normal magnetic curves. newlineMoreover, a characterization result is obtained for the Frenet curve to be a slant curve. Also, we gave some results on the curvature and torsion. newline newlineIn Chapter 3, we investigate the properties of the contact CR-submanifold with Killing tensor field and obtained some results in Kenmotsu manifolds. Furthermore, we gave some examples in a Kenmot