Executive Summary : | Geometry, derived from the Greek words "Geo" meaning Earth and "Metry" meaning measurement, is a branch of mathematics that studies the geometry of curves, surfaces, and higher-dimensional smooth manifolds. It employs differential and integral calculus ideas, linear and multilinear algebras, topology, and differential equations. Riemannian geometry studies Riemannian manifolds, which are smooth manifolds with a Riemannian metric. Contact geometry studies a geometric structure on smooth manifolds that satisfies a condition known as "complete non-integrability." This field has broad applications in physics, such as geometrical optics, classical mechanics, thermodynamics, geometric quantization, super gravity, string theory, and applied mathematics. The primary objective of this Ph.D. research is to study odd dimensional differentiable manifolds, their properties, and their generalizations. The research will also explore submanifolds, hypersurfaces, curvature tensors, submersion, properties, and applications of odd dimensional differentiable manifolds. |