CERTAIN GEOMETRIC ASPECTS OF A CLASS OF ALMOSTCONTACT STRUCTURES ON A SMOOTH METRIC MANIFOLD

Abstract

The classification of Smooth Geometric Manifolds still remains an open problem. Theconcept of almost contact Riemannian manifolds provides neat descriptions and distinc-tions between classes of odd and even dimensional manifolds and their geometries. Amongthe classes that have been extensively studied in the past are the Hermitian, Symplectic,Khalerian, Complex, Contact and Almost Contact manifolds which have applications inM-Theory and supergravity among other areas. The differential geometry of contact andalmost contact manifolds and hence their applications can be studied via certain invariantcomponents: the structure tensors, connections, the metrics and the maps. The study ofalmost contacts 1,2,3-manifolds has been explored before to an extent. However, littleknown is the existence and the geometry of an almost contact 4-structure. In this thesis,we have constructed a class of almost contact structures which is related to almost con-tact 3-structure carried on a smooth Riemannian metric manifold (M,gM) of dimension(5n+ 4): gcd (2,n) = 1. Starting with the almost contact metric manifold (N4n+3,gN)endowed with structure tensors (φi,ξj,ηk) of types (1,1),(1,0),(0,1) respectively, for alli,j,k= 1,2,3, we have showed that there exists an almost contact structure (φ4,ξ4,η4) on(N4n+3⊗Rd)≈M5n+4;gcd(4,d) = 1 andd|(2n+ 1) constructed as a linear combinationsof the first three structures on (N4n+3,gN). We have studied the geometric properties ofthe tensors of the constructed almost contact structure, the properties of the characteristicvector fields of the two manifoldsM5n+4andN4n+3and the relationship between them viaanα-rotated submersion Π : (N4n+3⊗Rd)↪→(N4n+3) and the metricsgMrespectivegN.This provides new forms of Gauss-Weingartens’ equations, Gauss-Codazzi equations andthe Ricci equations incorporating the submersion other than the First and Second Funda-mental coefficients only. We have observed that the almost contact structure (φ4,ξ4,η4)is constructible if and only if it is carried on the hidden compartment of the manifoldM5n+4∼=(N4n+3⊗Rd) which is related to the manifoldN4n+3. The results of this studyestablish a strong basis upon which the study of almost contact structures can be extendedto more than 4-structures. Moreover, the fact that the vector field{ξi:i= 1,···,4}ob-tained is killing gives rise to integral geodesic curves which allow for smooth interpolationbetween two high-dimensional points with application in computer vision where smoothanimations can be constructed by travelling along the geodesics between two images. Thesemanifolds can thus be applied in the exploration of M-theory and supergravity.