Certain Geometric Aspects Of A Class Of Almostcontact Structures On A Smooth Metric Manifold

ABSTRACT

The classication of Smooth Geometric Manifolds still remains an open problem. The

concept of almost contact Riemannian manifolds provides neat descriptions and distinctions

between classes of odd and even dimensional manifolds and their geometries. Among

the classes that have been extensively studied in the past are the Hermitian, Symplectic,

Khalerian, Complex, Contact and Almost Contact manifolds which have applications in

M-Theory and supergravity among other areas. The dierential geometry of contact and

almost contact manifolds and hence their applications can be studied via certain invariant

components: the structure tensors, connections, the metrics and the maps. The study of

almost contacts 1; 2; 3-manifolds has been explored before to an extent. However, little

known 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 contact

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 (i; j ; k) of types (1,1),(1,0),(0,1) respectively, for all

i; j; k = 1; 2; 3, we have showed that there exists an almost contact structure (4; 4; 4) on

(N4n+3 

 Rd) M5n+4; gcd(4; d) = 1 and dj(2n + 1) constructed as a linear combinations

of the rst three structures on (N4n+3; gN). We have studied the geometric properties of

the tensors of the constructed almost contact structure, the properties of the characteristic

vector elds of the two manifolds M5n+4 and N4n+3 and the relationship between them via

an -rotated submersion : (N4n+3 

 Rd) ,! (N4n+3) and the metrics gM respective gN.

This provides new forms of Gauss-Weingartens' equations, Gauss-Codazzi equations and

the Ricci equations incorporating the submersion other than the First and Second Fundamental

coecients only. We have observed that the almost contact structure (4; 4; 4)

is constructible if and only if it is carried on the hidden compartment of the manifold

M5n+4 = (N4n+3 

 Rd) which is related to the manifold N4n+3. The results of this study

establish a strong basis upon which the study of almost contact structures can be extended

to more than 4-structures. Moreover, the fact that the vector eld fi : i = 1; ; 4g obtained

is killing gives rise to integral geodesic curves which allow for smooth interpolation

between two high-dimensional points with application in computer vision where smooth

animations can be constructed by travelling along the geodesics between two images. These

manifolds can thus be applied in the exploration of M-theory and supergravity.