Legendre and Isotropic Submanifolds in Contact and Riemannian Geometry with Modern Developments

Abstract

The aim of this thesis is to study interconnection among various branches of differential geometry and its application to dynamical systems. At first, we discuss about various branches of differential geometry and study interconnection among these branches. Then some recent development on the application of differential geometry to dynamical system, called Brusselator model, is studied. Organization of the thesis is as follows. Chapter-1 provides some background materials on which the rest of the thesis is based. In this chapter basic definitions and theorems of real and complex manifolds are provided. This chapter is mainly a review. In Chapter-2, a brief review on connections on manifolds and Riemannian manifolds is first of all provided. Given a connection on a manifold we can define geodesic, Riemannian curvature tensor, Ricci tensor, Ricci scalar on it. It should be mentioned here that, hand calculations become extremely tedious to evaluate components of connection, Riemannian curvature tensor, Ricci tensor etc. on higher dimensional manifolds. In this chapter we have developed some computer codes for computing these components. Using computer techniques, the components of connection, Riemannian curvature tensor, Ricci tensor etc. can be computed easily. The work is original. In Chapter-3, interconnections among manifolds with symplectic structure are reviewed. This chapter is mainly a review. But there are some original calculations also. In this chapter we have studied connections of symplectic geometry with the contact geometry, Riemannian geometry and Kähler geometry using existing theorems. In Chapter-4, a review on symplectic geometry and contact geometry with complex manifold is provided. Here we have developed a special comparison between complex symplectic geometry and complex contact geometry.

Chapter-5 is mainly a review on Kodaira, Legendre and isotropic moduli spaces. However, there are some original calculations also. Here we have studied the existence and stability of Kodaira and Legendre moduli spaces and also the existence, completeness and maximality of isotropic moduli spaces. Also, interconnection among Kodaira, Legendre and isotropic moduli spaces is established in this chapter. Chapter-6 is original. It provides the main result. Here we analyze two slow-fast dynamical systems named Brusselator model and Lorenz-Haken model through differential geometry. First, we investigate the temporal and spatiotemporal Brusselator model, respectively and find periodic traveling wave solutions. As a result, we obtain a spot pattern of the model. Then, we investigate the Lorenz-Haken model. Next, we apply an old strategy called the Geometric Singular Perturbation Theory and another newly developed strategy that reflects the applications of differential geometry in the slow-fast dynamical system called the flow curvature method to the two models named as temporal Brusselator model and Lorenz-Haken model. According to the Flow Curvature Method, we determine the curvature of the trajectory curve analytically called flow curvature manifold by estimating the solution or trajectory curve of the dynamical system as a curve in Euclidean space. Since this manifold comprises the time derivatives of the velocity vector field and hence it receives knowledge about the dynamics of the corresponding system. In Model 1 named Brusselator model where we consider the temporal Brusselator model as a two dimensional slow-fast dynamical system. According to the Flow Curvature Method, we determine the flow curvature manifold which directly provides the slow invariant manifold where the Darboux invariance theorem is then used to show the invariance of the slow manifold. On the other hand, since the temporal Brusselator model has no singular approximation and hence, Geometric Singular Perturbation Theory fails to provide the slow invariant manifold associated with temporal Brusselator model. After that, we describe the effect of growth and curvature with surface deformation on pattern formation of the spatiotemporal Brusselator model. In Model 2 named Lorenz-Haken model, we consider as a three dimensional slow-fast dynamical system. By using Flow curvature method, we determined the flow curvature manifold which directly provides the third order approximation of the slow manifold where the Darboux invariance theorem is then used to show the invariance of the slow manifold. Then, we analyze the stability of the fixed point of the L-H model using the flow curvature manifold. On the other hand, since L-H model has singular approximation and it can be considered as a singularly perturbed system. Hence, by using Geometric Singular Perturbation Theory we determine the order by order approximation in the small multiplicative parameter of the slow manifold where the Fenichel’s invariance theorem is used to show the invariance of the slow manifold. After that, we compare the two geometric methods applied to the two slow-fast dynamical systems and highlight the significant results. Finally, some concluding remarks and scope for future work in this direction are given in Chapter-7.