Fractional differential calculus on riemannian manifolds and convexity problems

A Riemannian manifold embodies differential geometry science. Moreover, it has many important applications in physics and some other branches of sciences. Based on the above perspectives, the present thesis focuses on the study of some new results relating concept geodesic-ray property, convexity and star shapedness in complete simply connected smooth Riemannian manifold without conjugate points. In addition, the above terms are studied in the Cartesian product of two complete simply connected smooth Riemannian manifolds without conjugate points. Furthermore, this thesis introduces the concept of geodesic strongly E-convex functions and geodesic E-b-vex functions and discusses some of their properties. Moreover, examples in nonlinear programming problems are used to illustrate the applications of the results. Fractional calculus is a field of mathematics study that grows out of the traditional definitions of calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. This thesis shows some results related to fractional Riemannian manifolds such as fractional connection, Torsiontensor of a fractional connection and difference tensor of two fractional connections. Moreover, area and volume on fractional differentiable manifolds are studied. In conclusion, some new integral inequalities of generalized Hermite Hadamard’s type integral inequalities for generalizeds-convex functions in the second sense on fractal sets are discussed. In addition, a new class of generalizeds-convex functions in both senses on real linear fractal sets is defined. The definition of generalizeds-convex functions in both senses on the co-ordinates on fractal sets and some of its properties are studied. Some new inequalities for product of generalizeds-convex functions on the co-ordinates on fractal sets are presented.

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