Multiply Warped Products With A Semi-symmetric And Quarter-symmetric Connection

In this paper,we study connection,curvature and Killing vector field on multiply warped products with a semi-symmetric and quarter-symmetric connection.We also study Einstein and quasi-Einstein multiply warped products with a semi-symmetric and quarter-symmetric connection.This paper is divided into four parts.The first part introduces the background of Riemannian manifold.It includes Riemannian manifold,warped product,multiply warped product,operators on multiply warped product and so on.In the second part,we study Killing vector fields on multiply warped products with a semi-symmetric metric connection.We define a semi-symmetric metric Killing vector field,then study semi-symmetric metric Killing vector fields on warped and multiply warped products with a semi-symmetric metric connection.We also study Killing and 2-Killing vector fields on multiply warped products.In the third part,we study quasi-Einstein warped products with a semi-symmetric non-metric connection.We give the expressions of the Ricci tensors and scalar curvatures for the bases and fibres.In some cases we give some obstructions to the existence of the quasi-Einstein warped products with a semisymmetric non-metric connection.In the fourth part,we study the Einstein warped products and multiply warped products with a quartersymmetric connection.We also study warped products and multiply warped products with a quartersymmetric connection with constant scalar curvature.Then apply our results to generalized RobertsonWalker spacetimes with a quarter-symmetric connection and generalized Kasner space-times with a quartersymmetric connection.