Multiply twisted products

In this paper, we compute the index form of the multiply twisted products. We study the Killing vector fields on the multiply twisted product manifolds and determine the Killing vector fields in some cases. We compute the curvature of the multiply twisted products with a semi-symmetric metric connection and show that the mixed Ricci-flat multiply twisted products with a semi-symmetric metric connection can be expressed as multiply warped products. We also study the Einstein multiply warped products with a semi-symmetric metric connection and the multiply warped products with a semi-symmetric metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric metric connection and generalized Kasner spacetimes with a semi-symmetric metric connection and find some new examples of Einstein affine manifolds and affine manifolds with constant scalar curvature. We also consider the multiply twisted product Finsler manifolds and we get some interesting properties of these spaces.