Special multiply Einstein warped products with an affine connection

The aim of this paper is to study special multiply Einstein warped products having an affine connection. Let M = I x (f1) F-1 x ... x (fm) F-m be a multiply warped product such that I subset of R is an open interval, dim I = 1, f(i) : I -> (0,infinity), f(i) is an element of C-infinity(I), dim F-i = k(i) >= 1 for every i is an element of {1,...,m}, m >= 1, dim M = (n) over bar = 1+ Sigma(m)(i=1) k(i) and (del) over bar an affine connection on M. We compute the warping functions that make M an Einstein space in the following cases: (a) (del) over bar is a semi-symmetric metric/non-metric connection and all the fibers are Ricci flat. (b) (del) over bar is a quarter-symmetric metric/non-metric connection and all the fibers are Ricci flat.