Orientifolds of matrix theory and noncommutative geometry

We study explicit solutions for orientifolds of matrix theory compactified on a noncommutative torus. As quotients of the torus, a cylinder, Klein bottle, and Mobius strip are applicable as orientifolds. We calculate the solutions using the Connes-Douglas-Schwarz projective module solution, and investigate the twisted gauge bundle on quotient spaces as well. These solutions are of Yang-Mills theory on a noncommutative torus with proper boundary conditions which define the geometry of the dual space. [S0556-2821(99)02512-6].