LOS ANGELES (CNN) -- Actor Arnold Schwarzenegger will not run for California's governor as part of the October vote on whether to recall incumbent Gov. Gray Davis, state Republican sources said Wednesday.Posted by Greg RansomSchwarzenegger is expected to back Richard Riordan, a former Los Angeles mayor who unsuccessfully sought the GOP gubernatorial nomination in 2002. Riordan said yesterday that he would not run if the star of the Terminator movies, who turned 56 on Wednesday, entered the race.