  The authors define a class of functions on Riemannian manifolds, which is
called geodesic semilocal E-preinvex functions, as a generalization of geodesic
semilocal E-convex and geodesic semi E-preinvex functions and some of its
properties are established. Furthermore, a nonlinear fractional multiobjective
programming is considered, where the functions involved are geodesic E-$ \eta
$-semidifferentiability, sufficient optimality conditions are obtained. A dual
is formulated and duality results are proved using concepts of geodesic
semilocal E-preinvex functions, geodesic pseudo-semilocal E-prinvex functions
and geodesic quasi-semilocal E-preinvex functions.
