Perturbations of $C^{1}$ norms and associated isometry groups (Researches on isometries from various viewpoints)

Abstract

This note announces a recent result on isometries of C^{1}-function spaces over compact Riemannian manifolds [7]. We characterize isometries (with respect to certain norms inducing the C^{1}-topology) of the C^{1}-function spaces over compact Riemannian manifolds as generalized weighted composition operators, under some regularity assumption. We also apply the characterization to study continuous deformations of isometry groups induced by perturbations of norms on the function spaces.