In this paper, we study the extension of isometries between the unit spheres of ome Banach spaces E and the spaces C(Ω): We obtain that if the set sm:S1(E) of all smooth points of the unit sphere S1(E) is dense in S1(E), then under some condition, every surjective isometry V0 from S1(E) onto S1(C(Ω)) can be extended o be a real linearly isometric map V of E onto C(Ω): From this result we also obtain ome corollaries. This is the first time to study this problem on the differently typical paces, and the method of proof is also very different too.