Suppose that <i></i>₀ is an unknotted simple closed curve contained in the 3-sphere which happens to be invariant under a subgroup <i>G</i> of the Mbius group of <i>S³</i> = the group (generated by inversions in 2-spheres). It is shown that there is an equivariant isotopy <i></i>_<i>t</i>, 0 <i>t</i> 1, from <i></i>₀ to a round circle <i></i>₁.

Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.

We study the spherical cone metrics on surfaces from the point of view of inner angles. A rigidity result is obtained. The existence of spherical cone metric of Delaunay type is also established.