We study the rigidity of polyhedral surfaces and the moduli
space of polyhedral surfaces using variational principles. Curvaturelike
quantities for polyhedral surfaces are introduced and are shown
to determine the polyhedral metric up to isometry. The action
functionals in the variational approaches are derived from the cosine
law. They can be considered as 2-dimensional counterparts of
the Schlaefli formula.
Kaimanovich (2003)  introduced the concept of augmented tree on the symbolic space of a selfsimilar
set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009)  that
under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In
the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such
trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected
self-similar sets which has been undergoing some extensive development recently.