  Using the Steiner-Weyl expansion formula for parallel manifolds and the so
called gonihedric principle we find a large class of discrete integral
invariants which are defined on simplicial manifolds of various dimensions.
These integral invariants include the discrete version of the Hilbert-Einstein
action found by Regge and alternative actions which are linear with respect to
the size of the manifold. In addition the concept of generalized deficit angles
appear in a natural way and is related to higher order curvature terms. These
angles may be used to introduce various aspects of rigidity in simplicial
quantum gravity.
