  An earlier proposed theory with linear-gonihedhic action for quantum gravity
is reviewed. One can consider this theory as a "square root" of classical
gravity with a new fundamental constant of dimension one. We demonstrate also,
that the partition function for the discretized version of the Einstein-Hilbert
action found by Regge in 1961 can be represented as a superposition of random
surfaces with Euler character as an action and in the case of linear gravity as
a superposition of three-dimensional manifolds with an action which is
proportional to the total solid angle deficit of these manifolds. This
representation allows to construct the transfer matrix which describes the
propagation of space manifold. We discuss the so called gonihedric principle
which allows to defind a discrete version of high derivative terms in quantum
gravity and to introduce intrinsic rigidity of spacetime. This note is based on
a talk delivered at the II meeting on constrained dynamics and quantum gravity
at Santa Margherita Ligure.
