Quantum gravity with linear action. Intrinsic rigidity of spacetime

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.