It is conjectured that there exist states of the physical continuum which include regions with different signatures of the metric and that the observed Universe and an infinite number of other Universes arose as a result of quantum transitions with a change in the signature of the metric. The Lagrangian in such a theory must satisfy conditions of non-negativity in the regions with even signature. Signature here means the number of time coordinates. The induced gravitational Lagrangian in a conformally invariant theory of Kaluza--Klein type evidently satisfies this requirement and leads to effective equations of the gravitational theory of macroscopic space identical to the equations of the general theory of relativity. It is suggested that in our Universe there exist in addition to the observable (macroscopic) time dimension two or some other even number of compactified time dimensions. It is suggested that the formation of a Euclidean region in the center of a black hole or in the cosmological contraction of the Universe (if it is predetermined by the dynamics) is a possible outcome of gravitational collapse.