Effective action in multidimensional (Super)gravities

In the present paper we have obtained unique effective action ford=5 gravity on the backgroundM₄×s₁, whereM₄ is the Minkowski space,S₁ is a one-sphere. The effective action is calculated forN=1d=11 supergravity on the backgroundR_nxT_11–n whereR_n is then-dimensional curved space, T_d is thed-dimensional torus. A mechanism of inducing the Einstein gravity with zero--term is proposed.We thank A. O. Barvinsky, E. S. Fradkin, I. V. Tyutin and A. A. Tseytlin for helpful discussions. We are grateful to L. G. Fonova for technical assistance.