First eigenvalue of the Laplace operator and mean curvature

The main theorem of this paper states a relation between the first nonzero eigenvalue of the Laplace operator and the squared norm of mean curvature in irreducible compact homogeneous manifolds under spatial conditions. This statement has some consequences presented in the remainder of paper. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 1000–1003, July, 2008.