On Instability of Squashed Spheres in the Kaluza-Klein Theory

We study in Kaluza-Klein theories stability of the extra space against "squashing", in other words, the homogeneous deformation. Quantum fluctuations of matter fields at one-loop level are taken into consideration. We calculate the effective potential in models of the type, $M^4\times S^3$ and $M^4\times S^7$. It is found that in the case of scalar matter fields the stability depends on the coupling to the scalar curvature.