On inhomogeneous spherical space forms

Although much is known about minimal isometric immersions into spheres of homogeneous spherical space forms, up to very recently, [5], there were no results in the literature about such immersions in the dominant case of inhomogeneous space forms. For a large class of these, the pq-space forms, we extend the results of [5] to a necessary and sufficient condition for the existence of such an immersion of a given degree. This condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function. We are thus able to construct the first known minimal isometric immersion of an inhomogeneous lens space into a sphere. Moreover, we give a global lower bound for the degree of a minimal isometric immersion of a pq-space form into a sphere. In doing so, we discover several additional conditions for the existence of such an immersion relating the degree of the immersion and the order of the fundamental group.