We will give necessary
conditions for a compact hypersurface to be totally-geodesic in a manifold of strictly
positive curvatures. These conditions relate the volume of the hypersurface to
the pinching of the manifold. The method consists of obtaining estimates
from above and below for the volume of the manifold. Comparison of these
estimates gives inequalities for the volume of the hypersurface. The method
appears on the surface to apply to totally-geodesic submanifolds of arbitrary
codimension with but a little modification. We are grateful to R. L. Bishop for
pointing out that below the surface are snags that we have not yet been able to
avoid.
