Lower bound for Ln/2 curvature norm and its application

We control the number of critical points of a height function arising from the Nash isometric embedding of a compact Riemanniann-manifoldM. The L^n/2 curvature norm ∥R∥ and a similar scalar ∥R∥ are introduced and their integralR(M) andR(M) overM. We prove thatR(M) is bounded below by a constant depending only onn and the Betti numbers ofM. Thus a new sphere theorem is proved by eliminating allith Betti numbers fori = 1, .…n −1. The emphasis is that our sphere theorem imposes no restriction on the range of curvature. Research partially supported by Grant-in-Aid for General Scientific Research, grant no. 07454018.