The Pinching Family

In1982, R.Hamilton introduced Ricci flow to study the geometry of compact Riemannian manifold . He found that compact 3-Riemannian manifolds with positiveRicci curvature must be spherical space forms .Recently, C.B(o|¨)hm and B.Wilking defined a new linear operator and constructed a pinching family,they then got a more general result.that's in all dimensions compact Riemannian manifold with 2-positive curvature must be spherical space forms .In this paper we consider that (?) is a closed convexO(n)-invariant cone in the space of algebraic curvature operators,which has following properties: (a) (?) is invariant under the ODE (?)R = R² + R^#; (b) The cone{R∈S_B²(so(n)), R＞0} is in the interior of (?); (c) For any R∈(?) has nonnegative sectional curvature, we can get a new pinching family C(s)_s∈[0,1) with C(0) = (?).For its application.we quote the cone constructed in S.Brendle and R.Schoen's paper,which satisfies all the propertiesabove.