Ricci deformation of the metric on a Riemannian manifold

The interaction between algebraic properties of the curvature tensor and the global topology and geometry of a Riemannian manifold has been studied extensively.Of particular interest is the question under which conditions on its curvature tensor a Riemannian manifold is homeomorphic or diffeomorphic to a space of constant positive sectional curvature.A Riemannian manifold M with sectional curvature K is said to be δ-pinched if δ < K < 1 holds globally on M. The famous sphere theorem then states that a complete, simply connected \ -pinched manifold is homeomorphic to the standard sphere [1], [6], [7].It is also known that the homeomorphism theorem can be sharpened to a diffeomorphism theorem, if a more restrictive pinching condition is imposed, [3], [8].Recently, Ruh [9] was able to show with the help of the Calderon-Zygmund inequalities, that the global pinching condition can be weakened to a local one: If the curvature ratios of a compact Riemannian manifold of positive sectional curvature are close to one, then the manifold is diffeomorphic to a spherical space form.In this paper we use the heat flow method developed by Hamilton in [4] to give a new proof of Ruh's result and to obtain a more precise pointwise condition for the curvature tensor which ensures the existence of a diffeomorphism to a spherical space form.In [4] Hamilton showed that on a threedimensional manifold of strictly positive Ricci curvature the metric can be deformed into a metric of constant positive curvature.We show that this heat flow method works for any dimension n > 4, provided the norm of the Weyl conformal curvature tensor and the norm of the traceless Ricci tensor are not too large compared to the scalar curvature at each point.