On the Betti and Tachibana Numbers of Compact Einstein Manifolds

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold ( M , g ) with an Einstein constant α > 0 is a homological sphere when the minimum of its sectional curvatures > α / ( n + 2 ) ; in particular, ( M , g ) is a spherical space form when the minimum of its sectional curvatures > α / n . Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with α < 0 .

Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

A note on a three-dimensional sphere theorem with integral curvature condition

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold [Formula: see text] with constant positive scalar curvature, if a certain combination of [Formula: see text] norm of the Ricci curvature and [Formula: see text] norm of the scalar curvature is positive, then [Formula: see text] is diffeomorphic to a spherical space form.

The curvature and topological properties of hypersurfaces with constant scalar curvature

In this paper, we consider n (n ≥ 3)-dimensional compact oriented connected hypersurfaces with constant scalar curvature n(n − 1)r in the unit sphere Sn+1(1). We prove that, if r ≥ (n − 2)/(n − 1) and S ≤ (n − 1)(n(r − 1) + 2)/(n − 2) + (n − 2)/(n(r − 1) + 2), then either M is diffeomorphic to a spherical space form if n = 3; or M is homeomorphic to a sphere if n ≥ 4; or M is isometric to the Riemannian product , where c2 = (n − 2)/(nr) and S is the squared norm of the second fundamental form of M.

Metrics of Positive Scalar Curvature on Spherical Space Forms

We use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.