Closed manifolds with small excess. (Q5950193)

The excess \(e(M)\) of a metric space \(M\) is defined as some measure of the deviation in the triangle inequality from an equality as \[ e(M)=\inf_{p,q\in M} \left( \sup_{x\in M} d(p,x)+d(q,x)-d(p,q) \right). \] The authors show that a closed connected Riemannian manifold \(M\) whose Ricci curvature and injectivity radius are bounded from below is homeomorphic to a sphere if it has sufficiently small excess. They also show that if such an \(M\) has weakly bounded geometry, then it is a homotopy sphere, provided its excess is small enough. In this paper two theorems are proved. The first one shows that given \(c, \kappa>0\) and an integer \(n\geq 2\), there is an \(\varepsilon=\varepsilon(n, c,\kappa)>0\) such that any closed connected \(n\)-dimensional Riemannian manifold \(M\) with \(\text{Ric}_M\geq-(n-1)\kappa^2, \text{inj}_M\geq c\), and the excess \(e(M)\leq \varepsilon\) is homeomorphic to an \(n\)-sphere. The second theorem says the following. Given \(v,\kappa>0\) and an integer \(n\geq 2\), there is an \(\varepsilon=\varepsilon(n,v,\kappa)>0\) such that any closed connected \(n\)-dimensional Riemannian manifold \(M\) with sectional curvature \(K_M\geq -\kappa^2\), \(\inf_{x\in M}\text{vol}(B(x,1))\geq v\), and \(e(M)\leq \varepsilon\) is a homotopy sphere. Here \(\text{vol}(B(x,1))\) means the volume of the ball of center \(x\) and radius \(1\).