Rigidity for closed manifolds with positive curvature
Let \(M\) be an \(n\)-dimensional Riemannian manifold with sectional curvature \(\mathrm{sec}(M)\geq 1\), let \(d\) be the distance function and \(\mathrm{rad}(M)\) be the radius of \(M\). Wang proved that if \(\mathrm{rad}(M)> \pi/2\) then the length of any closed geodesic of \(M\) is bigger or equal to \(2\cdot \mathrm{rad}(M)\). In this paper, rigidity of positively curved manifolds with large radius is studied by the existence of special geodesics. Let \(\mathrm{conj}(x)\), the conjugate locus of \(M\) at \(x\), be the set of the first conjugate points to \(x\), for all geodesics starting at \(x\). Denote by \(S_xM\) the unit tangent sphere of \(M\) at \(x\). For any \(v\in S_xM\), the conjugate value \(c_v\) is the first number \(r>0\) such that thre is a nonzero Jacobi field \(J\), along the geodesic from \(x\) in the direction of \(v\), satisfying \(J(0)=J(r)=0\). It is shown that any \(n\)-dimensional complete connected Riemannian manifold \(M\) with \(\mathrm{sec}(M)\geq 1\) and \(\mathrm{rad}(M)>\pi/2\), such that for any \(x\in M\), the first conjugate locus of \(x\) is a single point and such that \(M\) contains a geodesic loop of length \(2\cdot\mathrm{rad}(M)\), is isometric to a round \(n\)-sphere. Further, it is shown that any \(n\)-dimensional complete connected Riemannian manifold \(M\) with \(\mathrm{sec}(M)\geq 1\) and \(\mathrm{rad}(M)>\pi/2\), such that for any \(x\in M\) the first conjugate value function \(c_x: S_xM\rightarrow {\mathbb R}\) is a constant function, is isometric to a round \(n\)-sphere.
scientific article; zbMATH DE number 5594049
scientific article