Rigidity of complete Riemannian manifolds without conjugate points (Q2862162)

In the present monograph some rigidity results for complete two-dimensional Riemannian planes and Riemannian cylinders are reviewed, based on the author's previous work [\textit{V. Bangert} and the author, J. Differ. Geom. 94, No. 3, 367--385 (2013; Zbl 1278.53038)]. The crucial assumptions aside from the completeness are the non-existence of conjugate points (non-existence of a Jacobi field vanishing at two distinct points) and a requirement on the area growth for balls of large radii. The method of proof follows the original one, introduced by \textit{E. Hopf} [Proc. Nat. Acad. Sci. USA 34, 47--51 (1948; Zbl 0030.07901)] and recalled in the present work, for proving a similar rigidity result on Riemannian 2-tori.NEWLINENEWLINENEWLINEIn the case of the plane, the area of large balls is shown to be asymptotically not smaller than the areas of corresponding Euclidean balls, whereas the equality characterizes the Euclidean plane. In the cylinder case instead, a ``subquadratic area growth'' is assumed, and enforces the cylinder to be flat. A different version of the theorem for cylinders also holds, when, in place of the ``subquadratic area growth'', the author assumes that both cylindrical ends open ``less than linearly''. Finally, Hopf's method is applied for higher-dimensional cylinders of the form \(\mathbb T ^{n-1} \times\mathbb R\), more precisely those which are complete, conformally flat and do not admit conjugate points. A further condition on the conformal factor enables the author to obtain a new rigidity result for the flat cylinder.