The holonomy in open manifolds of nonnegative curvature
Let \(V^n\) be a complete noncompact manifold of nonnegative sectional curvature \(K_\sigma\geq 0\). Let \(S\) be a soul of \(V^n\), i.e., \(S\) is a compact, totally convex, totally geodesic submanifold of \(V^n\), without boundary, such that \(0\leq\dim S\leq\dim V^n\). \textit{J. Cheeger} and \textit{D. Gromoll} [Ann. Math., II. Ser. 96, 413-443 (1972; Zbl 0246.53049)] proved that \(V^n\) is diffeomorphic to the normal bundle \(\nu(S)\) and, if V\(^n\) is isometric to the direct product \(V^n=S\times W\) (where \(W\) is a complete noncompact manifold of nonnegative sectional curvature, diffeomorphic to the Euclidean space), then the holonomy operator is the identity, i.e., \(\nu(S)\) has trivial holonomy. The principal theorem in this article states that the converse of this last statement is also true, i.e., if \(\nu(S)\) has trivial holonomy, then \(V^n\) is isometric to the direct product \(V^n=S\times W\) where \(W\) is a complete noncompact manifold, diffeomorphic to the Euclidean space of the corresponding dimension. According to the author, this theorem has already been stated [\textit{V. B. Marenich}, Sov. Math. Dokl. 24, 595-597 (1981; Zbl 0495.53037)]. In the present article a clear and detailed proof is given. The second part of this work includes new proofs of three theorems. The original ones were given by the author [in ``The holonomy in open manifolds of nonnegative curvature'', MSRI preprint No. 003-94], but were very long. G. Perel'man gave a new proof for one of these statements, more precisely, for the theorem that states that for every point \(p\in S\) and every 2-dimensional direction \(\sigma\) that is normal to \(S\) at this point (i.e., \(\sigma\subset\nu_p(S)\)), then \(I_\omega= \text{id}\) for every contractible curve \(\omega\) on \(S\), if \(K_\sigma=0\), and the universal cover \(\widetilde V^n\) of \(V^n\) is isometric to the direct product (\(I_\omega\) is the parallel translation along this curve). The author includes the proof given by G. Perel'man which involves elementary properties of the curvature tensor, and adapts his reasoning to simplified versions for the two other statements: for every closed curve \(\omega\) contractible in some flat domain \(D\) in \(S\), the holonomy along \(\omega\) vanishes, i.e., for every point \(p\) of \(\omega\) and every \(v\) of \(\nu_p(S)\), \(I_\omega v=v\); and if for every point \(p\in S\), and every \(e\), \(v\), and \(w\), \(K_{\sigma(p,e,v,w,\rho)}=o(\rho^2)\), as \(\rho\to 0\), then \(I_\omega= \text{id}\) for every contractible curve \(\omega \) on \(S\), the universal cover \(\widetilde V^n\) of \(V^n\) is isometric to the direct product, and \(K_{\sigma(p,e,v,w,\rho)}=0\).
scientific article; zbMATH DE number 934197