Applications of critical point theory of distance functions to geometry
The author uses the critical point theory of distance functions to obtain two results. First he considers a complete open manifold \(M^ n\) of nonnegative sectional curvature and rapidly increasing volume and gives a new proof for the following Theorem A [the reviewer and \textit{V. Toponogov}, Sib. Mat. Zh. 26, No. 4(152), 191-194 (1985; Zbl 0578.53030)]: ``If for some point \(p\) of \(M\) \[ \lim_{r\to \infty}{v(B(p,r))\over v_ 0(r)}>0 \] where \(v(B(p,r))\) denotes the volume of the \(r\)-ball in \(M\) with a centre \(p\) and \(v_ 0(r)\) the volume of the \(r\)-ball in the Euclidean space \(\mathbb{R}^ n\), then \(M\) is diffeomorphic to \(\mathbb{R}^ n\). According to the well-known Cheeger- Gromoll soul-theorem \(M\) is diffeomorphic to the total space of the normal bundle of some totally geodesic submanifold \(S\subset M\) (called a soul of \(M\)) and in fact the author proves that under conditions of theorem A the distance function \(d(p,\cdot)\) has no critical points for every \(p\) of \(S\). This leads to the conclusion that \(M\) is contractible and \(S\) is a point. Recall that a point \(q\) is called \(\epsilon\)-critical to \(p\) (\(0\leq \varepsilon \leq \pi/2\)) if for every vector \(v\) of \(T_ qM\) there exists some minimal geodesic \(\gamma(s)\) from \(q\) to \(p\) making an angle \(\angle (v,\dot\gamma(0))\leq \varepsilon\) with \(v\); and \(q\) is called a critical to \(p\) if \(q\) is \(\pi/2\)-critical to \(p\). The second statement which is proved in the article is Theorem C: ``Given any number \(\Lambda>0\), there is an explicit positive number \(\varepsilon=\varepsilon(\Lambda)\) such that if a complete Riemannian \(n\)-manifold \(M\) contains two points \(p\) and \(q\) with the properties: 1) \(p\), \(q\) are mutually \(\varepsilon\)-critical points, 2) \(d(p,q)^ 2K_ M\geq -\Lambda\), then \(M\) is homeomorphic to \(S^ n\) ''. In this case the author shows that if \(\Lambda\) and \(d(p,q)\) are fixed then for every \(\varepsilon\) sufficiently close to \(\pi/2\), \(M\) is a union of two contractible balls and therefore, according to the generalized Schoenflies theorem, \(M\) is homeomorphic to the sphere \(S^ n\).
scientific article; zbMATH DE number 23989
scientific article