The extremal case in Toponogov's comparison theorem and gap-theorems (Q1310504)

The author uses the extremal case in \textit{V. A. Toponogov}'s comparison theorem [Dokl. Akad. Nauk SSSR 124, 282-284 (1959; Zbl 0092.146)]to prove some gap theorems conjectured or proved by \textit{K. Sugahara} [Hokkaido Math. J. 18, No. 3, 459-468 (1989; Zbl 0685.53035)]. For some proofs the author also uses some results of \textit{V. Schroeder} and \textit{W. Ziller} [Trans. Am. Math. Soc. 320, No. 1, 145-160 (1990; Zbl 0724.53033)]. Theorem 2. If the Riemannian manifold \(M^ n\) with continuous sectional curvature \(K_ \sigma \geq k>0\) contains some isometrically embedded open neighborhood of \(S_ k^{n-1}\) in \(S^ n_ k\), then \(M^ n\) is isometric to \(S^ n_ k\). From this theorem follows: Sugahara's Conjecture 4. Let \(M^ n\) be an open hemisphere of the \(S^ n_ 1\). Then its structure cannot be changed in any compact subset with \(K_ M \geq 1\). Theorem 3. If the Riemannian manifold \(M^ n\), \(n \geq 3\), with continuous sectional curvature \(K_ \sigma \leq k\) contains some isometrically embedded open neighborhood of \(S_ k^{n-1}\) in \(S^ n_ k\), then \(M^ n\) is isometric to \(S^ n_ k\). For \(n=2\) this result is false. For negatively curved manifolds another conjecture of Sugahara is proved: Theorem 4. Let \(M^ n\) be a complete Riemannian manifolds of dimension \(\geq 3\). Suppose that \(K_ \sigma \geq-1\) and \(M^ n\) is isometric at infinity to the \(n\)-dimensional hyperbolic space \(H^ n(-1)\) of constant curvature \(-1\) (that is: \(M^ n \backslash W\) is isometric to \(H^ n(- 1) \backslash V\) outside some compact sets \(W\) and \(V)\). Then \(M^ n\) is isometric to \(H^ n(-1)\). Finally a new proof is given for the following Sugahara's Theorem 5. Let \(M^ n\) be a complete Riemannian manifold of dimension \(\geq 3\). Suppose that \(K_ \sigma \leq-1\) and \(M^ n\) is isometric at infinity to the \(n\)-dimensional hyperbolic space \(H^ n(-1)\) of constant curvature \(-1\). Then \(M^ n\) is isometric to \(H^ n(-1)\).