Application of the extremal case in the congruence theorem for obtaining gap theorems (Q1204699)

The paper contains the proof of the theorem [hypothesis in \textit{K. Sugahara}, Hokkaido Math. J. 18, No. 3, 459-468 (1989; Zbl 0685.53035)]: If a Riemannian manifold \(M^ n\) with sectional curvature \(K_ \sigma \geq k > 0\) contains an isometrical embedded neighbourhood of a Euclidean sphere \(S^{n-1}_ k\) in \(S^ n_ k\), then \(M^ n\) is isometric to \(S^ n_ k\). Analogous results (also interpreted as gap theorems) are obtained for the case \(K_ \sigma \leq k\), \((k > 0)\) and for the hyperbolic space \(H^ n\). The proofs are based on an extremal case of Toponogov's triangle comparison theorem and a method of \textit{V. Schroeder} and \textit{W. Ziller} [Trans. Am. Math. Soc. 320, No. 1, 145-160 (1990; Zbl 0724.53033)] for continuation of isometries of symmetric spaces.