A differentiable sphere theorem by curvature pinching. II (Q1893800)

[For the first part see J. Math. Soc. Japan 43, No. 3, 527-553 (1991; Zbl 0790.53041).] Answers to the \(\delta ( > 1/4)\)-pinched problem are improved here to the extent of \(\delta = 0.654\), namely, the following theorem is proved: Let \((M,g)\) be a complete, simply connected and 0.654-pinched Riemannian manifold. Then \(M\) is diffeomorphic to the standard sphere. The proof proceeds with the aid of the following two propositions together with their numerical outcomes. 1. Let \(f: \mathbb{Z}^{n - 1} \to S^{n - 1}\) be a diffeomorphism, and assume that there exists a differentiable map \(F : [0,1] \times S^{n - 1} \to S^{n - 1}\) satisfying the conditions: (1) \(F(0, \cdot) = f\); (2) \(F_1 = F(1, \cdot)\) is an isometry of \(S^{n-1}\); (3) \(F_t = F(t,\cdot): S^{n-1} \to S^{n-1}\) is a diffeomorphism for each \(t \in [0,1]\). Then \(M\) is diffeomorphic to \(S^n\). 2. Let \(f\) be a diffeomorphism of \(S^{n-1}\) and \(\alpha\) a differentiable map of \(S^{n-1}\) into \(\text{SO} (n;\mathbb{R})\) with \(f(x) = \alpha_X x\). Choose numbers \(N_1\), \(\theta_1\) and \(\theta_2\) such that \(|(d_X \alpha)V|< N_1\), \(d_s(\alpha_x V, \alpha_{-x} V) < \theta_1\), \(d_s (\alpha_x X, (df)_x X/|(df)_x X|) < \theta_2\) for any \(x \in S^{n - 1}\) and any unit vector \(X \in T_X (S^{n -1})\) and \(V \in \mathbb{R}^n\). Then one has that, if \(N_1 \pi + \theta_1 + 2\theta_2 < 2\pi\), there exists a diffeotopy \(F\) constructed from \(f\). This may be referred to as a diffeotopy theorem.