On the second equation of Obata (Q717739)

Let \((M,g)\) be a connected, simply connected, complete \(n(\geq 2)\)-dimensional Riemannian manifold. In [J. Math. Soc. Japan 30, 509--531 (1978; Zbl 0387.53015)], \textit{S. Tanno} proved that \((M,g)\) is isometric to an Euclidean sphere if and only if there exists a non-constant function \(f\) on \(M\) satisfying an Obata equation [\textit{M. Obata}, Proc. United States-Japan Semin. Diffet. Geom., Kyoto 1965, 101--114 (1966; Zbl 0144.20903)]. Slightly generalizing the Obata equation, the authors of the present paper give several characterizations for the sectional curvature of \(M\) to be constant.