Characterizations of isoparametric hypersurfaces in a sphere (Q2917733)

The authors give characterizations of isoparametric hypersurfaces in spheres both by means of covariant derivatives of their shape operators and by extrinsic properties of their geodesics. Namely they give a proof of the following result: let \(M\) be a connected hypersurface in a sphere \(S^{n+1}\) (whose shape operator is denoted by \(A\) and Levi-Civita connection by \(\nabla\)). Then the following conditions are equivalent:\newline (1) \(M\) is locally congruent to an isoparametric hypersurface in a sphere \(S^{n+1}\). \newline (2) The tangent bundle \(TM\) splits as the direct sum of the principal eigendistributions of the shape operator \(V_\lambda=\{X\in TM:AX=\lambda X\}\), such that \((\nabla_XA)Y=0\) for any \(X,Y\in V_\lambda\) and any principal curvature \(\lambda\) of \(M\). \newline (3) For each point \(p\in M\), there exists an orthonormal basis \(\{ v_1,\dots,v_{m_p}\}\) of the orthogonal complement of \(\operatorname{ker}A_p\) in \(T_pM\) (\(m_p=\operatorname{rank} A_p\)) such that every geodesic of \(M\) through \(p\) with initial vector \(v_i\) is a circle of positive curvature in \(S^{n+1}\). \newline Actually, the equivalence between (1) and (3) was shown by \textit{M. Kimura} and the first author [Can. Math. Bull. 43, No. 1, 74--78 (2000; Zbl 0964.53044)].\newline Moreover, as consequences of this result, the authors review characterizations of the Clifford hypersurfaces and the minimal Cartan isoparametric hypersurfaces with \(3\) principal curvatures which were proved by \textit{T. Adachi} and the first author [Colloq. Math. 105, No. 1, 143--148 (2006; Zbl 1118.53037)]. These are performed by the existence at every point of a certain finite set of geodesics that are small circles in the ambient sphere.