Geodesic hyperspheres in complex projective space (Q1340408)

We characterize a geodesic hypersphere by a certain condition on the second fundamental form: Theorem 4.1. Let \(M\) be a real hypersurface in \(P^n\mathbb{C}\), \(n\geq 3\). If the shape operator \(A\) satisfies \[ (R(Y,Z)A)X=0 \] for all tangent vectors \(X\), \(Y\), \(Z\) perpendicular to \(\xi\), then \(M\) is locally congruent to a geodesic hypersphere. Theorem 4.2. Let \(M\) be a real hypersurface in \(P^n\mathbb{C}\), \(n\geq 2\). If the shape operator \(A\) satisfies \[ (\nabla^2A)(X;Y;Z)= f\{g(X,\phi Y)\phi Z+g(X,\phi Z)\phi Y\} \] for all tangent vectors \(X\), \(Y\), \(Z\) perpendicular to \(\xi\), where \(f\) is a \(C^\infty\)-function on \(M\), then \(f\) is nonzero constant and \(M\) is locally congruent to a geodesic hypersphere.