Rigidity theorems for real hypersurfaces in a complex projective space (Q1355155)

Let \(M\) be a \((2n-1)\)-dimensional Riemannian manifold, and let \(\iota\), \(\widehat \iota\) be two isometric immersions of \(M\) into \(P_n (\mathbb{C})\). The authors prove that \(\iota\) and \(\widehat \iota\) are congruent if the type number of \(\iota\) and \(\widehat\iota\) is not equal to 2 everywhere, and, moreover, (a) two structure vector fields coincide up to sign, or (b) there exists an \(m\)-dimensional \((2\leq m\leq n-1)\) subspace of the tangent space of \(M\), at each point invariant under the action of the shape operators of \(\iota\) and \(\widehat \iota\).