The rigidity for real hypersurfaces in a complex projective space
Let \(M\) be a \((2n-1)\)-dimensional connected Riemannian manifold and \(\iota, \hat\iota\) two isometric immersions from \(M\) into the \(n\)-dimensional complex projective space \(P_n(\mathbb{C})\), \(n\geq 4\). The authors first prove that the two structure vector fields on \(M\) induced by these immersions coincide up to sign. Combining this with a previous result by the authors they conclude that if the type number of \(\iota\) or \(\hat\iota\) is not equal to 2 at every point of \(M\), then there exists an isometry \(\phi\) of \(P_n(\mathbb{C})\) such that \(\phi\circ \iota= \hat\iota\). As an application, they obtain that, if \(M\) is a \((2n-1)\)-dimensional connected homogeneous Riemannian manifold and if it admits an isometric immersion \(\iota\) into \(P_n(\mathbb{C})\), then \(\iota(M)\) is extrinsically homogeneous. The analogous statements for complex hyperbolic space are also valid.
scientific article; zbMATH DE number 1275224