The rigidity for \(K\)-contact real hypersurfaces in a complex projective space
If \(f:M\to\mathbb{C} P^n\) is an isometric immersion from an oriented \((2n-1)\)-dimensional Riemannian manifold into the complex projective space \(\mathbb{C} P^n\) (equipped with the Fubini-Study metric), then an almost contact structure \((\varphi,\xi)\) is induced on \(M\) by the following conditions: \(Jf_* \xi\) is a unit normal of \(f\) and \(\varphi\) is the tensor field with kernel \(\mathbb{R} \xi\) satisfying \(f_*\varphi(X) =Jf_*X\) for \(X\perp\xi\). The immersion \(f\) is said to be \(K\)-contact iff \(\varphi=\nabla\xi\). The authors show that two such isometric immersions \(f_1,f_2: M\to\mathbb{C} P^n\) are congruent, if \(n\geq 3\) and one of the following conditions is fulfilled: (i) \(f_1\) and \(f_2\) are \(K\)-contact or (ii) \(f_1\) is \(K\)-contact, the angle between \(\xi_1\) and \(\xi_2\) is constant and the type number of \(f_2\) is different from 2 everywhere.
scientific article; zbMATH DE number 1725444