1-type submanifolds of the complex projective space
Let \(f: M^ n\to E^ N\) be an isometric immersion of a compact Riemann manifold into a Euclidean space and let \(\Delta\) be the Laplace operator acting on smooth functions on \(M\). An immersion \(f\) is said to be of 1- type, if it can be decomposed as \(f=f_ 0+f_ t\), where \(f_ 0=\text{const}\) and \(\Delta f_ t=\lambda_ tf_ t\) [\textit{B. Y. Chen}, Total mean curvature and submanifolds of finite type (World Scientific 1984; Zbl 0537.53049)]. Now, let \(\varphi\) be the standard embedding of \(\mathbb{C} P^ m\) into the set \(H(m+1)\) of Hermitian matrices of degree \(m+1\), which is a Euclidean space \(E^ N\), where \(N=(m+1)^ 2\). \textit{A. Ros} gave a classification of \(CR\)-minimal submanifolds of \(\mathbb{C} P^ m\) which are of 1-type via the standard embedding [Kodai Math. J. 6, 88-99 (1983; Zbl 0518.53057)]. In this paper the author considers \(x: M^ n\to \mathbb{C} P^ m\) to be an isometric immersion of a compact Riemannian manifold into \(\mathbb{C} P^ m\) with parallel mean curvature vector and proves that \(f=\varphi\circ x\) is of 1-type if and only if one of the following cases occurs: 1) \(n\) is even and \(M^ n\) is congruent to the complex projective space \(\mathbb{C} P^{n/2}\) immersed as a totally geodesic complex submanifold of \(\mathbb{C} P^ m\). 2) \(M^ n\) is a totally real minimal submanifold of a complex totally geodesic \(\mathbb{C} P^ n\) in \(\mathbb{C} P^ m\). 3) \(n\) is odd, and \(M^ n\) is congruent to the geodesic hypersphere \[ M^ c_{0,(n- 1)/2}=\pi(S^ 1(\sqrt{1/(n+3)}\times S^ n(\sqrt{(n+2)/(n+3)}) \] of the complex projective space \(\mathbb{C} P^{(n+1)/2}\) immersed as a totally geodesic complex submanifold of \(\mathbb{C} P^ m\).
scientific article; zbMATH DE number 60674
scientific article