\(n\)-dimensional real submanifolds with \((n-1)\)-dimensional maximal holomorphic tangent subspace in complex projective spaces
Let \(\overline {M}\) be an almost Hermitian manifold and let \(M\) be a real submanifold of \(\overline {M}\) with \(\dim M = n (> 1)\), \(\text{codim } M = p (\geq 1)\) and \((n - 1)\)-dimensional maximal holomorphic tangent subspace. Then \(M\) admits a naturally induced almost contact metric structure. After obtaining general results, the authors assume that \(\overline {M}\) is the complex projective space and the naturally distinguished normal field \(\xi_ 1\) of \(M\) is parallel in the normal bundle. It is proved that the shape operator \(A_{\xi_ 1}\) has at most three distinct eigenvalues which are all constants. For the Hopf fibration \(\pi : S^{n + p + 1} (1) \to \mathbb{C} P^{(n + p)/2}\), it is shown that the shape operator \(\pi^{-1} (M)\) with respect to the horizontal lift of \(\xi_ 1\) has two distinct eigenvalues with involutive eigendistributions. Thus \(\pi^{-1}(M)\) is locally the Riemannian product of the corresponding integral submanifolds. The case \(\overline {M} = \mathbb{C}^{(n + p)/2}\) will be studied by the authors in a forthcoming paper.
scientific article; zbMATH DE number 760328
scientific article