Real Hypersurfaces in Complex Grassmannians of Rank Two

It is known that there does not exist any Hopf hypersurface in complex Grassmannians of rank two of complex dimension 2m with constant sectional curvature for m≥3. The purpose of this article is to extend the above result, and without the Hopf condition, we prove that there does not exist any locally conformally flat real hypersurface for m≥3.

Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator

We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn (c) satisfying the two conditions φ l = l φ , $\varphi l=l\varphi ,$ where l = R ( ⋅ , ξ ) ξ and φ $l=R(\cdot ,\xi )\xi \text{ and }\varphi $ is the almost contact metric structure of M, and ( ∇ ξ l ) X = $\left( {{\nabla }_{\xi }}l \right)X=$ ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0.

HYPERSURFACES IN P2 AND H2 WITH TWO DISTINCT PRINCIPAL CURVATURES

It is known that hypersurfaces in ${\mathbb C}$Pn or ${\mathbb C}$Hn for which the number g of distinct principal curvatures satisfies g ≤ 2, must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that n ≥ 3. In this paper, we construct a two-parameter family of non-Hopf hypersurfaces in ${\mathbb C}$P2 and ${\mathbb C}$H2 with g=2 and show that every non-Hopf hypersurface with g=2 is locally of this form.