On minimal two-spheres immersed in complex Grassmann manifolds with parallel second fundamental form

On Conformal Minimal Immersions of Two-Spheres in a Complex Hyperquadric with Parallel Second Fundamental Form

“…Since the second fundamental form of the map ϕ is parallel, its Gauss curvature is a constant (cf. [6], Theorem 4.5). Thus, the curvature of f p in (Ia) and f 0 in (Ib) are both constants.…”

“…Lemma 3.1 [6] Let ϕ : S 2 → G(k, n; C) be a conformal minimal immersion with the second fundamental form B. Suppose that B is parallel.…”

“…Here we suppose that the metric on G(2, n; R) is given by Sect. 2 of [6]; then the metric is twice as much as the standard metric on Q n−2 induced by the inclusion τ : Q n−2 → CP n−1 , where this latter space is given the Fubini-Study metric of constant holomorphic sectional curvature 4. (B) In this section we give the general expression of some geometric quantities about conformal minimal immersions from S 2 to G(2, n; R), or equivalently, a complex hyperquadric Q n−2 .…”

“…Generally, the study of conformal minimal two-spheres immersed in various Riemannian symmetric spaces N with parallel second fundamental form is very difficult. Recently we studied conformal minimal immersions of two-spheres in G(k, n; C) and Q 2 with parallel second fundamental form, and obtained some geometric properties of them [6,8]. And paper [5] gave a classification theorem of linearly full conformal minimal immersions of S 2 in H P n with parallel second fundamental form.…”

Classification of conformal minimal immersions from $S^2$ to $G(2,N;\mathbb{C})$ with parallel second fundamental form

“…where P = ∂ Az λ 2 with A z = (2φ − I)∂φ, I is the identity matrix (cf. [12,13]). In the following, we review the rigidity theorem of conformal minimal immersions with constant curvature from S 2 to CP N .…”