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

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Abstract

In this paper, we study some properties of the linearly full conformal minimal immersions φ : S 2 → G(k, n) with second fundamental form B. At first we compute the Laplacian of square length ||B||2 of B and the relations of Gaussian curvature K and normal curvature K N. Then we obtain a necessary and sufficient condition of the parallel second fundamental form, and prove that K must be constant if B is parallel. Moreover, if it is not totally geodesic, K ≤ ||B||2/2, especially, K = ||B||2/2 when it is holomorphic. We also consider the pseudo-holomorphic curve in G(k, n) with parallel second fundamental form and compute its Gaussian curvature and Kähler angle.

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Authors and Affiliations

  1. Department of Mathematics, Graduate University, Chinese Academy of Sciences, Beijing, 100049, China

    Xiaoxiang Jiao

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  1. Xiaoxiang Jiao
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Correspondence to Xiaoxiang Jiao.

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Communicated by D. V. Alekseevsky.

Supported by the National Natural Science Foundation of China (Grant No. 11071248).

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Jiao, X. On minimal two-spheres immersed in complex Grassmann manifolds with parallel second fundamental form. Monatsh Math 168, 381–401 (2012). https://doi.org/10.1007/s00605-012-0403-z

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