Abstract
In this short paper, we study compact Lagrangian submanifolds of the homogeneous nearly Kähler 6-dimensional unit sphere \({\mathbb {S}}^6(1)\). Following a strategy in Hu et al. (J Geom Phys 144:199–208, 2019), we shall establish an optimal pinching theorem in terms of the Ricci curvature so that a new characterization of the totally geodesic \({\mathbb {S}}^3(1)\) and the Dillen–Verstraelen–Vrancken’s Berger sphere \(S^3\) (described in J Math Soc Jpn 42:565–584, 1990) in \({\mathbb {S}}^6(1)\) can be given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antić, M., Djorić, M., Vrancken, L.: Characterization of totally geodesic totally real \(3\)-dimensional submanifolds in the \(6\)-sphere. Acta Math. Sin. (Engl. Ser.) 22, 1557–1564 (2006)
Chern, S.S., Do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length. In: Browder, F.E. (ed.) Functional Analysis and Related Fields, pp. 59–75. Springer, New York (1970)
Dillen, F., Opozda, B., Verstraelen, L., Vrancken, L.: On totally real \(3\)-dimensional submanifolds of the nearly Kaehler \(6\)-sphere. Proc. Am. Math. Soc. 99, 741–749 (1987)
Dillen, F., Verstraelen, L., Vrancken, L.: Classification of totally real \(3\)-dimensional submanifolds of \(S^6(1)\) with \(K\ge 1/16\). J. Math. Soc. Jpn. 42, 565–584 (1990)
Djorić, M., Vrancken, L.: On \(J\)-parallel totally real three-dimensional submanifolds of \(S^6(1)\). J. Geom. Phys. 60, 175–181 (2010)
Ejiri, N.: Compact minimal submanifolds of a sphere with positive Ricci curvature. J. Math. Soc. Jpn. 31, 251–256 (1979)
Ejiri, N.: Totally real submanifolds in a \(6\)-sphere. Proc. Am. Math. Soc. 83, 759–763 (1981)
Foscolo, L., Haskins, M.: New \(G_2\)-holonomy cones and exotic nearly Kähler structures on \({\mathbb{S}}^6\) and \({\mathbb{S}}^3\times {\mathbb{S}}^3\). Ann. Math. 185, 59–130 (2017)
Hu, Z., Yin, J., Yin, B.: Rigidity theorems of Lagrangian submanifolds in the homogeneous nearly Kähler \({\mathbb{S}}^6(1)\). J. Geom. Phys. 144, 199–208 (2019)
Li, A.M., Li, J.M.: An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math. 58, 582–594 (1992)
Li, H.: Curvature pinching for odd-dimensional minimal submanifolds in a sphere. Publ. Inst. Math. (Beograd) 53, 122–132 (1993)
Li, H.: The Ricci curvature of totally real \(3\)-dimensional submanifolds of the nearly Kaehler \(6\)-sphere. Bull. Belg. Math. Soc. 3, 193–199 (1996)
Shen, Y.: Curvature pinching for three-dimensional minimal submanifolds in a sphere. Proc. Am. Math. Soc. 115, 791–795 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This project was supported by NSF of China, Grant Number 11771404.
Rights and permissions
About this article
Cite this article
Hu, Z., Yao, Z. & Yin, J. On Ricci Curvature Pinching of Lagrangian Submanifolds in the Homogeneous Nearly Kähler \(\pmb {\mathbb {S}}^6(1)\). Results Math 75, 52 (2020). https://doi.org/10.1007/s00025-020-1179-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-020-1179-4
Keywords
- Ricci curvature pinching
- nearly Kähler 6-sphere
- Lagrangian submanifold
- Dillen–Verstraelen–Vrancken’s Berger sphere