The rigidity of minimal submanifolds in a locally symmetric space
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In the present paper, we discuss the rigidity phenomenon of closed minimal submanifolds in a locally symmetric Riemannian manifold with pinched sectional curvature. We show that if the sectional curvature of the submanifold is no less than an explicitly given constant, then either the submanifold is totally geodesic, or the ambient space is a sphere and the submanifold is isometric to a product of two spheres or the Veronese surface in S4.
Abstract
In the present paper, we discuss the rigidity phenomenon of closed minimal submanifolds in a locally symmetric Riemannian manifold with pinched sectional curvature. We show that if the sectional curvature of the submanifold is no less than an explicitly given constant, then either the submanifold is totally geodesic, or the ambient space is a sphere and the submanifold is isometric to a product of two spheres or the Veronese surface in S4.
참고문헌 (16)
- S. S. Chern, "Minimal submanifolds of a sphere with second fundamental form of constant length, In Functional analysis and relathed fields", pp.59~75 (1970), Springer
- Q. Ding, "On Chern’s problem for rigidity of minimal hypersurfaces in the spheres", Adv. Math, 227(1), pp.131~145 (2011)
- J. Q. Ge, "A proof of the DDVV conjecture and its equality case", Pacific J. Math, 237(1), pp.87~95 (2008)
- S. I. Goldberg, "Curvature and Homology" (1998), Academic Press
- J. R. Gu, "On Yau rigidity theorem for minimal submanifolds in spheres", ~
- T. Itoh, "On Veronese manifolds", J. Math. Soc. Japan, 27(3), pp.497~506 (1975)
- T. Itoh, "Addendum to my paper “On Veronese manifolds”", J. Math. Soc. Japan, 30(1), pp.73~74 (1978)
- B. Lawson, "Local rigidity theorems for minimal hypersurfaces", Ann. of Math, 89(2), pp.179~185 (1969)
- A. M. Li, "An intrinsic rigidity theorem for minimal submanifolds in a sphere", Arch. Math, 58(6), pp.582~594 (1992)
- Z. Lu, "Proof of the normal scalar curvature conjecture", ~
- J. Simons, "Minimal varieties in Riemannian manifolds", Ann. of Math, 88(2), pp.62~105 (1968)
- W. D. Song, "On minimal submanifolds in a locally symmetric space", Chinese Ann. Math. Ser. A, 19, pp.693~698 (1998)
- S. M. Wei, "Scalar curvature of minimal hypersurfaces in a sphere", Math. Res. Lett, 14(3), pp.423~432 (2007)
- H. W. Xu, "On closed minimal submanifolds in pinched Riemannian manifolds", Trans. Amer. Math. Soc, 347(5), pp.1743~1751 (1995)
- H. C. Yang, "Chern’s conjecture on minimal hypersurfaces", Math. Z, 227(3), pp.377~390 (1998)
- S. T. Yau, "Submanifolds with constant mean curvature I, II", Amer. J. Math, 96, pp.346~366 (1974)