On the mean curvature of submanifolds with nullity


In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical n-dimensional minimal submanifolds with index of relative nullity \(n-2\), fully described by Dajczer and Florit (Ill J Math 45:735–755, 2001) in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal n-dimensional submanifolds in space forms of any codimension are locally cylinders provided that they carry a totally geodesic distribution of rank \(n-2\ge 2,\) which is contained in the relative nullity distribution, such that the length of the mean curvature vector field is constant along each leaf. The case of dimension \(n=3\) turns out to be special. We show that there exist elliptic three-dimensional submanifolds in spheres satisfying the above properties. In fact, we provide a parametrization of three-dimensional submanifolds as unit tangent bundles of minimal surfaces in the Euclidean space whose first curvature ellipse is nowhere a circle and its second one is everywhere a circle. Moreover, we provide several applications to submanifolds whose mean curvature vector field has constant length, a much weaker condition than being parallel.

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The first named author would like to acknowledge financial support by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI), Grant No: 133, Action: Support for Postdoctoral Researchers.

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Correspondence to Th. Vlachos.

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Kanellopoulou, A.E., Vlachos, T. On the mean curvature of submanifolds with nullity. Ann Glob Anal Geom 58, 79–108 (2020). https://doi.org/10.1007/s10455-020-09717-6

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  • Index of relative nullity
  • Relative nullity distribution
  • Mean curvature
  • Cylinder
  • Elliptic submanifolds
  • Minimal surfaces
  • Curvature ellipse

Mathematics Subject Classification

  • Primary 53C42; Secondary 53C40
  • 53B25