Geometric properties of normal submanifolds

Abstract

This paper deals with normal submanifolds immersed in a Riemannian manifold \({\overline{M}}\). We generalized some recent results of surfaces in space forms obtained by Hernández-Lamoneda and Ruiz-Hernández (Bull Braz Marh Soc (NS) 49:447–462, 2018) to arbitrary submanifolds. More precisely, given a submanifold M in \({\overline{M}}\), we study the submanifolds formed by orthogonal geodesics to M, and call it a ruled normal submanifold to M. In the first part of this paper, we analyze these submanifolds and establish some geometric properties of them. Furthermore, we extend some properties about the lines of curvature and using the ideas of [3] also give an extension of the classical Theorem of Bonnet to hypersurfaces of \({\overline{M}}\).

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Acknowledgements

The authors would like to thank the anonymous referee for her/his remarks and suggestions which have helped us to improve this paper. Josué Meléndez was partially supported by Programa Especial de Apoyo a la Investigación: Sistemas Hamiltonianos, Mecánica y Geometría, UAM.

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Correspondence to Josué Meléndez or Mario Hernández.

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Meléndez, J., Hernández, M. Geometric properties of normal submanifolds. Bol. Soc. Mat. Mex. 26, 1273–1288 (2020). https://doi.org/10.1007/s40590-020-00284-1

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Keywords

  • Submanifolds
  • Mean curvature
  • Geodesic
  • Lines of curvature

Mathematics Subject Classification

  • 53B25