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On complete submanifolds with parallel normalized mean curvature in product spaces

Part of: Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  27 January 2022

Fábio R. dos Santos Open the ORCID record for Fábio R. dos Santos [Opens in a new window]  and
Sylvia F. da Silva
Fábio R. dos Santos
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, 50.740-540 Recife, Pernambuco, [email protected]@gmail.com
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Abstract

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$, where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$, it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.

Keywords

Submanifolds with parallel normalized mean curvature vector fieldSimons type formulaconstant second mean curvaturetotally umbilical submanifolds

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature 53C20: Global Riemannian geometry, including pinching
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 331 - 355
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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