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Characterization of totally geodesic foliations with integrable and parallelizable normal bundle

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  10 May 2022

Euripedes C. da Silva Open the ORCID record for Euripedes C. da Silva [Opens in a new window] ,
David C. Souza Open the ORCID record for David C. Souza [Opens in a new window]  and
Fernando P.P. Reis Open the ORCID record for Fernando P.P. Reis [Opens in a new window]
Euripedes C. da Silva*
Affiliation:
Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Av. Parque Central, 1315, Maracanaú, Ceará, CEP 61939-140, Brazil
David C. Souza
Affiliation:
Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Av. Parque Central, 1315, Maracanaú, Ceará, CEP 61939-140, Brazil
Fernando P.P. Reis
Affiliation:
Universidade Federal do Espríto Santo, Centro Universitário Norte do Espríto Santo, Rodovia Governador Mário Covas, Km 60, 1315, São Mateus, ES, CEP 29932-540, Brazil
*
*Corresponding author. E-mail: euripedescarvalhomat@gmail.com
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Abstract

In this work, we study foliations of arbitrary codimension $\mathfrak{F}$ with integrable normal bundles on complete Riemannian manifolds. We obtain a necessary and sufficient condition for $\mathfrak{F}$ to be totally geodesic. For this, we introduce a special number $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ that measures when the foliation ceases to be totally geodesic. Furthermore, applying some maximum principle we deduce geometric properties for $\mathfrak{F}$. We conclude with a geometrical version of Novikov’s theorem (Trans. Moscow Math. Soc. (1965), 268–304), for Riemannian compact manifolds of arbitrary dimension.

Keywords

Complete Riemannian manifoldOrthogonal foliationsTotally geodesic and umbilical

MSC classification

Primary: 53C12: Foliations (differential geometric aspects)
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53B50: Applications to physics

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 1 , January 2023 , pp. 128 - 137
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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