A Hopf Theorem for Ambient Spaces of Dimensions Higher than Three

Hilário Alencar (), Manfredo do Carmo () and Renato Tribuzy ()
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Hilário Alencar: Universidade Federal de Alagoas, Instituto de Matemática
Manfredo do Carmo: Universidade Federal de Alagoas, Instituto de Matemática
Renato Tribuzy: Universidade Federal de Alagoas, Instituto de Matemática

A chapter in Manfredo P. do Carmo – Selected Papers, 2012, pp 471-487 from Springer

Abstract: Abstract We consider surfaces M 2 immersed in $$ E_{c}^{n} \times \mathbb{R}$$ , where $$ E_{c}^{n}$$ is a simply connected n-dimensional complete Riemannian manifold with constant sectional curvature $$ E_{c}^{n}$$ , and assume that the mean curvature vector of the immersion is parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M 2, where the complex structure of M 2 is compatible with the induced metric. It is not hard to check that Q is holomorphic (see [3], p.289). We will use this fact to give a reasonable description of immersed surfaces in $$ E_{c}^{n} \times \mathbb{R}$$ that have parallel mean curvature vector.

Keywords: Riemannian Manifold; Tangent Vector; Fundamental Form; Normal Bundle; Integral Curve (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-25588-5_34

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DOI: 10.1007/978-3-642-25588-5_34

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