Constant isotropic surfaces in 5-dimensional space forms

Sakamoto, Kunio

doi:10.1007/bf00572445

Cited by 4 publications

(5 citation statements)

References 10 publications

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“…So R ? ðzÞ 0 0 for all z A M, hence f is l-isotropic with l constant and the conclusion follows from the classification of such surfaces in [32].…”

Section: Pseudo-parallel Surfacesmentioning

confidence: 60%

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Pseudo-parallel submanifolds of a space form

Asperti¹,

Lobos²,

Mercuri³

2001

Advg

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Pseudo-parallel immersions into space forms are defined as extrinsic analogues of pseudo-symmetric manifolds (in the sense of R. Deszcz) and as a direct generalization of semiparallel immersions. In this paper we obtain a description of pseudo-parallel hypersurfaces of a space form as quasi-umbilic hypersurfaces or cyclids of Dupin. Moreover, we study pseudoparallel immersions of surfaces and pseudo-parallel immersions with maximal first normal bundle in space forms. Finally, we give a topological classification of some complete, simply connected manifolds admitting a pseudo-parallel immersion into a space form.

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“…So R ? ðzÞ 0 0 for all z A M, hence f is l-isotropic with l constant and the conclusion follows from the classification of such surfaces in [32].…”

Section: Pseudo-parallel Surfacesmentioning

confidence: 60%

“…A direct calculation shows that T is superminimal with l 2 ¼ c 2 (see [32]). In particular, by Proposition 4.1 T is P.P.…”

Section: Pseudo-parallel Surfacesmentioning

confidence: 99%

Pseudo-parallel submanifolds of a space form

Asperti¹,

Lobos²,

Mercuri³

2001

Advg

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“…An isometric immersion φ : M →M is said to be constant isotropic if H (X, X) 2 is constant on the unit tangent bundle of M. In the case that M is a surface, we easily see that φ is constant isotropic if and only if it is pseudo-umbilical ( γ , η = 0), the curvature ellipses are circles ( γ , γ = 0) and η 2 + |γ | 2 /2 is constant. In [23], we determined constant isotropic PROOF. Since ∇ ⊥ η, γ = 0, we have, from (4.12),…”

Section: Corollary If the Normal Curvature Tensor Is Parallel Then The Immersion φ Is Critical For The Functionalmentioning

confidence: 99%

“…Let φ : M → S n (c) be a minimal immersion of compact surface M. If φ satisfies (3.10) and the curvature ellipses are circles everywhere, then the Gauss curvature of M is constant and the immersion is a standard minimal immersion of a sphere or a constant isotropic minimal immersion of a flat torus (cf. [6,17,23]).…”

Section: Corollary If the Normal Curvature Tensor Is Parallel Then The Immersion φ Is Critical For The Functionalmentioning

confidence: 99%

Variational problems of normal curvature tensor and concircular scalar fields

Sakamoto

2003

Tohoku Math. J. (2)

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We consider the integral of (the square of) the length of the normal curvature tensor for immersions of manifolds into real space forms, especially into spheres. The first variation formula is given and the Euler-Lagrange equation is expressed in terms of the isothermal coordinates when the submanifold is two-dimensional. The relations between the critical surfaces and Willmore surfaces are discussed. We also give formulas concerning the residue of logarithmic singularities of S-Willmore points or estimate it by a conformal invariant.We show that if a compact critical surface satisfies certain conditions and the immersion is minimal, then the Gauss curvature is a non-negative constant and the immersion is a standard minimal immersion of a sphere or a constant isotropic minimal immersion of a flat torus. To prove this result, we study two-dimensional Riemannian manifolds admitting concircular scalar fields whose characteristic functions are polynomials of degree 2. Moreover, the case that the characteristic functions are polynomials of degree 3 is studied.

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“…2 y. This example, that appears in Sakamoto (1989), is a minimal λ-isotropic flat torus with λ = c 2 and non vanishing normal curvature. In particular, f is a pseudo-parallel immersion in S 5 c with φ = −c 2 .…”

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confidence: 95%

Pseudo-parallel surfaces of $$\mathbb {S}_c^n \times \mathbb {R}$$ S c n × R and $$\mathbb {H}_c^n \times \mathbb {R}$$ H c n × R

Lobos

Tassi

Hancco

2018

Bull Braz Math Soc, New Series

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In this work we give a characterization of pseudo-parallel surfaces in S n c ×R and H n c ×R, extending an analogous result by Asperti-Lobos-Mercuri for the pseudo-parallel case in space forms. Moreover, when n = 3, we prove that any pseudo-parallel surface has flat normal bundle. We also give examples of pseudo-parallel surfaces which are neither semi-parallel nor pseudo-parallel surfaces in a slice. Finally, when n ≥ 4 we give examples of pseudo-parallel surfaces with non vanishing normal curvature.

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Constant isotropic surfaces in 5-dimensional space forms

Cited by 4 publications

References 10 publications

Pseudo-parallel submanifolds of a space form

Pseudo-parallel submanifolds of a space form

Variational problems of normal curvature tensor and concircular scalar fields

Pseudo-parallel surfaces of $$\mathbb {S}_c^n \times \mathbb {R}$$ S c n × R and $$\mathbb {H}_c^n \times \mathbb {R}$$ H c n × R

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