Illinois Journal of Mathematics

Almost conformally flat hypersurfaces

Christos-Raent Onti and Theodoros Vlachos

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Abstract

We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the round sphere and extend a result due to Shiohama and Xu (J. Geom. Anal. 7 (1997) 377–386) for compact hypersurfaces in any space form.

Article information

Source
Illinois J. Math., Volume 61, Number 1-2 (2017), 37-51.

Dates
Received: 8 December 2016
Revised: 11 August 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1520046208

Digital Object Identifier
doi:10.1215/ijm/1520046208

Mathematical Reviews number (MathSciNet)
MR3770835

Zentralblatt MATH identifier
1387.53014

Subjects
Primary: 53C40: Global submanifolds [See also 53B25] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Citation

Onti, Christos-Raent; Vlachos, Theodoros. Almost conformally flat hypersurfaces. Illinois J. Math. 61 (2017), no. 1-2, 37--51. doi:10.1215/ijm/1520046208. https://projecteuclid.org/euclid.ijm/1520046208


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