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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3458–3476 | Cite as

Harmonic Manifolds and Tubes

  • Balázs CsikósEmail author
  • Márton Horváth
Article
  • 90 Downloads

Abstract

Csikós and Horváth (J Lond Math Soc (2) 94(1):141–160, 2016) showed that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed only for tubes about geodesic segments. As a consequence, we obtain similar characterizations of harmonic manifolds in terms of the total mean curvature and the total scalar curvature of tubular hypersurfaces about curves. We find simple formulae expressing the volume, total mean curvature, and total scalar curvature of tubular hypersurfaces about a curve in a harmonic manifold as a function of the volume density function.

Keywords

Harmonic manifold D’Atri space Weyl’s tube formula Steiner’s formula Total mean curvature Total scalar curvature 

Mathematics Subject Classification

Primary 53C25 Secondary 53B20 

Notes

Acknowledgements

The authors are grateful to professor Gudlaugur Thorbergsson for helpful discussions at the University of Cologne in 2013, that motivated the research presented above. They are also indebted to professor Eduardo García Río who proposed them the study of the total scalar curvature of tubes in harmonic manifolds during a personal communication at the Conference on Differential Geometry and its Applications in Brno in 2016. Balázs Csikós was supported by the National Research Development and Innovation Office (NKFIH) Grant Nos. ERC_HU_15 118286 and OTKA K112703. Márton Horváth was supported by the National Research Development and Innovation Office (NKFIH) Grant No. OTKA K112703.

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.Budapest University of Technology and EconomicsBudapestHungary

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