The Journal of Geometric Analysis

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

Harmonic Manifolds and Tubes

  • Balázs CsikósEmail author
  • Márton Horváth


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.


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 



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.


  1. 1.
    Copson, E.T., Ruse, H.S.: Harmonic Riemannian spaces. Proc. R. Soc. Edinb. 60, 117–133 (1940)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Lichnerowicz, A.: Sur les espaces riemanniens complètement harmoniques. Bull. Soc. Math. France 72, 146–168 (1944)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ledger, A.J.: Symmetric harmonic spaces. J. Lond. Math. Soc. 32, 53–56 (1957)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Walker, A.G.: On Lichnerowicz’s conjecture for harmonic 4-spaces. J. Lond. Math. Soc. 24, 21–28 (1949)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Nikolayevsky, Y.: Two theorems on harmonic manifolds. Comment. Math. Helv. 80(1), 29–50 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Szabó, Z.I.: The Lichnerowicz conjecture on harmonic manifolds. J. Differ. Geom. 31(1), 1–28 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Allamigeon, A.-C.: Propriétés globales des espaces de Riemann harmoniques. Ann. Inst. Fourier (Grenoble) 15(fasc. 2), pp. 91–132 (1965)Google Scholar
  8. 8.
    Knieper, G.: New results on noncompact harmonic manifolds. Comment. Math. Helv. 87(3), 669–703 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Damek, E., Ricci, F.: A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc. (N.S.) 27(1), 139–142 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Berndt, J., Tricerri, F., Vanhecke, L.: Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, vol. 1598. Lecture Notes in Mathematics. Springer, Berlin (1995)Google Scholar
  11. 11.
    Heber, J.: On harmonic and asymptotically harmonic homogeneous spaces. Geom. Funct. Anal. 16(4), 869–890 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Michel, D.: Comparaison des notions de variétés riemanniennes globalement harmoniques et fortement harmoniques. C. R. Acad. Sci. Paris Sér. A-B, 282(17), pp. Aiii, A10007–A1010 (1976)Google Scholar
  13. 13.
    Csikós, B., Horváth, M.: On the volume of the intersection of two geodesic balls. Differ. Geom. Appl. 29(4), 567–576 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Csikós, B., Horváth, M.: A characterization of harmonic spaces. J. Differ. Geom. 90(3), 383–389 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hotelling, H.: Tubes and spheres in n-spaces, and a class of statistical problems. Am. J. Math. 61(2), 440–460 (1939)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gray, A., Vanhecke, L.: The volumes of tubes about curves in a Riemannian manifold. Proc. Lond. Math. Soc. (3) 44(2), 215–243 (1982)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Csikós, B., Horváth, M.: Harmonic manifolds and the volume of tubes about curves. J. Lond. Math. Soc. (2) 94(1), 141–160 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gray, A., Vanhecke, L.: The volumes of tubes in a Riemannian manifold. Rend. Sem. Mat. Univ. Politec. Torino 39(3), 1–50 (1983) (1981)Google Scholar
  20. 20.
    Besse, A.L.: Manifolds All of Whose Geodesics are Closed. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93. Springer, Berlin. With appendices by Epstein, D.B.A., Bourguignon, J.-P., Bérard-Bergery, L., Berger, M., Kazdan, J.L. (eds.) (1978)Google Scholar
  21. 21.
    Gray, A.: Tubes. In: Miquel, V. (ed.) Progress in Mathematics, 2nd ed., vol. 221. Birkhäuser Verlag, Basel (2004)Google Scholar
  22. 22.
    DeTurck, D.M., Kazdan, J.L.: Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. (4) 14(3), 249–260 (1981)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Vanhecke, L.: A note on harmonic spaces. Bull. Lond. Math. Soc. 13(6), 545–546 (1981)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vanhecke, L., Willmore, T.J.: Interaction of tubes and spheres. Math. Ann. 263(1), 31–42 (1983)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Encyclopedia of Mathematics and its Applications, vol. 61. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  26. 26.
    Gray, A., Vanhecke, L.: Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142(3–4), 157–198 (1979)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Abbena, E., Gray, A., Vanhecke, L.: Steiner’s formula for the volume of a parallel hypersurface in a Riemannian manifold. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(3), 473–493 (1981)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Gheysens, L., Vanhecke, L.: Total scalar curvature of tubes about curves. Math. Nachr. 103, 177–197 (1981)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kowalski, O., Vanhecke, L.: Ball-homogeneous and disk-homogeneous Riemannian manifolds. Math. Z. 180(4), 429–444 (1982)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Günther, P., Prüfer, F.: Mean value operators, differential operators and D’Atri spaces. Ann. Glob. Anal. Geom. 17(2), 113–127 (1999)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Szabó, Z.I.: Spectral theory for operator families on Riemannian manifolds. In: Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, pp. 615–665, American Mathematical Society, Providence, RI (1993)Google Scholar

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Authors and Affiliations

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

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