Skip to main content
SpringerLink
Log in

Power series expansions, differential geometry of geodesic spheres and tubes, and mean-value theorems

  • II. Differential Geometry
  • Conference paper
  • First Online:
Geometry and Differential Geometry

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 792))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. ABBENA, A. GRAY & L. VANHECKE, "Steiner's formula for the volume of a parallel hypersurface in a Riemannian manifold," preprint I.H.E.S., Buressur-Yvette, 1979.

    Google Scholar 

  2. C.B. ALLENDOERFER, "Steiner's formula on a general Sn+1," Bull. Amer. Math. Soc. 54 (1958), 128–135.

    Article  MathSciNet  Google Scholar 

  3. R. BISHOP, "A relation between volume, mean curvature, and diameter," Amer. Math. Soc. Notices 10 (1963), 364.

    Google Scholar 

  4. A.L. BESSE, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik, vol. 93, Springer-Verlag, 1978.

    Google Scholar 

  5. R. COURANT & D. HILBERT, Methods of mathematical physics, vol. 2, Interscience, 1962.

    Google Scholar 

  6. B.-Y. CHEN & L. VANHECKE, "Total curvatures of geodesic spheres," Arch. Math.(Basel) 32 (1979), 404–411.

    Article  MathSciNet  MATH  Google Scholar 

  7. B.-Y. CHEN & L. VANHECKE, "Differential geometry of geodesic spheres," to appear.

    Google Scholar 

  8. A. GRAY, The volume of a small geodesic ball in a Riemannian manifold," Michigan Math. J. 20 (1973), 329–344.

    MathSciNet  MATH  Google Scholar 

  9. A. GRAY, "Comparison theorems for volumes of tubes," to appear.

    Google Scholar 

  10. A. GRAY & L. VANHECKE, "Riemannian geometry as determined by the volumes of small geodesic balls," Acta Math. 142 (1979), 157–198.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. GRAY & L. VANHECKE, "Oppervlakten van geodetische cirkels op oppervlakken," to appear in Med. Konink. Acad. Wetensch. Lett. Schone Kunst. België Kl. Wetensch..

    Google Scholar 

  12. A. GRAY & L. VANHECKE, "The volume of tubes about curves in a Riemannian manifold," to appear.

    Google Scholar 

  13. A. GRAY & L. VANHECKE, "The volume of tubes in a Riemannian manifold," to appear.

    Google Scholar 

  14. A. GRAY & T.J. WILLMORE, "Mean-value theorems for Riemannian manifolds," to appear.

    Google Scholar 

  15. S. HELGASON, "The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds," Acta Math. 113 (1965), 153–180.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. PIZZETTI, "Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera," Atti R. Accad. Rend. Cl. Sci. Fis. Mat. Natur. ser. 5, 18 (1909), 182–185.

    MATH  Google Scholar 

  17. P. PIZZETTI, "Sull significato geometrica del secundo parametro differenziale di una funzione sopra una superficie qualunque," Atti R. Accad. Rend. Cl. Sci. Fis. Mat. Natur. ser. 5, 18 (1909), 309–316.

    MATH  Google Scholar 

  18. J. STEINER, "Über parallele Flächen," Monatsbericht der Akademie der Wissenschaften zu Berlin (1840), 114–118. Also Werke, vol. 2 (1882), 171–176.

    Google Scholar 

  19. H. WEYL, "On the volumes of tubes," Amer. J. Math. 61 (1939), 461–472.

    Article  MathSciNet  MATH  Google Scholar 

  20. T.J. WILLMORE, "Mean-value theorems in harmonic Riemannian spaces," J. London Math. Soc. 25 (1950), 54–57.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, Maryland, U.S.A.

    A. Gray & L. Vanhecke

  2. Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3030, Leuven, Belgium

    A. Gray & L. Vanhecke

Authors
  1. A. Gray
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Vanhecke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rafael Artzy Izu Vaisman

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Gray, A., Vanhecke, L. (1980). Power series expansions, differential geometry of geodesic spheres and tubes, and mean-value theorems. In: Artzy, R., Vaisman, I. (eds) Geometry and Differential Geometry. Lecture Notes in Mathematics, vol 792. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0088683

Download citation

  • DOI: https://doi.org/10.1007/BFb0088683

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09976-5

  • Online ISBN: 978-3-540-39214-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation