Skip to main content
SpringerLink
Log in

General Normal Cycles and Lipschitz Manifolds of Bounded Curvature

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Closed Legendrian (d − 1)-dimensional locally rectifiable currents on the sphere bundle in \(\mathbb{R}\)d are considered and the associated index functions are studied. A topological condition assuring the validity of a local version of the Gauss–Bonnet formula is established. The case of lower-dimensional Lipschitz submanifolds in \(\mathbb{R}\)d and their associated normal cycles is examined in detail.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston, 1990.

    Google Scholar 

  2. Bernig, A.: Aspects of curvature, preprint.

  3. Bernig, A. and Bröcker, L.: Lipschitz–Killing invariants, Math. Nachr. 245 (2002), 5–25.

    Google Scholar 

  4. Cheeger, J., Müller, W., and Schrader, R.: Kinematic and tube formulas for piecewise linear spaces, Indiana Univ. Math. J. 35 (1986), 737–754.

    Google Scholar 

  5. Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

    Google Scholar 

  6. Federer, H.: Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.

    MATH  Google Scholar 

  7. Federer, H.: Geometric Measure Theory, Springer-Verlag, Berlin, 1969.

    Google Scholar 

  8. Fu, J.: Monge-Ampèere functions I, Indiana Univ. Math. J. 38 (1989), 745–771.

    Google Scholar 

  9. Fu, J.: Curvature measures of subanalytic sets, Amer. J. Math. 116 (1994), 819–880.

    Google Scholar 

  10. Hug, D. and Schätzle, R.: Intersections and translative integral formulas for boundaries of convex bodies, Math. Nachr. 226 (2001), 99–128.

    Google Scholar 

  11. Rataj, J. and Zähle, M.: Curvatures and currents for unions of sets with positive reach, II, Ann. Global Anal. Geom. 20 (2001), 1–21.

    Google Scholar 

  12. Rataj, J. and Zähle, M.: Normal cycles of Lipschitz manifolds by approximation with parallel sets, Differential Geom. Appl. 19 (2003), 113–126.

    Google Scholar 

  13. Schneider, R.: Parallelmengen mit Vielfachheit und Steiner-Formeln, Geom. Dedicata 9 (1980), 111–127.

    Google Scholar 

  14. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  15. Whitehead, J. H. C.: Manifolds with transverse fields in euclidean space, Ann. Math. 73 (1961), 154–212.

    Google Scholar 

  16. Zähle, M.: Curvatures and currents for unions of sets with positive reach, Geom. Dedicata 23 (1987), 155–171.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Charles University, Sokolovská 83, 18675, Praha 8, Czech Republic

    J. Rataj

  2. Mathematical Institute, Friedrich-Schiller-University, D-07740, Jena, Germany

    M. Zähle

Authors
  1. J. Rataj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Zähle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Rataj.

Additional information

Supported by the Grant Agency of Charles University, Project No. 283/2003/B-MAT/MFF, and by the Czech Ministry of Education, project MSM 113200007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rataj, J., Zähle, M. General Normal Cycles and Lipschitz Manifolds of Bounded Curvature. Ann Glob Anal Geom 27, 135–156 (2005). https://doi.org/10.1007/s10455-005-5218-x

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-005-5218-x

Search

Navigation