Skip to main content
SpringerLink
Log in

Euler characteristic and Lipschitz–Killing curvatures of closed semi-algebraic sets

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We prove a formula that relates the Euler–Poincaré characteristic of a closed semi-algebraic set to its Lipschitz–Killing curvatures.

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. Bekka K.: Regular stratification of subanalytic sets. Bull. London Math. Soc. 25(1), 7–16 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernig A.: Scalar curvature of definable Alexandrov spaces. Adv. Geom. 2(1), 29–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernig A.: Scalar curvature of definable CAT-spaces. Adv. Geom. 3(1), 23–43 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernig A.: The normal cycle of a compact definable set. Israel J. Math. 159, 373–411 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bernig A., Bröcker L.: Courbures intrinsèques dans les catégories analytico-géométriques. Ann. Inst. Fourier (Grenoble) 53(6), 1897–1924 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bröcker L., Kuppe M.: Integral geometry of tame sets. Geometriae Dedicata 82, 285–323 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohn-Vossen S.: Kuerzeste Wege und Totalkruemmung auf Flaechen. Compositio Math. 2, 69–133 (1935)

    MathSciNet  MATH  Google Scholar 

  9. Comte G., Merle M.: Equisingularité réelle II: invariants locaux et conditions de régularité. Ann. Sci. Éc. Norm. Supér. (4) 41(2), 221–269 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Dillen F., Kühnel W.: Total curvature of complete submanifolds of Euclidean space. Tohoku Math. J. (2) 57(2), 171–200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dutertre N.: A Gauss–Bonnet formula for closed semi-algebraic sets. Adv. Geom. 8(1), 33–51 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Federer H.: Geometric Measure Theory, Die Grundlehren Der Mathematischen Wissenschaften. Band 153. Springer, New York (1969)

    Google Scholar 

  13. Fu J.H.G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39(4), 1115–1154 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fu J.H.G.: Curvature measures of subanalytic sets. Amer. J. Math. 116(4), 819–880 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fu J.H.G., Mccrory C.: Stiefel-Whitney classes and the conormal cycle of a singular variety. Trans. Amer. Math. Soc. 349(2), 809–835 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goresky M., Mac-Pherson R.: Stratified Morse Theory. Springer, Berlin (1988)

    MATH  Google Scholar 

  17. Hardt R.M.: Topological properties of subanalytic sets. Trans. Amer. Math. Soc. 211, 57–70 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Langevin, R.: Courbures, feuilletages et surfaces, Dissertation, Université Paris-Sud, Orsay, 1980. Publications Mathématiques d’Orsay 80, 3. Université de Paris-Sud, Département de Mathématiques, Orsay (1980)

  19. Shiohama K.: Total curvatures and minimal areas of complete surfaces. Proc. Amer. Math. Soc. 94(2), 310–316 (1985)

    MathSciNet  MATH  Google Scholar 

  20. Solanes, G.: Total curvature of complete surfaces in hyperbolic space, Preprint arXiv:0902.0672v1 [math.DG]

  21. Wintgen, P.: Normal cycle and integral curvature for polyhedra in Riemannian manifolds, Colloquia Math. Soc. J. Bolyai 31 Differential Geometry, Budapest 1979, North-Holland Publishing Comp. (1982)

  22. Zähle M.: Curvatures and currents for unions of sets with positive reach. Geom. Dedicata 23(2), 155–171 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Provence, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453, Marseille Cedex 13, France

    Nicolas Dutertre

Authors
  1. Nicolas Dutertre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Dutertre.

Additional information

Supported by Agence Nationale de la Recherche (reference ANR-08-JCJC-0118-01).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dutertre, N. Euler characteristic and Lipschitz–Killing curvatures of closed semi-algebraic sets. Geom Dedicata 158, 167–189 (2012). https://doi.org/10.1007/s10711-011-9627-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-011-9627-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation