Deux types de courbures sont associés à un sous-ensemble compact et définissable d'une variété riemannienne analytique réelle. Si la variété est de courbure constante, il y a des relations linéaires entre ces mesures. Comme application, nous démontrons une formule cinématique, définissons des densités locales, et nous étudions les volumes des simplexes réguliers.
Two types of curvatures are associated to a compact, definable subset of a real analytic Riemannian manifold. If the manifold has constant curvature, there are some linear relations between these measures. As application, a kinematic formula is proved, local densities are defined and volumes of regular simplexes are studied.
Mot clés : courbures, espaces sous-analytiques, formule cinématique, densités
Keywords: curvatures, subanalytic spaces, kinematic formula, densities
@article{AIF_2003__53_6_1897_0,
author = {Bernig, Andreas and Br\"ocker, Ludwig},
title = {Courbures intrins\`eques dans les cat\'egories analytico-g\'eom\'etriques},
journal = {Annales de l'Institut Fourier},
pages = {1897--1924},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {53},
number = {6},
year = {2003},
doi = {10.5802/aif.1995},
mrnumber = {2038783},
zbl = {1053.53053},
language = {fr},
url = {http://www.numdam.org/articles/10.5802/aif.1995/}
}
TY - JOUR AU - Bernig, Andreas AU - Bröcker, Ludwig TI - Courbures intrinsèques dans les catégories analytico-géométriques JO - Annales de l'Institut Fourier PY - 2003 SP - 1897 EP - 1924 VL - 53 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1995/ DO - 10.5802/aif.1995 LA - fr ID - AIF_2003__53_6_1897_0 ER -
%0 Journal Article %A Bernig, Andreas %A Bröcker, Ludwig %T Courbures intrinsèques dans les catégories analytico-géométriques %J Annales de l'Institut Fourier %D 2003 %P 1897-1924 %V 53 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1995/ %R 10.5802/aif.1995 %G fr %F AIF_2003__53_6_1897_0
Bernig, Andreas; Bröcker, Ludwig. Courbures intrinsèques dans les catégories analytico-géométriques. Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1897-1924. doi : 10.5802/aif.1995. http://www.numdam.org/articles/10.5802/aif.1995/
[1] On metrics properties of stratified sets, Manuscripta Math, Volume 111 (2003), pp. 71-95 | DOI | MR | Zbl
[2] Scalar Curvature of definable Alexandrov spaces, Advances in Geometry, Volume 2 (2002), pp. 29-55 | DOI | MR | Zbl
[3] Scalar Curvature of definable CAT spaces, Advances in Geometry, Volume 3 (2003), pp. 23-43 | DOI | MR | Zbl
[4] Lipschitz-Killing invariants, Mathematische Nachrichten, Volume 245 (2002), pp. 5-25 | DOI | MR | Zbl
[5] Géométrie Algébrique Réelle, Springer-Verlag, 1987 | MR | Zbl
[6] Metric Spaces of Non-Positive Curvature, Springer-Verlag, 1999 | MR | Zbl
[7] Integral geometry of tame sets, Geom. Dedicata, Volume 82 (2000), pp. 285-323 | DOI | MR | Zbl
[8] On the curvature of piecewise flat spaces, Comm. Math. Phys, Volume 92 (1984), pp. 405-454 | DOI | MR | Zbl
[9] Kinematic and tube formulas for piecewise flat spaces, Indiana Univ. Math. I, Volume 35 (1986), pp. 737-754 | DOI | MR | Zbl
[10] Équisingularité réelle : invariants locaux et conditions de régularité (2001) (preprint Université de Nice)
[11] Tame Geometry with Applications in Smooth Analysis (livre à paraître) | Zbl
[12] An introduction to o-minimal geometry (2000) (Universitá di Pisa, Dipartimento di Matematica)
[13] Geometric Measure Theory, Springer Verlag, Berlin-Heidelberg-New York, 1968 | MR | Zbl
[14] About geodesic distance on Riemannian stratified spaces (1997) (preprint)
[15] Curvature measures of subanalytic sets, Amer. J. Math, Volume 116 (1994), pp. 819-880 | DOI | MR | Zbl
[16] Kinematic formulas in integral geometry, Indiana Univ. Math. J, Volume 39 (1990), pp. 1115-1154 | DOI | MR | Zbl
[17] Monge-Ampère functions I, Indiana Univ. Math. J, Volume 38 (1989), pp. 745-771 | DOI | MR | Zbl
[18] Stratified Morse Theory, Springer Verlag, Berlin-Heidelberg, 1988 | MR | Zbl
[19] On stratified Morse theory, Topology, Volume 38 (1999), pp. 427-438 | DOI | MR | Zbl
[20] Sheaves on Manifolds, Springer Verlag, Berlin-Heidelberg-New York, 1990 | MR | Zbl
[21] Integralgeometrie Whitney-Stratifizierter Mengen, Dissertation Münster (1999) | Zbl
[22] Densité des ensembles sous-analytiques, Ann. Inst. Fourier, Volume 39 (1989) no. 3, pp. 753-771 | DOI | EuDML | Numdam | MR | Zbl
[23] Tame Topology and O-Minimal Structures, London Math. Soc. Lecture Notes Ser, 248, Cambridge University Press, 1998 | MR | Zbl
[24] Geometric Categories and o-minimal structures, Duke Math. Journal, Volume 84 (1996), pp. 497-540 | DOI | MR | Zbl
[25] Introduction to Integral Geometry, Publications de l'Institut Mathématique de l'Université de Nanc, Paris, 1953 | MR | Zbl
[26] Topology of singular spaces and constructible sheaves (, www.math.uni-hamburg.de/home/schuermann) | Zbl
[27] Differentialgeometrie und Faserbündel, Birkhäuser Verlag, Basel-Stuttgart, 1972 | MR | Zbl
[28] Espaces stratifiés réels, Stratifications, singularities and differential equations (travaux en cours), Volume 55 (1997), pp. 93-107 | Zbl
[29] On the volume of tubes, Amer. J. Math, Volume 61 (1939), pp. 461-472 | DOI | JFM | MR
[30] Curvature and currents for unions of sets with positive reach, Geom. Dedicata, Volume 23 (1987), pp. 155-171 | MR | Zbl
[31] Approximation and characterization of generalized Lipschitz-Killing curvatures, Ann. Global Anal. Geom, Volume 8 (1990), pp. 249-260 | DOI | MR | Zbl
Cité par Sources :
