[Intrinsic curvatures in analytic-geometric categories]
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.
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.
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, Volume 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
Cited by Sources:
