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Cours spécialisés - Parutions - 23 (2016)

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Integral geometry from Buffon to geometers of today
Rémi Langevin
Cours spécialisés 23 (2016), 284 pages
Acheter l'ouvrage

Résumé :
La géométrie intégrale de Buffon jusqu'aux géomètres d'aujourd'hui
La géométrie intégrale, aussi appelée théorie des probabilités géométriques, a accompagné pendant plus de deux siècles le développement des probabilités, de la théorie de la mesure et de la géométrie. Elle commence pour nous en 1777, date de la publication du «traité d'arithmétique morale» de Buffon. Ce n'est que presque un siècle plus tard que Crofton explicitera ce que veut dire mettre une mesure sur un ensemble continu comme l'ensemble des droites. Le sens de la formule de Cauchy-Crofton «la longueur d'une courbe plane est proportionnelle à la mesure pondérée de l'ensemble des droites qui la coupent», est maintenant clair. Au début du vingtième siècle, la géométrie intégrale commence à considérer les formes des objets étudiés. Minkowski, en 1901, a trouvé la première relation entre une intégrale de courbure sur Q et la mesure de l'ensemble des plans coupant Q. Steiner, Blaschke, Chern ont obtenu de nombreux résultats dans le même esprit concernant les courbes et les surfaces. La topologie est entrée dans le jeu pendant la seconde moitié du vingtième siècle. Toutefois le premier résultat de géométrie intégrale faisant intervenir la topologie est celui montré en 1929 par Fenchel : la courbure totale d'une courbe fermée plongée dans l'espace euclidien R^3 est supérieure ou égale à 2. Nous savons maintenant que si, de plus, la courbe est nouée, cette courbure totale est supérieure à 4 , ce qui indique qu'une topologie plus compliquée exige une géométrie plus compliquée. Nous montrerons bien d'autres exemples, dans des espaces euclidiens ou de courbure constante. Nous présenterons ensuite des résultats du même type concernant les courbes complexes et les feuilletages de variétés de courbure constante. Jusqu'à maintenant, il fallait que l'espace ambient soit muni d'une métrique. La géométrie locale des courbes, surfaces ou feuilletages que nous observions était mesurée en utilisant des fonctions de courbure. La géométrie conforme se comporte souvent différemment. Il faut souvent considérer la géométrie locale de paires de points. La contrepartie globale sera mesurée en considérant la position de ces courbes ou surfaces relativement à des sphères. La non-compacité de l'ensemble des 2-sphères de S^3 est une difficulté supplémentaire. Nous décrirons les premiers résultats obtenus dans cette direction.

Mots-clefs : Géométrie intégrale, courbes, surfaces, projections, sections, feuilletages, singularités algébriques, géométrie de Möbius, géométrie lorentzienne, noeuds, entrelacs.

Abstract:
Integral geometry, also called theory of geometric probabilities, followed during more than two centuries the development of probability, measure theory and geometry. The birthdate of integral geometry is for us the publication of Buffon's “traité d'arithmétique morale” in 1777. It is only almost a century later that Crofton will explicit what is a measure on a continuous set like the set of lines of the plane. The meaning of Cauchy-Crofton formula: “the length of a plane curve is proportional to the weighted measure of the set of lines intersecting it”, is now clear. At the beginning of the twentieth century, integral geometry considers shapes. Minkowski, in 1901, found the first relation between a curvature integral on the boundary Q of a convex body Q, and the measure of the planes intersecting Q. Steiner, Blaschke, Chern and Santaló provided many results about curves and surfaces in the same spirit. Topology played a role mainly during the second half of twentieth century. A seminal example is Fenchel's statement: the total curvature of a closed curve embedded in Euclidean space R^3 is larger than 2. Moreover, we know now that, if the curve is knotted, the total curvature is larger than 4, showing that ``more complicated topology demands more complicated geometry''. We will present many more examples, in Euclidean space or in space-forms. Complex curves, foliations of space-forms, also provide results of the same type. Until now, we considered ambient spaces endowed with a Riemannian metric. The geometry of the curves, surfaces or foliation was observed locally via curvature functions. Conformal geometry behaves differently: we will often need to involve the local geometry of pairs of points. The counterpart will be measured observing the relative position of curves or surfaces in ^3 with spheres. The non-compacity of the set of 2-spheres of S^3 also contributes to change the rules of the game. We will describe the first results obtained in this direction.

Keywords: Integral geometry, curves, surfaces, projections, sections, foliations, algebraic singularities, Möbius geometry, Lorentz geometry, knots, links.

Class. math. : 14H20, 32S55, 52A22, 52C65, 52M25, 53A30, 53C12, 53C42, 53C45, 53C50, 53C56, 53C99


ISBN : 978-2-85629-822-0
ISSN : 1284-6090

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