Suchergebnisse

We give a detailed proof of an old result of N. Wallach and ourselves [1] which never appeared and which was asked by several colleagues in the last year.

Soit C une courbe voisine d'une courbe donnée C dans un espace riemannien à n dimensions, telle que le point P de C correspondant à un point quelconque P de C se trouve sur le vecteur unitaire ξ(3) de la seconde normale à C en P et à une distance infinitésimale ε de P. Soit ξ(3) le vecteur obtenu à partir du vecteur unitaire ξ(3) de la seconde normale à C en P en faisant subir à ce dernier un déplacement parallèle infinitésimal le long de ξ(3) jusqu'en P. On obtient n-2 conditions nécessaires et suffisantes pour que ξ'(3), coïncide avec ξ(3) au premier ordre près en ε.

Résumé. — Comme il a été annoncé dans [6], on démontre ici qu'un espace à segments qui est localement de dimension deux en un point, possède un voisinage homéomorphe au plan euclidien. Il s'en suit qu'un espace à segments, globalement deuxdimensionnel est une variété de dimension deux.

Un espace à segments est un espace topologique muni d'une famille de parties, les segments, satisfaisant à des axiomes choisis parmi les propriétés des arcs géodésiques minimaux d'une variété riemannienne connexe complète. On démontre que, si un espace à segments est unidimensionnel en un point, il est unidimensionnel en tout point et est en fait une variété de dimension 1.

In Indonesia, polymusic – juxtaposed music – is performed during major ceremonies. Several groups play simultaneously, in the same space, but different tunes. If many tunes are performed at the same time in the same space, who can listen to them ? Seven cases of Indonesian polymusic, from South Sulawesi, West Kalimantan, and Bali, recorded between 1991 and 2001 are described : a Sa'dan Toraja funeral (pasonglo), house ceremony (bua' sangrapu), and trance ritual (maro pabalikan), a Dayak Taman ritual for the ancestors (gawai mamandung), a Balinese temple ceremony (odalan). The seven examples are compared through an analysis of space and time, in order to disclose their common aspects and relevant meanings. Polymusic presents the musicologist with a paradox of temporal and spatial perception. While assembling groups of singers at the same time and in the same space, these rituals prescribe their separation and individualization. The analysis of space and time points to a double perception : that of a differentiated and an undifferentiated world. This is translated on the acoustic level by the simultaneous juxtaposition and synthesis of different repertoires and different groups, which reveals – both sensorily and intellectually – either the groups' differences or the achievement of a macro-unity expressing the ritual expenditure.

In Riemannian space Hayden [1] studied the asymptotic lines of order p and Srivastava [4] obtained geodesies of order p for the same space. For the Finsler space Srivastava and Sinha studied the asymptotic lines of order p in one of their papers [5]. In this paper we define geodesies of order p, investigate their properties and establish relationship with the asymptotic lines of order p.

We prove that every cone in a Hilbert space satisfies an optimal quasinormality condition and obtain an eigenvalue theorem for k-set contractions on cones.

Asymptotic lines of order p have been defined by Hayden in [1]. In this paper we define geodesies of order p, investigate their properties and establish their relationship with the asymptotic lines of order p.

A space with which is associated a non-symmetric fundamental tensor gij has been termed «a generalised Riemannian space » by Eisenhart (1951). The object of this paper is to study the subspaces of a generalised Riemannian space.