Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-18T19:06:17.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic Riemann manifolds

Published online by Cambridge University Press:  22 January 2016

Kazuo Yamato
Kazuo Yamato*
Affiliation:
Department of Mathematics, College of General Education, Nagoya University, Chikusa-ku, Nagoya 464, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present paper, we are concerned with the problem to know whether two algebraic Riemann manifolds are isometric or not, where we mean Riemann manifolds of class CΩ (real algebraic smoothness or Nash category) simply by algebraic Riemann manifolds.

For this purpose we introduce notions of minimal differential polynomials and singular base points, both of which are isometry invariants. With the aid of these invariants, we give the following isometry theorem for algebraic Riemann manifolds:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 115 , September 1989 , pp. 87 - 104
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Atiyah, M., Bott, R. and Patodi, V. K., On the heat equation and the index theorem, Invent. Math., 19 (1973), 279–330.CrossRefGoogle Scholar
[2] Berger, M., Gauduchon, P. and Mazet, E., Le spectre d’une variété riemannienne, Lecture Notes in Math., 194, Springer, 1971.Google Scholar
[3] Cartan, E., Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, 1946.Google Scholar
[4] Cheeger, J. and Ebin, D., Comparison Theorems in Riemannian Geometry, North-Holland Publ., 1975.Google Scholar
[5] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Interscience, 1963 (Vol. I), 1969 (Vol. II).Google Scholar
[6] Nash, J., Real algebraic manifolds, Ann. of Math., 56 (1952), 405–421.CrossRefGoogle Scholar
[7] Nomizu, K., Sur les algebres de Lie de générateurs de Killing et l’homogénéité d’une variété Riemannienne, Osaka J. Math., 14 (1962), 45–51.Google Scholar
[8] Palais, R., Equivariant real algebraic differential topology, Part I, Smoothness categories and Nash manifolds, Notes, Brandeis Univ., 1972.Google Scholar
[9] Shiota, M., Nash manifolds, Lecture Notes in Math., 1269, Springer, 1987.Google Scholar
[10] Singer, I. M., Infinitesimally homogeneous spaces, Comm. Pure Appl. Math., 13 (1960), 685–697.CrossRefGoogle Scholar