Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-21T15:41:45.723Z Has data issue: false hasContentIssue false
  • English
  • Français

On Automorphisms of A Kählerian Structure

Published online by Cambridge University Press:  22 January 2016

Shoshichi Kobayashi  and
Katsumi Nomizu
Shoshichi Kobayashi
Affiliation:
University of Washington, Nagoya University
Katsumi Nomizu
Affiliation:
University of Washington, Nagoya University
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.

Is every isometry, or more generally, every affine transformation of a Kählerian manifold a complex analytic transformation? The answer is certainly negative in the case of a complex Euclidean space. This question has been recently studied by Lichnerowicz [8] and Schouten-Yano [11] from the infinitesimal point of view; they have found some conditions in order that every infinitesimal motion of a Kählerian manifold preserve the complex structure. (As a matter of fact, [11] has dealt with the case of a pseudo-Kählerian manifold, which does not differ essentially from a Kählerian manifold as far as the question at hand is concerned.)

In the present paper, we generalize their results by a different approach. In order to explain our main idea, we shall first give a few definitions (1 and 2) and state our main results (3). The proofs are given in the subsequent sections.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 11 , February 1957 , pp. 115 - 124
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

[1] Eckmann, B., Sur les structures complexes et presque complexes, Colloque de Géométrie Différentielle, Strasbourg (1953),151–159.Google Scholar
[2] Hano, J., On affine transformations of a Riemannian manifold, Nagoya Math. J. 9 (1955),99–109.CrossRefGoogle Scholar
[3] Hano, J. and Matsushima, Y., Some studies on Kählerian homogeneous spaces, this journal.Google Scholar
[4] Iwamoto, H., On the structure of Riemannian space whose holonomy groups fix a null-system, Tohoku Math. J. 2nd ser. 1 (1950),109–125.Google Scholar
[5] Kobayashi, S., A theorem on the affine transformation group of a Riemannian manifold, Nagoya Math. J. 9 (1955),39–41.CrossRefGoogle Scholar
[6] Kobayashi, S., Principal fiber bundles with the 1-dimensinal toroidal group, to appear in Tohoku Math. J. Google Scholar
[7] Lichnerowicz, A., Espaces homogènes kählériens, Colloque de Géométrie Différentielle, Strasbourg (1953),171–184.Google Scholar
[8] Lichnerowicz, A., Sur les groupes d’automorphismes de certaines variétés kählérien-nes, C. R. Acad. Sci. Paris, 239 (1954), 1344–1345.Google Scholar
[9] Nomizu, K., Studies on Riemannian homogeneous spaces, Nagoya Math. J. 9 (1955),43–56.CrossRefGoogle Scholar
[10] de Rham, G., Sur la réductibilité d’un espace de Riemann, Comm. Math. Helv. 26 (1952),328–344.CrossRefGoogle Scholar
[11] Schouten, J. A. and Yano, K., On pseudo-Kählerian spaces admitting a continuous group of motions, Indagationes Math. 17 (1955),565–570.CrossRefGoogle Scholar