Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T22:56:39.118Z Has data issue: false hasContentIssue false
  • English
  • Français

On Certain Mappings of Riemannian Manifolds

Published online by Cambridge University Press:  22 January 2016

Minoru Kurita
Minoru Kurita*
Affiliation:
Mathematical Institute, 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.

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965 , pp. 121 - 142
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Adati, T., On subprojective spaces I-VI, Tôhoku Math. J. 3 (1951), pp. 159–159, 330–330, 343–343; Tensor I (1951), pp. 105–105, 116–116, 130–130.Google Scholar
[2] Chern, S. S., Topics in differential geometry. Mimeographed note. The institute of advanced study, Princeton (1951).Google Scholar
[3] Hano, J., On affine transformations of a Riemannian manifold. Nagoya Math. J. 9 (1955), pp. 99–99.CrossRefGoogle Scholar
[4] Kobayashi, S., A theorem on the affine transformation group of a Riemannian manifold. Nagoya Math, J. 9 (1955), pp. 39–39.Google Scholar
[5] Kurita, M., On conformal Riemann spaces.- J. Math. Soc. Japan 7 (1955), pp. 13–13.Google Scholar
[6] Kurita, M., Geodesic correspondence of Riemann spaces. J. Math. Soc. Japan (1956),pp. 22–22.Google Scholar
[7] Yano, K., Concircular geometry, I Concircular transformations. Proc. Imp. Acad. Japan (1940), pp. 195–195.Google Scholar
[8] Yano, K. Subprojective transformations, subprojective spaces and subprojective collineations. Proc. Imp. Acad. Japan (1944), pp. 701–701.Google Scholar
[9] Yano, K., On harmonic and Killing vector fields. Annals of Math. 55 (1952), pp. 38–38.Google Scholar