Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T01:53:34.458Z Has data issue: false hasContentIssue false
  • English
  • Français

XX.—Differential Geometry on Hypersurfaces in a Cayley Space

Published online by Cambridge University Press:  14 February 2012

K. Yano  and
T. Sumitomo
Get access Rights & Permissions [Opens in a new window]

Extract

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.

It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.

For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 66 , Issue 4 , 1964 , pp. 216 - 231
Copyright
Copyright © Royal Society of Edinburgh 1964

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References to Literature

Calabi, E., 1958. “Construction and Properties of some 6-dimensional almost complex Manifolds”, Trans. Amer. Math. Soc, 87, 407–438.Google Scholar
Cartan, E., 1939. “Sur quelques families remarquables d'hypersurfaces”, C. R. Congr. Math. Liége, 30–41.Google Scholar
Eckmann, B., and Frölicher, A., 1951. “Sur l'intégrabilité des structures presque complexes”, C. R. Acad. Sci. Paris, 232, 2284–2286.Google Scholar
Ehresmann, C, 1952. “Sur les variétés presque complexes”, Proc. Int. Congr. Math., 2, 412–419.Google Scholar
Ehresmann, C, and Libermann, P., 1951. “Sur les structures presque her-mitiennes isotropes”, C. R. Acad. Sci. Paris, 232, 1281–1283.Google Scholar
Freudenthal, H., 1951. Oktaven, Ausnahmengruppen und Oktavengeometrie. Utrecht.Google Scholar
Frölicher, A., 1955. “Zur Differentialgeometrie der komplexen Strukturen”, Math. Ann., 129, 50–95.Google Scholar
Fukami, T., and Ishihara, S., 1955. “Almost-Hermitian structure on S6”, Tohoku Math. J., 7,151–156.Google Scholar
Koto, S., 1960. “Some Theorems on almost Kaehlerian Spaces”, J. Math. Soc. japan, 12, 422–433.Google Scholar
Schouten, J. A., and Yano, K., 1955. “On Invariant Subspaces in the almost complex X2n”, Indag. Math., 17, 261–269.Google Scholar
Tachibana, S., 1959. “On almost Analytic Vectors in certain almost Hermitian Manifolds”, Tohoku Math. J., II, 351–363.Google Scholar
Yano, K., 1957. “The theory of Lie derivatives and its applications” Amsterdam.Google Scholar
Yano, K., and Bochner, S., 1953. “Curvature and Betti numbers”, Ann. Math. Study., 32.Google Scholar