Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T19:46:09.336Z Has data issue: false hasContentIssue false

Note on Almost Product Manifolds and their Tangent Bundles

Published online by Cambridge University Press:  20 November 2018

Chorng Shi Houh*
Affiliation:
University of Manitoba
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.

Let Mn be an n-dimensional manifold of differentiability class C with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hsu, C.J., On some structures which are similar to the quaternion structure. TɄhoku Math. J. 12, (1960), pages 403-428.Google Scholar
2. Hsu, C. J., Remarks on certain almost product spaces. Pacific J. Math., 14, (1964), pages 163-176.Google Scholar
3. Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds. TɄhoku Math. J. 10, (1958), pages 338-354.Google Scholar
4. Tachibana, S. and Okumura, M., On the almost-complex structure of tangent bundles of Riemannian spaces. T?hoku Math. J. 14, (1962), pages 156-161.Google Scholar
5. Yano, K., Affine connections in an almost product space. Kadai Math. Sem. Rep. 11, (1959), pages 1-24.Google Scholar
6. Yano, K., Differential geometry on complex and almost complex spaces. Pergamon Press, (1965).Google Scholar