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The Equality of a Manifold's Rank and Dimension

Published online by Cambridge University Press:  20 November 2018

R. S. D. Thomas
R. S. D. Thomas*
Affiliation:
University of Zambia, Lusaka
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T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 2 , April 1969 , pp. 183 - 184
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Willmore, T.J., Connexions for systems of parallel distributions. Quart. J. Math. Oxford (2) 7 (1956) 269–276.Google Scholar
2. Gottschalk, W. H. and Hedlund, G. A., Topological dynamics. American Mathematical Society Colloquium Publications Volume XXXVI Providence, R.I., 1955.Google Scholar