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On the rank of horizontal maps

Published online by Cambridge University Press:  24 October 2008

J. H. Rawnsley
J. H. Rawnsley
Affiliation:
University of Warwick
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Extract

Let M, N be smooth manifolds and ℋ ⊂ TN a smooth sub-bundle. A smooth map φ:M → N will be called horizontal if

At points where φ(M) is a submanifold of N, the tangent spaces to φ(M) will be sub-spaces of the fibres of ℋ. In other words where φ(M) is a submanifold it is an integral submanifold of ℋ. If ℋ is not integrable it will follow that the rank of Æ must be less than dim ℋx. But we can often place much stricter bounds on the rank of φ by examining the integrability tensor of ℋ. This we shall do in this note.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 3 , November 1982 , pp. 485 - 488
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCE

(1)Eells, J. and Wood, J. C. Harmonic maps from surfaces to complex projective spaces (To appear in Advances in Mathematics).Google Scholar