On the rank of horizontal maps

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....

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Bibliographic Details
Published in:Mathematical proceedings of the Cambridge Philosophical Society 1982-11, Vol.92 (3), p.485-488
Main Author: Rawnsley, J. H.
Format: Article
Language:English
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Summary: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.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004100060187