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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on a differential concomitant

Authors: P. R. Eiseman and A. P. Stone
Journal: Proc. Amer. Math. Soc. 53 (1975), 179-185
MSC: Primary 58A10
MathSciNet review: 0383445
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Abstract: If $ {\mathbf{h}}$ and $ {\mathbf{k}}$ are vector $ 1$-fotms, the vanishing of the concomitant $ [{\mathbf{h}},\;{\mathbf{k}}]$ is an integrability condition fot certain problems on manifolds. In the case that $ {\mathbf{h}} = {\mathbf{k}}$ the vanishing of the Nijenhuis tensor $ [{\mathbf{h}},\;{\mathbf{h}}]$ implies $ d(\operatorname{tr} {\mathbf{h}})$ is a conservation law for $ {\mathbf{h}}$, provided that $ \operatorname{tr} {\mathbf{h}}$ is not constant. When the trace of $ {\mathbf{h}}$ is constant, a conservation law for $ {\mathbf{h}}$ exists if one can find a vector $ 1$-form $ {\mathbf{k}}$ with nonconstant trace such that $ [{\mathbf{h}},\;{\mathbf{k}}] = 0$.

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Keywords: Vector $ 1$-form, differential concomitant, differential form, conservation law
Article copyright: © Copyright 1975 American Mathematical Society