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A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers
We prove a sufficient condition for the existence of explicit first integrals
for vector fields which admit an integrating factor. This theorem recovers and
extends previous results in the literature on the integrability of vector
fields which are volume preserving and possess nontrivial normalizers. Our
approach is geometric and coordinate-free and hence it works on any smooth
orientable manifold
Contact structures and Beltrami fields on the torus and the sphere
We present new explicit tight and overtwisted contact structures on the
(round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly
compatible. Our proofs are based on the construction of nonvanishing curl
eigenfields using suitable families of Jacobi or trigonometric polynomials. As
a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk
and Massot (2012) does not hold for weakly compatible metric as it was
conjectured. We also establish a geometric rigidity for tight contact
structures by showing that any contact form on the 3-sphere admitting a
compatible metric that is the round one is isometric, up to a constant factor,
to the standard (tight) contact form.Comment: 19 pages; version accepted for publication (Indiana University
Mathematics Journal
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