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