Prof. Székelyhidi's lecture
Let $M$ be an $n$-dimensional compact Riemannian manifold with $n\geq 5$. As is well known, a unit-length vectorfield $V$ on $M$ exists if and only if the Euler characteristic $\chi(M)$ vanishes. Thurston showed that under this condition one can also find a codimension-1 foliation on $M$: that is, a unit-length vectorfield $V$ whose perpendicular hyperplane-distribution is integrable.
In the talk we discuss the possibility that the vectorfield is in addition divergence-free. If the perpendicular hyperplane-distribution is integrable, such vectorfields $V$ correspond to codimension-1 foliations by minimal surfaces and in this case $V$ is called a calibration. Such foliations are rather special and their existence depends very much on the metric, not just the topology. It turns out however, that in the opposite ``contact'' case, when the perpendicular hyperplane-distribution is non-integrable, the existence of unit-length divergence-free vectorfields is determined solely by the Euler characteristic, provided $n\geq 5$. This can be proved using a method of construction very similar to Nash's construction of smooth isometric embeddings.