Existence questions — Question 2.4

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct.

Statement recorded in the supplied source.

Given a vector field $X$ on a $3$-manifold $M$, characterize existence of a codimension-one foliation $\mathcal F$ satisfying $X(p)\notin T_p\mathcal F$ at every $p\in M$. In particular $X$ must be nowhere zero. The requested foliation is not additionally required to be taut. [S2]

Short English statement

Characterize vector fields admitting a foliation everywhere transverse to them.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300004 (AMR-102-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 2.4; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.