Vector field cycles in the tangent bundle

TL;DR

This paper studies vector fields on Riemannian manifolds by embedding them into tangent bundles with Sasaki metrics, characterizing minimal fields, especially on spheres, and analyzing their geometric properties and deformations.

Contribution

It introduces a new framework for analyzing vector fields via their embeddings into tangent bundles, characterizing minimal and contact structures, and linking them to curvature functionals.

Findings

Minimal unit vector fields on spheres are contact structures with specific Levi forms.

The zero section is a critical cycle if the scalar curvature is constant.

Certain vector fields are characterized as eigenvectors of a perturbed Laplacian.

Abstract

Given a closed Riemannian manifold $(M^m,g)$ and a vector field $v$ on $M$, we form the Sasaki metric $g_S$ on $TM$, and restrict it to the image of the cross section map of $M$ into $TM$ defined by $v$, whose pull back to $M$ defines a new metric $g(v)$ on $M$. We then view the cross section as an isometric embedding $f_{g(v)}: (M,g(v))\rightarrow (TM,g_S)$, which when $\| v\|_g=1$, ranges into the unit sphere bundle $(S^1(TM),g_S)$. $v$ is minimal or minimal unit if these embeddings have null mean curvature vectors, conditions that occur if, $v$ is in the kernel or is an eigenvector, respectively, of a first order perturbation of a weighted rough Laplacian, the weights and perturbation determined by the covariant derivatives $\nabla^g_{e}v$ along unit directions $e$ in suitable normal frames that include $v$ when $\| v\|_g=1$, and curvature tensor of $g$. A minimal unit field must be Killing, and other than parallel fields, $v=0$ is the only minimal one. We characterize the minimal unit vector fields on the standard sphere $(\mb{S}^{2n+1},g) \hookrightarrow (\mb{R}^{2n+2},\| \, \|^2)$ as those defining contact strictly pseudoconvex CR structures whose Levi form and sign are determined by $g$ and the orientation. If $\Theta_{f_{g(v)}}(M)$ and $\Phi_{f_{g(v)}}(M)$ are the total exterior scalar curvature and squared $L^2$ norm of the mean curvature vector functionals, and $m>2$, a canonical cycle $f_{g(v)}(M)$ is a critical point of the functional $(m/m-1) \Theta_{f_{g(v)}}(M) +\Phi_{f_{g(v)}}(M)$ under conformal deformations, notion conveniently defined also when $m\leq 2$. The zero section of $TM$ is a canonical cycle if, and only if, the scalar curvature of $g$ is constant. We describe some examples of these vector fields and cycles, and analyze their deformations under dilations of the field.