2025/08/24 by Simanca, Santiago R.
#53C20 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.17441
Given a closed Riemannian manifold (Mm,g) and a vector field v on M, we form the Sasaki metric gS 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 fg(v): (M,g(v))→ (TM,gS), which when ‖ v‖g=1, ranges into the unit sphere bundle (S1(TM),gS). 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 ∇gev 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 (\mbS2n+1,g) \hookrightarrow (\mbR2n+2,‖ ‖2) as those defining contact strictly pseudoconvex CR structures whose Levi form and sign are determined by g and the orientation. If Θ_fg(v)(M) and Φ_fg(v)(M) are the total exterior scalar curvature and squared L2 norm of the mean curvature vector functionals, and m>2, a canonical cycle fg(v)(M) is a critical point of the functional (m/m-1) Θ_fg(v)(M) +Φ_fg(v)(M) under conformal deformations, notion conveniently defined also when m≤ 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.