2025/09/29 by Grabowska, Katarzyna, Grabowski, Janusz · 1 citation
#32Q60 #53C15 #53C30 #58B12 #58B20 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2509.25405
In the infinite-dimensional Banach setting, we consider general smooth Banach fibrations τ:M→ M0 and `(1,1)-tensors' N:TM→ TM that are projectable (in the obvious sense) onto Nijenhuis operators N0:TM0→ TM0 on M0. We prove that the vanishing of the Nijenhuis torsion of N0 is equivalent to the fact that the Nijenhuis torsion of N takes only vertical values, i.e., values in ker(Tτ). Consequences for almost complex structures on (real) Banach manifolds are also derived. As canonical examples, we define tangent lifts dT(N0):TT M0→ TT M0 of Nijenhuis operators N0 in the Banach category, and prove that they are automatically projectable for the canonical fibrations τM0:TM0→ M0. Finally, we comment on the projectability in the case of Banach homogeneous manifolds τ:G→ G/K, studied recently by some authors.