Nijenhuis operators on Banach fibration
Katarzyna Grabowska, Janusz Grabowski
TL;DR
The paper extends Nijenhuis operator theory to infinite-dimensional Banach fibrations by introducing projectable ${\mathcal N}$-operators and proving that ${\mathcal T}_{N_0}=0$ on the base iff ${\mathcal T}_N$ is vertical on the total space. It further develops a tangent lift construction $d_{\mathsf T}(N)$, showing it preserves Nijenhuis-ness and remains projectable, thereby enabling integrable structures to be lifted to higher tangent bundles. The work also analyzes homogeneous Banach manifolds, establishing criteria for projecting invariant Nijenhuis operators from a Banach Lie group to its homogeneous space and linking these to complex-structure integrability via Lie-algebra conditions. Collectively, these results provide a robust framework for studying integrable, complex, and geometric structures in infinite-dimensional settings and apply to Banach homogeneous spaces through a Lie-algebraic reduction. The findings have potential implications for the geometry of infinite-dimensional manifolds and the study of integrable systems in Banach settings.
Abstract
In the infinite-dimensional Banach setting, we consider general smooth Banach fibrations $τ:M\to M_0$ and `$(1,1)$-tensors' $N:TM\to TM$ that are projectable (in the obvious sense) onto Nijenhuis operators $N_0:TM_0\to TM_0$ on $M_0$. We prove that the vanishing of the Nijenhuis torsion of $N_0$ 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 $d_T(N_0):TT M_0\to TT M_0$ of Nijenhuis operators $N_0$ in the Banach category, and prove that they are automatically projectable for the canonical fibrations $τ_{M_0}:TM_0\to M_0$. Finally, we comment on the projectability in the case of Banach homogeneous manifolds $τ:G\to G/K$, studied recently by some authors.