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Rank of the Nijenhuis tensor on parallelizable almost complex manifolds

Lorenzo Sillari, Adriano Tomassini

TL;DR

This work investigates how the rank of the Nijenhuis tensor N_J measures the nonintegrability of almost complex structures on parallelizable manifolds. It develops a deformation framework that reduces the problem to solving first-order PDEs for smooth deformation data, enabling explicit construction of constant-rank structures on Kodaira-Thurston and Iwasawa manifolds and analysis of rank changes along curves. The authors then classify, for 6-dimensional nilpotent Lie algebras, which admit left-invariant almost complex structures with ranks 1, 2, or 3, providing explicit models and a large database of examples; they also derive a topological bound for left-invariant ranks on solvmanifolds. The results yield a detailed panorama of possible ranks across 6-nilmanifolds, including a complete table of algebras and ranks, and demonstrate how rank variations occur through deformations in both left-invariant and non-left-invariant settings.

Abstract

We study almost complex structures on parallelizable manifolds via the rank of their Nijenhuis tensor. First, we show how the computations of such rank can be reduced to finding smooth functions on the underlying manifold solving a system of first order PDEs. On specific manifolds, we find an explicit solution. Then we compute the Nijenhuis tensor on curves of almost complex structures, showing that there is no constraint (except for lower semi-continuity) to the possible jumps of its rank. Finally, we focus on $6$-nilmanifolds and the associated Lie algebras. We classify which $6$-dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for left-invariant structures on $6$-nilmanifolds. We also find a topological upper-bound for the rank of the Nijenhuis tensor for left-invariant almost complex structures on solvmanifolds of any dimension, obtained as a quotient of a completely solvable Lie group. Our results are complemented by a large number of examples.

Rank of the Nijenhuis tensor on parallelizable almost complex manifolds

TL;DR

This work investigates how the rank of the Nijenhuis tensor N_J measures the nonintegrability of almost complex structures on parallelizable manifolds. It develops a deformation framework that reduces the problem to solving first-order PDEs for smooth deformation data, enabling explicit construction of constant-rank structures on Kodaira-Thurston and Iwasawa manifolds and analysis of rank changes along curves. The authors then classify, for 6-dimensional nilpotent Lie algebras, which admit left-invariant almost complex structures with ranks 1, 2, or 3, providing explicit models and a large database of examples; they also derive a topological bound for left-invariant ranks on solvmanifolds. The results yield a detailed panorama of possible ranks across 6-nilmanifolds, including a complete table of algebras and ranks, and demonstrate how rank variations occur through deformations in both left-invariant and non-left-invariant settings.

Abstract

We study almost complex structures on parallelizable manifolds via the rank of their Nijenhuis tensor. First, we show how the computations of such rank can be reduced to finding smooth functions on the underlying manifold solving a system of first order PDEs. On specific manifolds, we find an explicit solution. Then we compute the Nijenhuis tensor on curves of almost complex structures, showing that there is no constraint (except for lower semi-continuity) to the possible jumps of its rank. Finally, we focus on -nilmanifolds and the associated Lie algebras. We classify which -dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for left-invariant structures on -nilmanifolds. We also find a topological upper-bound for the rank of the Nijenhuis tensor for left-invariant almost complex structures on solvmanifolds of any dimension, obtained as a quotient of a completely solvable Lie group. Our results are complemented by a large number of examples.
Paper Structure (16 sections, 24 theorems, 140 equations)

This paper contains 16 sections, 24 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Almost complex structures of prescribed rank on parallelizable manifolds
  3. Outline of the general procedure.
  4. The Kodaira-Thurston manifold.
  5. Holomorphically parallelizable complex -solvmanifolds
  6. Torus.
  7. Iwasawa Manifold.
  8. Nakamura manifold.
  9. Curves of almost complex structures
  10. Left-invariant almost complex structures on real -nilmanifolds
  11. Structures of rank 3 (maximally non-integrable).
  12. Structures of rank 2.
  13. Structures of rank 1.
  14. Complex structures.
  15. Consequences on homogeneous manifolds.
  16. ...and 1 more sections

Key Result

Theorem A

Let $(M,J_0)$ be a parallelizable almost complex manifold. Let $\{ \phi^j \}_{j=1}^m$ be a co-frame of $(1,0)$-forms for $J_0$. Assume that $J_1$ is an almost complex structure defined by a co-frame of $(1,0)$-forms of the form where $f^j_k \in C^\infty(M)$. Consider the curve of almost complex structures $J(s)$, $s\in \mathbb{C}$, defined by Then we have that at every point $x \in M$.

Theorems & Definitions (46)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 36 more