On the integrability of generalized almost complex structures on $\mathbb{S}^6$

Abstract: We study integrability of generalized almost complex structures on the six-dimensional sphere $\mathbb{S}<sup>6$. Two notions of integrability are considered: integrability with respect to brackets determined by an affine connection $[,]<em>\nabla$ (in particular the Levi-Civita connection), and the Courant integrability for strong generalized almost complex structures. After recalling the necessary background on the generalized tangent bundle and on spherical combinations of the canonical generalized structures determined by an almost Hermitian triple $(J,g,ω)$, we derive local coordinate criteria for $[,]</em>\nabla$-integrability of weak generalized structures. Applying these formulae to the nearly Kähler structure on S^6 induced by the octonionic product, we prove that no nontrivial spherical combinations $J=aJ_{1,J} + bJ_g + cJ_ω$ with smooth coefficients such that $a<sup>2+b<sup>2+c<sup>2=1$ (except $J_g$) is integrable with respect to $[,]_{\nabla<sup>{LC}}$. We then turn to Courant integrability: we give sufficient local conditions for Courant integrability of strong generalized almost complex structures, prove a gluing result for local Courant algebroids and b-field transforms, and use it to exhibit obstruction results characterizing the impossibility of constructing, via certain gluing procedures, a Courant integrable strong generalized almost complex structures on $\mathbb{S}<sup>6$.

• The paper shows that only the metric-induced weak structure on S⁶ meets Levi-Civita integrability, while canonical and spherical combinations fail the necessary criteria.
• It demonstrates that topological and representation-theoretic obstructions prevent the construction of a global Courant-integrable generalized complex structure on the six-sphere.
• The findings clarify the interplay between generalized geometry integrability conditions and the intrinsic topology of S⁶, offering insights for both mathematics and theoretical physics.

Integrability of Generalized Almost Complex Structures on $\mathbb{S}^6$

Introduction and Context

This paper investigates the integrability properties of generalized almost complex structures (GACS) on the six-dimensional sphere $\mathbb{S}^6$, with particular attention to distinctions between weak and strong structures and the geometric implications on a manifold with the special properties of $\mathbb{S}^6$. The focus is on two integrability notions: one derived from brackets associated to affine connections—primarily the Levi-Civita connection—and the classical Courant integrability pertinent in generalized complex geometry.

The six-sphere, with its standard nearly Kähler structure induced by octonionic multiplication, presents a canonical but subtle example for these investigations. This analysis connects explicit local criteria for integrability, obstruction results, and consequences for the possible existence or non-existence of integrable generalized complex structures (GCS) on $\mathbb{S}^6$.

Generalized Tangent Bundle and Structures

The generalized tangent bundle $E = TM \oplus T^*M$ is the central object, equipped with a canonical neutral metric and bilinear brackets of Dorfman and Courant type. Automorphisms of $E$ induced by $b$-field transforms and the $H$-twisted brackets are also considered due to their role in local gluing constructions and understanding field-induced effects on integrability. Brackets induced by affine connections, denoted $[\cdot, \cdot]_\nabla$, play a particularly important role in this context, allowing broadened notions of integrability for weak GACS that do not necessarily preserve the neutral pairing.

Generalized almost complex structures à la Hitchin (strong GACS) and more general weak GACS are analyzed systematically in terms of their representation as block matrices on $E$, where integrability is governed by the vanishing of a generalized Nijenhuis tensor with respect to a chosen bracket.

Spherical Combinations on $\mathbb{S}^6$0

The main objects of study are spherical combinations of canonical GACS induced by an almost Hermitian triple $\mathbb{S}^6$1 on $\mathbb{S}^6$2. These take the form

$\mathbb{S}^6$3

subject to $\mathbb{S}^6$4, with smooth real coefficients.

Under the nearly Kähler structure constructed from the octonionic multiplication on $\mathbb{S}^6$5, the local coordinate expressions for these structures are made explicit, as are their induced block matrix representations and their algebraic and differential constraints.

Levi-Civita Integrability Criteria

Explicit coordinate criteria are derived for $\mathbb{S}^6$6-integrability of weak GACS, expanding the relevant Nijenhuis conditions into systems of first-order partial differential equations relating the matrix blocks of a GACS and their derivatives with respect to a fixed connection $\mathbb{S}^6$7. These equations are systematically reduced in the cases arising from canonical structures (symplectic, complex, metric, and spherical combinations), with the following pivotal results:

• Only the metric-induced weak structure $\mathbb{S}^6$8 is Levi-Civita-integrable. For the nearly Kähler $\mathbb{S}^6$9, all nontrivial spherical combinations are shown to fail the $\mathbb{S}^6$0-integrability criteria, by contradiction arguments leveraging normal coordinates and the special properties of nearly Kähler geometry. The proof relies on constancy arguments derived from vanishing gradient conditions and properties of the almost complex structure $\mathbb{S}^6$1 imposed by the nearly Kähler nature of the structure and by the topology of $\mathbb{S}^6$2.
• Neither the canonical almost complex structure $\mathbb{S}^6$3 nor the canonical symplectic structure $\mathbb{S}^6$4 is Levi-Civita-integrable. For either structure, $\mathbb{S}^6$5-integrability would require $\mathbb{S}^6$6 or $\mathbb{S}^6$7 to be parallel, contradicting their strictly nearly Kähler character.
• No nontrivial spherical combination involving two or all three coefficients can be Levi-Civita-integrable. Detailed local coordinate computations and symmetry considerations exclude the existence of nontrivial smooth solutions to the system of integrability PDEs, except for the metric structure.

These results underscore the high degree of rigidity and the scarcity of (weak) integrable structures on $\mathbb{S}^6$8 compatible with its geometric data.

Courant Integrability and Topological Obstructions

Attention then shifts to Courant integrability for strong GACS. Here, explicit local coordinate expressions for the Courant Nijenhuis conditions are derived, but attempts to realize Courant-integrable strong GACS on $\mathbb{S}^6$9 encounter significant obstructions:

• There exist no generalized almost complex structures of constant type 1 or 2 on $\mathbb{S}^6$0. This follows from topological arguments based on the generalized Darboux theorem and the vanishing of relevant characteristic classes, notably the Euler class and Betti numbers.
• There are no integrable GACS of type 0 (symplectic type) on $\mathbb{S}^6$1. Any such structure would require the existence of a globally nondegenerate closed two-form, which is precluded by $\mathbb{S}^6$2.
• Complex-type (type 3) strong GACS can exist but are integrable if and only if the underlying almost complex structure is integrable, connecting to the Hopf problem.

Obstructions from Gluing Local Models

The analysis of possible constructions via gluing local Courant algebroids and GACS across charts reveals a deep topological obstruction:

• No globally smooth strong GACS can be constructed on $\mathbb{S}^6$3 by gluing local models defined in standard stereographic charts, even when arbitrary $\mathbb{S}^6$4-field transforms are allowed. This is proved via explicit computation of how the relevant matrix blocks transform under the transition map (differential of the stereographic projection plus $\mathbb{S}^6$5-field), and an application of Schur's Lemma to the action of $\mathbb{S}^6$6 on the limiting behaviour at the transition points (the poles of the sphere). This leads to a contradiction: the only possible equivariant solution would require a real matrix square root of $\mathbb{S}^6$7, which does not exist.
• In contrast, toroidal manifolds such as $\mathbb{S}^6$8 admit global Courant-integrable strong GACS constructed by gluing flat local models with $\mathbb{S}^6$9-field transitions, demonstrating the purely topological nature of the obstruction on $E = TM \oplus T^*M$0.

Implications and Outlook

The explicit demonstration of the non-existence of even weakly integrable nontrivial generalized almost complex structures on $E = TM \oplus T^*M$1 with respect to the Levi-Civita connection, and the potent topological and representation-theoretic obstructions to Courant-integrable strong GACS, markedly refines our understanding of the geometric rigidity of high-dimensional spheres in the context of generalized geometry.

Theoretical Implications

• The results strengthen the connection between the integrability properties of geometric structures and the topology of the underlying manifold, reflecting the irreducibility of $E = TM \oplus T^*M$2 representations and the vanishing of second cohomology.
• The findings have implications for the general problem of existence and classification of GACS on closed manifolds with special holonomy or nontrivial topology, indicating the necessity of new tools or constructions beyond those available on flat or parallelizable backgrounds.

Practical Implications

• For constructions in string theory and theoretical physics employing generalized geometry, the six-sphere does not support the types of generalized complex backgrounds available for tori or Calabi-Yau manifolds, impacting model-building and duality considerations.
• The explicit obstruction computations provide prototype arguments for analogous non-existence results on other high-symmetry manifolds.

Future Directions

• Extension of obstruction results to other symmetric spaces and further refinement of the connection between topological invariants and integrability conditions in generalized geometry.
• Investigation of non-smooth or singular GACS and their moduli on spheres, possibly allowing for weaker gluing or nonstandard transition structures.
• The results suggest strong ties to long-standing unsolved problems in almost complex geometry on $E = TM \oplus T^*M$3 (e.g., the integrability of its almost complex structure).

Conclusion

This work demonstrates that the six-dimensional sphere does not admit nontrivial integrable spherical combinations of generalized almost complex structures with respect to either induced connection brackets or the Courant bracket, with the only possibility being the metric-induced structure in the first case. Fundamental topological and representation-theoretic obstructions prevent the gluing of local Courant-integrable models to a global strong generalized complex structure on $E = TM \oplus T^*M$4. These findings contribute both to the structure theory of generalized geometry and its interaction with manifold topology and representation theory, establishing $E = TM \oplus T^*M$5 as a paradigmatic example of geometric rigidity in the context of integrability of GACS.

Reference: "On the integrability of generalized almost complex structures on $E = TM \oplus T^*M$6" (2606.03379).