An almost Kähler Cheeger--Gromoll splitting theorem with applications

Abstract: In this paper, we establish an almost Kähler analogue of the Cheeger--Gromoll splitting theorem for complete almost Kähler manifolds with nonnegative Ricci curvature. As applications, we use the splitting to obtain Goldberg-type integrability results and establish a relation between symplectic non-hyperbolicity and nonnegative Ricci curvature via a theorem of Bangert.

• The paper proves that a complete almost Kähler manifold with nonnegative Ricci curvature and a Riemannian line splits on its universal cover with a complex Euclidean factor, using harmonicity and Bochner’s identity to show that the almost complex image of a parallel field is also parallel.
• The paper applies this structure theorem to show that closed symplectically aspherical examples are flat Kähler manifolds finitely covered by complex tori, while in dimension four nonnegative Ricci curvature forces integrability outside the rational-surface case.
• The paper establishes broad symplectic non-hyperbolicity results, proving the existence of nonconstant entire holomorphic curves for all tame almost complex structures on closed symplectically aspherical manifolds satisfying the curvature hypothesis.

Overview

This paper by Nguyen, Ning, and Pusey-Nazzaro establishes an almost Kähler analogue of the Cheeger–Gromoll splitting theorem and derives two families of consequences: Goldberg-type integrability results for the compatible almost complex structure, and a criterion for symplectic non-hyperbolicity under nonnegative Ricci curvature. The work is motivated by the broader program of finding symplectic counterparts of classical Riemannian rigidity theorems, following precedents such as McDuff's symplectic Cartan–Hadamard theorem in the Kähler setting (2608.18477) and its recent extension to the almost Kähler setting by Cristofaro-Gardiner.

The central technical observation is that on an almost Kähler manifold $(X,\omega,J,g)$ with $\operatorname{Ric}_g \geq 0$, every parallel vector field $V$ has a parallel image $JV$. This fails to be automatic when $J$ is not parallel, and it is precisely what forces the Euclidean factor produced by the Riemannian splitting theorem to be $J$-invariant.

The splitting theorem

An almost Kähler manifold is called split if it is isomorphic to a product $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$, where $C \cong \mathbb{R}_s\times\mathbb{R}_t$ carries the standard complex structure. The main theorem states:

Let $(X,\omega,J,g)$ be an almost Kähler manifold such that $g$ is complete with $\operatorname{Ric}_g \geq 0$0. If $\operatorname{Ric}_g \geq 0$1 contains a Riemannian line, then the universal cover $\operatorname{Ric}_g \geq 0$2 splits.

The proof proceeds in three steps. First, the classical Cheeger–Gromoll theorem applied to the line yields a Riemannian product decomposition with Busemann function gradient $\operatorname{Ric}_g \geq 0$3 parallel. Second, the key lemma shows that $\operatorname{Ric}_g \geq 0$4 is harmonic: $\operatorname{Ric}_g \geq 0$5 handles the $\operatorname{Ric}_g \geq 0$6 term (using $\operatorname{Ric}_g \geq 0$7), while Cartan's formula gives $\operatorname{Ric}_g \geq 0$8, and since a parallel field is Killing, its flow commutes with the Hodge star, so $\operatorname{Ric}_g \geq 0$9. The Bochner–Weitzenböck identity then gives $V$0, forcing $V$1; hence $V$2 is parallel. Third, since $V$3, the field $V$4 is tangent to the level sets of $V$5 and descends to a parallel unit field on the complementary factor; de Rham's decomposition theorem then splits the universal cover as $V$6, and the compatibility relation $V$7 propagates the splittings of $V$8 and $V$9 from that of $JV$0.

Iterating the theorem yields a structural corollary: for a closed almost Kähler manifold with $JV$1, the maximal Euclidean factor of the universal cover has even real dimension $JV$2, and the full splitting is compatible with the almost Kähler structure:

$JV$3

with $JV$4 closed and containing no line. This refines Oprea's cohomologically symplectic observation that the Euclidean rank of a closed c-symplectic manifold with nonnegative Ricci curvature is even; here the conclusion holds at the level of the symplectic form itself rather than merely cohomologically.

Automatic integrability

The splitting theorem feeds into two Goldberg-type results. Recall the Goldberg conjecture: a compact almost Kähler Einstein manifold is Kähler, proved by Sekigawa under nonnegative scalar curvature.

Dimension four. For closed almost Kähler four-manifolds with $JV$5, the paper obtains a trichotomy:

Diffeomorphism type	Integrability of $JV$6
Rational ($JV$7 or $JV$8, $JV$9)	Not forced
Irrational ruled ($J$0 or $J$1)	Forced
$J$2, Enriques, $J$3, hyperelliptic	Forced; $J$4 Ricci-flat

In the non-rational, non-ruled case, Kazdan–Warner conformal deformation to positive scalar curvature (when $J$5) combined with the Liu–Ohta–Ono theorem forces $J$6; Sekigawa's theorem then gives integrability, and LeBrun's classification of symplectic Einstein four-manifolds identifies the diffeomorphism types. In the irrational ruled case, virtual abelianness of $J$7 excludes genus $J$8 bases, and the splitting theorem applied to the universal cover — which contains a line by cocompactness — forces integrability of $J$9, hence of $J$0.

The rational case is genuinely exceptional: starting from any del Pezzo surface with a Kähler metric of positive Ricci curvature, one can perturb the complex structure to a non-integrable $J$1-compatible $J$2 arbitrarily close to $J$3; since positive Ricci curvature is $J$4-open, the resulting almost Kähler metric retains $J$5. This demonstrates sharply that nonnegative Ricci curvature alone does not force integrability in dimension four, and identifies exactly which topology obstructs the failure.

Symplectically aspherical manifolds. In all dimensions, if $J$6 is closed, symplectically aspherical (i.e., $J$7 for all smooth $J$8), and admits an almost Kähler metric with $J$9, then $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$0 is a flat Kähler manifold finitely covered by a complex torus. The argument is short: the splitting reduces the universal cover to $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$1 with $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$2 closed; asphericity forces $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$3 in de Rham cohomology via Hurewicz, so $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$4 is exact on the closed $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$5, contradicting non-degeneracy unless $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$6. The conclusion replaces delicate curvature hypotheses (Einstein, scalar curvature sign conditions) with a purely topological assumption — a complementary Goldberg-type mechanism.

Symplectic non-hyperbolicity

The paper defines a symplectic manifold $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$7 to be hyperbolic if every $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$8-holomorphic map $$(C \times X',\, ds\wedge dt + \omega',\, i\oplus J',\, ds^2+dt^2+g')$$9 is constant for every $C \cong \mathbb{R}_s\times\mathbb{R}_t$0-tame $C \cong \mathbb{R}_s\times\mathbb{R}_t$1, and non-hyperbolic otherwise — a symplectic analogue of Brody hyperbolicity. Two intermediate results are established:

First, an almost Kähler version: if $C \cong \mathbb{R}_s\times\mathbb{R}_t$2 is closed with $C \cong \mathbb{R}_s\times\mathbb{R}_t$3 and either $C \cong \mathbb{R}_s\times\mathbb{R}_t$4 is infinite or $C \cong \mathbb{R}_s\times\mathbb{R}_t$5, then $C \cong \mathbb{R}_s\times\mathbb{R}_t$6 admits a nonconstant $C \cong \mathbb{R}_s\times\mathbb{R}_t$7-holomorphic plane. The infinite fundamental group case follows directly from the splitting (a line exists by cocompactness, and the split factor $C \cong \mathbb{R}_s\times\mathbb{R}_t$8 supplies the plane). The finite-$C \cong \mathbb{R}_s\times\mathbb{R}_t$9 four-dimensional case combines the classification above with Kamenova–Lu–Verbitsky's non-hyperbolicity of K3 surfaces and symplectic uniruledness of rational surfaces via Gromov compactness.

Second, the main application: if $(X,\omega,J,g)$0 is closed, symplectically aspherical, and admits an almost Kähler metric with $(X,\omega,J,g)$1, then $(X,\omega,J,g)$2 is non-hyperbolic — i.e., every $(X,\omega,J,g)$3-tame almost complex structure admits a nonconstant $(X,\omega,J,g)$4-holomorphic plane. By the flat classification, the universal cover is $(X,\omega,J,g)$5, and the proof runs Bangert's scheme: the lifted tame structure $(X,\omega,J,g)$6 is bounded and uniformly tamed, so Bangert's Proposition 2.7 produces uniformly Lipschitz $(X,\omega,J,g)$7-holomorphic disks with $(X,\omega,J,g)$8; deck translations recenter them, and Arzelà–Ascoli together with Gromov's generalized Weierstrass theorem extracts a nonconstant entire curve. The result transfers Bangert's theorem for linear symplectic tori (2608.18477) from the flat setting to arbitrary compact flat Kähler quotients (generalized hyperelliptic manifolds in the sense of Catanese–Corvaja).

Limitations and open questions

Several boundaries of the results are explicit. The rational-surface exception in dimension four is sharp only up to $(X,\omega,J,g)$9: for $g$0, it remains unknown whether $g$1 admits an almost Kähler metric with $g$2 and non-integrable $g$3, because the perturbation argument requires a starting Kähler metric of positive Ricci curvature. Without symplectic asphericity, the non-hyperbolicity statement for all $g$4-tame structures is open; the authors note that Kamenova–Lu–Verbitsky's result covers only the specific compatible $g$5 in dimension four (via K3), not arbitrary tamed structures. Finally, the non-hyperbolicity conclusion applies only under the strong hypothesis that some almost Kähler metric of nonnegative Ricci curvature exists; whether weaker curvature or topological hypotheses suffice is not addressed.

Conclusion

The paper proves that on complete almost Kähler manifolds with $g$6, lines split off complex Euclidean factors, via the elementary but effective fact that parallelism is preserved by $g$7 when Ricci curvature is nonnegative. The applications are twofold: a complete four-dimensional answer to when nonnegative Ricci curvature forces integrability (everywhere except rational surfaces, where it demonstrably does not), and a reduction of symplectic non-hyperbolicity on symplectically aspherical manifolds to flat Kähler geometry, where Bangert's machinery applies. The results position the almost Kähler splitting theorem as a usable structural tool linking Riemannian rigidity, the Goldberg problem, and pseudo-holomorphic curve existence.