Weak Almost Hermitian Structures

• Weak almost Hermitian structures are generalized geometries where the usual complex structure J is replaced by a skew-symmetric endomorphism A satisfying A² = –Q, encompassing both classical and weighted cases.
• The framework employs a nonsymmetric tensor G = g + F and leverages torsion and Einstein metricity conditions to facilitate product decompositions and nearly Kähler rigidity.
• This theory bridges various approaches—including algebraic, PDE, and variational perspectives—highlighting its wide applicability and structural interplay with Hermitian and topological notions.

Weak almost Hermitian structures are an extension of almost Hermitian geometry in which the usual almost complex endomorphism $J$ is replaced by a skew-symmetric endomorphism $A$ whose square is not necessarily $-\mathrm{Id}$, but rather $-Q$ for a positive self-adjoint endomorphism $Q$. In the current geometric literature, this framework is developed most explicitly for generalized Riemannian manifolds with nonsymmetric basic tensor $G=g+F$, where $g$ is the symmetric metric part and $F$ is the skew-symmetric part encoded by $A$ through $F(X,Y)=g(AX,Y)$ (Zlatanović et al., 11 Aug 2025). It recovers ordinary almost Hermitian geometry when $A$0, but also permits weighted or rank-deficient geometries that lead to nearly Kähler-type rigidity and local product decompositions (Rovenski et al., 28 Jun 2025). The phrase is not completely univocal across the literature: it also appears, or is naturally interpreted, in stable-topological, variational, PDE, and torsion-theoretic senses, which makes the topic intrinsically plural rather than purely terminological (Gauduchon et al., 2010).

1. Algebraic model and generalized Riemannian setting

The basic generalized Riemannian datum is a nonsymmetric $A$1-tensor

$A$2

with

$A$3

Here $A$4 is assumed non-degenerate and $A$5 has constant rank. The skew part determines a unique $A$6-tensor $A$7 by

$A$8

and since $A$9 is skew-symmetric one has

$-\mathrm{Id}$0

This places weak almost Hermitian geometry inside the broader study of generalized Riemannian manifolds with linear connections and torsion (Zlatanović et al., 11 Aug 2025).

A weak almost Hermitian manifold $-\mathrm{Id}$1 is, in the even-dimensional case $-\mathrm{Id}$2, a Riemannian manifold endowed with a non-singular skew-symmetric endomorphism $-\mathrm{Id}$3 and a positive self-adjoint endomorphism $-\mathrm{Id}$4 satisfying

$-\mathrm{Id}$5

Because $-\mathrm{Id}$6, the endomorphisms commute: $-\mathrm{Id}$7 hence also

$-\mathrm{Id}$8

If $-\mathrm{Id}$9, the structure reduces to an ordinary almost Hermitian one; if $-Q$0, the geometry is governed by the spectral decomposition of $-Q$1, which acts as a weight operator on the Hermitian-type structure (Rovenski et al., 28 Jun 2025).

The Levi-Civita notion corresponding to nearly Kähler geometry also admits a weak version. A weak nearly Kähler structure is defined by

$-Q$2

while $-Q$3 defines the weak Kähler case (Zlatanović et al., 11 Aug 2025). In this sense, weak almost Hermitian geometry interpolates between standard almost Hermitian structures and weighted nearly Kähler geometries.

2. Einstein metricity, skew torsion, and the $-Q$4-torsion condition

A central feature of the modern theory is the use of linear connections with torsion. For a connection $-Q$5, the torsion tensor is

$-Q$6

In the Einstein-connection formulation, the main compatibility requirement is Einstein’s metricity condition

$-Q$7

Equivalently,

$-Q$8

In terms of $-Q$9 and $Q$0, this yields explicit formulas for $Q$1 and $Q$2 in terms of torsion and $Q$3 (Zlatanović et al., 11 Aug 2025).

The additional structural hypothesis is the $Q$4-torsion condition

$Q$5

which, for totally skew-symmetric torsion, becomes

$Q$6

This condition is one of the main rigidity mechanisms in the theory. For generalized Riemannian manifolds with totally skew-symmetric torsion, an Einstein connection satisfies the $Q$7-torsion condition if and only if it preserves the symmetric metric part: $Q$8 Under this hypothesis,

$Q$9

so both the Levi-Civita derivative of $G=g+F$0 and the Nijenhuis tensor are controlled directly by the torsion and the exterior derivative of the fundamental $G=g+F$1-form (Zlatanović et al., 11 Aug 2025).

The Einstein formulation also gives an explicit connection formula. The unique Einstein connection with totally skew-symmetric torsion is

$G=g+F$2

with contorsion

$G=g+F$3

When torsion is totally skew-symmetric,

$G=g+F$4

A parallel metric-connection formulation replaces Einstein metricity by the stronger requirement $G=g+F$5; in that setting,

$G=g+F$6

and the same torsion formalism leads to weak nearly Kähler conclusions under the $G=g+F$7-torsion symmetry (Rovenski et al., 28 Jun 2025).

3. Weak nearly Kähler rigidity

The main structural theorem in the weak almost Hermitian case states that if $G=g+F$8 carries an Einstein connection with totally skew-symmetric torsion satisfying the $G=g+F$9-torsion condition, then the manifold is weak nearly Kähler (Zlatanović et al., 11 Aug 2025). This is the precise sense in which the generalized Riemannian data $g$0, once coupled to Einstein metricity and skew torsion, collapses onto a nearly Kähler-type regime.

In the Einstein-connection formulation, the torsion is determined by the fundamental $g$1-form and the Nijenhuis tensor through

$g$2

Moreover, the structure tensors become parallel in the relevant senses: $g$3 is preserved by both the Levi-Civita and Einstein connections,

$g$4

and the relation

$g$5

shows that $g$6 is parallel modulo torsion (Zlatanović et al., 11 Aug 2025).

The metric-preserving version gives an equivalent conceptual picture. If $g$7 is weak almost Hermitian, $g$8, and the torsion is totally skew-symmetric, then $g$9 satisfies the $F$0-torsion condition if and only if $F$1 is weak nearly Kähler. In that case the torsion is expressed explicitly in terms of $F$2 and $F$3, so weak nearly Kählerity is not an auxiliary extra assumption but the geometric content forced by the torsion symmetry (Rovenski et al., 28 Jun 2025).

This rigidity should not be confused with weak Kählerity in the sense $F$4. The theory singles out a nearly Kähler analogue rather than a parallel complex analogue. A plausible implication is that the weak theory is designed less as a deformation of Kähler geometry than as a weighted enlargement of the nearly Kähler class.

4. Weighted products, eigen-distributions, and lower-rank splittings

When $F$5 is nondegenerate, so $F$6, the commutation relation $F$7 allows simultaneous diagonalization of $F$8 and $F$9 in an adapted basis. The spectral behavior of $A$0 then governs the local geometry. In the conformal case $A$1, the function $A$2 is constant and the rescaled structure becomes an ordinary nearly Kähler one. In the non-conformal case, $A$3 splits into mutually orthogonal even-dimensional eigen-distributions of $A$4 with constant eigenvalues, each $A$5-parallel and totally geodesic, and the manifold is locally a weighted product of nearly Kähler factors (Zlatanović et al., 11 Aug 2025).

For the degenerate case $A$6, the theory passes to weak almost contact metric or more general weak $A$7-structures. The weak almost contact metric model $A$8 satisfies

$A$9

Under the same Einstein-plus-$F(X,Y)=g(AX,Y)$0-torsion hypotheses, the Reeb field becomes Levi-Civita parallel,

$F(X,Y)=g(AX,Y)$1

the contact distribution $F(X,Y)=g(AX,Y)$2 is involutive because $F(X,Y)=g(AX,Y)$3, and the geometry splits off a line factor controlled by a nearly Kähler structure on $F(X,Y)=g(AX,Y)$4 (Zlatanović et al., 11 Aug 2025).

Setting	Condition on $F(X,Y)=g(AX,Y)$5	Local geometry
Weak almost Hermitian, $F(X,Y)=g(AX,Y)$6	$F(X,Y)=g(AX,Y)$7	$F(X,Y)=g(AX,Y)$8 constant; rescaled ordinary nearly Kähler structure
Weak almost Hermitian, $F(X,Y)=g(AX,Y)$9	$A$00	$A$01-weighted product of nearly Kähler manifolds
Weak almost contact, $A$02	$A$03	locally $A$04
Weak almost contact, $A$05	$A$06	$A$07-weighted product of $A$08 and nearly Kähler factors

The weak $A$09-structure version extends the same pattern to higher-dimensional kernels of $A$10: if $A$11, then the image distribution $A$12 splits into mutually orthogonal even-dimensional eigen-distributions, each involutive and totally geodesic, and the manifold is locally a weighted product of $A$13 and nearly Kähler $A$14-manifolds (Rovenski et al., 28 Jun 2025). The dominant geometric message is that weak almost Hermitian data become reducible into nearly Kähler blocks once the torsion-compatible connection exists.

5. Competing meanings of “weak” in the literature

The phrase “weak almost Hermitian structures” is not used uniformly across the literature. One major alternative is the stable-topological notion of weak almost complex or weakly complex structure: a real manifold is weakly complex if $A$15 is a complex vector bundle for some trivial real bundle $A$16. In this sense, compact quaternionic-Kähler manifolds of positive type admit no weak almost complex structure except the complex Grassmannians $A$17, and simply connected irreducible inner symmetric spaces of compact type are weakly complex if and only if they are spheres or Hermitian symmetric spaces (Gauduchon et al., 2010). This is a topological obstruction theory, not an $A$18 generalization.

A second usage is variational. A survey on harmonic almost Hermitian structures states that the term “weak” is not used explicitly there, but that a natural interpretation relevant in that context is a compatible almost complex structure that is harmonic as a section of the twistor bundle rather than harmonic as a map, or that minimizes section energy without being Kähler (Davidov, 2016). The basic Euler–Lagrange equation is

$A$19

and the survey emphasizes that harmonic section, harmonic map, energy-minimizing section, and minimal embedding are distinct notions.

A third usage is PDE-theoretic. Weakly $A$20-harmonic almost complex structures are Sobolev almost complex structures

$A$21

satisfying, in the distributional sense,

$A$22

This is the “weak” of weak solutions rather than weak metric geometry. In particular, weakly biharmonic almost complex structures are smooth in dimension four, obtained from the combination of Hölder regularity in the critical dimension and higher-order elliptic bootstrapping (He et al., 2019).

These senses are conceptually independent. The $A$23-formalism weakens the algebraic identity $A$24; stable weakly complex geometry weakens the existence problem to a stable bundle statement; harmonic and polyharmonic theories weaken the Euler–Lagrange equation to section-critical or distributional formulations.

6. Lie-theoretic realizations and related Hermitian classes

Lie groups and solvmanifolds provide explicit models in which weak or weakened almost Hermitian phenomena can be computed algebraically. On almost abelian Lie groups, harmonicity of a left-invariant almost Hermitian structure reduces to explicit matrix identities for the Lie algebra endomorphism $A$25, and the Gray–Hervella classification becomes unusually rigid. In that setting, Kähler, nearly Kähler, balanced, and locally conformally Kähler structures are automatically harmonic, and compact solvmanifold examples realize harmonic structures in many Gray–Hervella classes, including mixed classes and the full generic class $A$26 (Andrada et al., 2023). This demonstrates that harmonicity is a property of the pair $A$27, not of the Lie group alone.

A different but related class is the Gray–Hervella $A$28 class, the locally conformally Kähler class controlled by the Lee form $A$29. There the intrinsic torsion is entirely determined by $A$30, and minimality of the induced $A$31-structure inside $A$32 reduces to explicit equations involving $A$33, $A$34, and $A$35. The paper on minimal $A$36-structures explicitly treats these as minimal weak almost Hermitian structures in the sense that the intrinsic torsion is nonzero but confined to a single torsion class; its examples include globally conformally Kähler metrics on $A$37 and Hopf manifolds with parallel Lee form (Niedzialomski, 2016).

The torsion literature introduces yet another nearby notion: weak Kähler with torsion. For Hermitian Lie groups with two-dimensional commutator subgroups, the Bismut connection and its torsion $A$38-form $A$39 are computed explicitly, and the distinction between strong and weak KT is the closure or non-closure of $A$40: $A$41 That work gives concrete Type I and Type II four-dimensional abelian weak Kähler with torsion examples, alongside non-Kähler SKT and genuinely Kähler cases (Moghaddam, 25 Jan 2026). Although this is not the $A$42-based weak almost Hermitian theory, it shows how “weak” continues to mark controlled failure of a stronger Hermitian condition, here via torsion rather than via $A$43.

Taken together, these developments indicate that weak almost Hermitian structures are best understood as a family of closely related research directions rather than a single immutable definition. In the strict geometric sense now used for generalized Riemannian manifolds, the subject centers on the replacement of $A$44 by $A$45, on skew-torsion connections satisfying Einstein-type or metricity conditions, and on the resulting emergence of weak nearly Kähler geometry and weighted product decompositions (Zlatanović et al., 11 Aug 2025). In adjacent topological, analytic, and torsion-theoretic settings, the same adjective “weak” signals different relaxations of complex or Hermitian structure, and the distinction between these usages is itself part of the modern subject.