F-Structures in Differential Geometry

• F-structures are generalized (1,1)-tensor fields that satisfy polynomial identities like F³+F=0, leading to a natural splitting of the tangent bundle.
• They possess integrability conditions defined via the vanishing of Nijenhuis torsion, which underpins the formation of CR-structures and local product decompositions.
• F-structures extend to generalized and para-Hermitian geometries and play crucial roles in supersymmetric sigma models and the classification of geometric structures.

An F-structure is a generalization of almost complex and almost product structures with deep ramifications in differential geometry, generalized geometry, operad theory, and mathematical physics. It is typically realized as a (1,1)-tensor field subject to a polynomial constraint that generalizes the classical case $f^2 = -\mathrm{id}$, and its study encompasses the integrability, algebraic classification, and applications across geometry and mathematical physics.

1. Algebraic Definition and Basic Properties

Let $M$ be a smooth $n$-dimensional manifold. An F-structure is a (1,1)-tensor $F: TM \rightarrow TM$ of constant rank $r$ satisfying a polynomial identity. The Yano F-structure, the archetype, satisfies the cubic relation: $F^3 + F = 0$ This condition ensures that $F^2$ has eigenvalues $-1$ and $0$, inducing a direct sum splitting of the tangent bundle: $TM = \im F \oplus \ker F$ Here, $F^2 = -\mathrm{id}$ on $\im F$ (endowing it with an almost complex structure) and $F^2 = 0$ on $\ker F$. This framework recovers the almost complex case when $\ker F = 0$.

The structure admits two complementary projection operators: $\ell = -F^2,\quad m = 1 + F^2$ These satisfy $\ell^2 = \ell$, $m^2 = m$, $\ell + m = 1$, $\ell\,m = 0$, and $F\,\ell = \ell\,F = F$ with $F\,m = m\,F = 0$ (Lindström, 2022, Lindström, 2023).

Higher-degree F-structures are defined by more general minimal polynomials: $\alpha F^{K+1} + \beta F^K + F = 0$ for constants $\alpha,\beta$ and $K \geq 3$. The spectrum of $F$ consists of the roots of the characteristic polynomial, giving rise to a diverse algebraic classification depending on the choice of parameters (Zagane, 1 Apr 2024).

2. Integrability and Geometric Structures

Integrability of an F-structure is characterized by the vanishing of its Nijenhuis torsion: $$N_F(X, Y) = [F X, F Y] - F\bigl([F X, Y] + [X, F Y]\bigr) + F^2[X, Y]$$ Partial and complete integrability are defined via the involutivity of the distributions $D_\ell = \im \ell$ and $D_m = \im m$:

• Partial integrability: $D_\ell$ is involutive and $F$ is almost complex on its leaves.
• Complete integrability: Both $D_\ell$ and $D_m$ are involutive and $F$ is leafwise integrable on $D_\ell$.

The geometric interpretation is that the complex bundle $H = \{ X - i F X : X \in D_\ell \}$ defines a Cauchy–Riemann structure on $M$; $F$ is completely integrable if and only if $H$ is a CR-structure (Zagane, 1 Apr 2024). When $F$ is integrable, $M$ locally splits as a product of a complex $k$-fold with a real $(n-2k)$-fold.

3. F-Structures in Generalized and Para-Hermitian Geometry

Generalized F-structures extend the notion to the generalized tangent bundle $TM \oplus T^*M$, interpreted as orthogonal, skew-symmetric endomorphisms $\Phi$ satisfying $\Phi^3 + \Phi = 0$ and orthogonality with respect to the canonical pairing. A split generalized F-structure (SGF-structure) is an orthogonal almost complex structure on an even-rank subbundle, characterized by $J^2 = -\mathrm{id}$ and $J^T = -J$ (Aldi et al., 2015).

In para-Hermitian geometry, an almost para-Hermitian manifold $(\mathcal{M}, \eta, K)$ is a $4n$-dimensional manifold with a split signature metric $\eta$ and a (1,1)-tensor $K$ with $K^2 = 1$, $K^T \eta K = -\eta$. The tangent bundle splits into the $+1$ and $-1$ eigenbundles of $K$. Yano F-structures can be constructed on the doubled tangent bundle $TM \oplus TM$, with supersymmetric sigma models providing concrete realizations. The integrability and algebraic closure of such structures are central in the realization of extended supersymmetry and generalized complex geometry (Lindström, 2022, Lindström, 2023).

4. Operadic and Algebraic Formulations: F-Manifolds

F-manifolds, as introduced by Hertling and Manin, are smooth manifolds with a fiberwise commutative, associative product $\circ$ on the tangent sheaf and a Lie bracket of vector fields, satisfying the Hertling–Manin identity: $$P_{X \circ Y}(Z, W) = X \circ P_Y(Z, W) + Y \circ P_X(Z, W)$$ with $$P_X(Y, Z) = [X, Y \circ Z] - [X, Y] \circ Z - Y \circ [X, Z]$$.

Operadically, the FMan operad controls these structures. It arises as the associated graded of the pre-Lie operad under the filtration by the Lie bracket ideal. The FMan operad is generated by two binary operations (symmetric product $\circ$ and skew-symmetric bracket $[-, -]$), subject to associativity, Jacobi identity, and the Hertling–Manin cubic relation. This links F-manifold geometry intimately to pre-Lie, commutative, associative, and Lie algebraic structures (Dotsenko, 2017).

5. Applications in Geometry and Physics

Supersymmetric Sigma Models: (2,2) and (4,4) supersymmetries in sigma models naturally require target geometries that are captured by bi-quaternionic or para-hermitian structures. In these frameworks, off-shell closure of the extended supersymmetry algebra leads to the emergence of F-structures on doubled tangent bundles, which encode both manifest and hidden symmetry properties of the models (Lindström, 2022, Lindström, 2023).

Generalized Geometry: F-structures provide morphisms between complex and para-complex, as well as between complex and CR structures, with reductions of doubled structures corresponding to more classical geometries. SGF-structures and their integrability criteria yield abstract Morimoto–type theorems, providing unified frameworks for product and contact geometries (Aldi et al., 2015).

Classification Problems: The formal classification of (T)-structures (meromorphic connections) over irreducible germs of 2-dimensional F-manifolds elucidates the moduli of such structures, with distinguished cases for semisimple and nilpotent types, controlled by the behavior of higher-order terms in the connection matrices (David et al., 2018).

6. Generalizations, Weak and Framed Variants

Weakened (framed) f-structures generalize classical f-structures by replacing the complex structure on the image with a nonsingular skew-symmetric tensor $Q$. The algebraic relation is deformed to $f^3 + f Q = 0$, leading to subclasses such as weak $K$-, $C$-, and $S$-structures, each with distinct geometric and integrability properties, including rigidity results that show any weak $S$-structure is automatically classical. These variants are essential in the analysis of totally geodesic foliations and generalizations of almost contact and almost cosymplectic structures (Rovenski, 2022).

7. Selected Examples and Variants

Structure Type	Defining Relation	Characteristic Decomposition	Integrable Leaves
Yano F-structure	$F^3 + F = 0$	$\im F \oplus \ker F$	$\im F$ (almost complex)
Generalized F-structure	$\Phi^3 + \Phi = 0$ on $TM \oplus T^*M$	split subbundles via $\Phi$	$-i$-eigenbundle closure
Weakened f-structure	$f^3 + f Q = 0$ ($Q$ skew-symmetric, $Q \vert_{\im f}$ nonsingular)	$\im f \oplus \ker f$	determined by $Q$
Higher-degree F	$\alpha F^{K+1} + \beta F^K + F = 0$	by roots of characteristic	$D_l$, $D_m$ as above

Worked examples (e.g., explicit 2×2 and 3×3 matrices) exhibit complete integrability, realization of CR-structures, and decomposition into local products of complex and real manifolds (Zagane, 1 Apr 2024).

8. Connections and Further Directions

F-structures and their generalizations naturally link to generalized complex geometry, Courant algebroids, and T-duality frameworks. Their algebraic underpinnings drive the structure theory for F-manifolds, operadic deformations, and flat connection classifications in the theory of Frobenius manifolds and singularity theory.

In mathematical physics, the appearance of F-structures is fundamentally tied to supersymmetry, the geometry of sigma models, and the structure of target spaces with both manifest and non-manifest symmetries, providing a unifying language across extended supersymmetry, generalized Kähler and para-hermitian geometries, and beyond (Lindström, 2022, Lindström, 2023, Aldi et al., 2015, Dotsenko, 2017, Rovenski, 2022, Zagane, 1 Apr 2024, David et al., 2018).