GAUGE-COMPATIBLE TENSORS IN STATISTICAL
MANIFOLDS

Stéphane Puechmorel

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Stéphane Puechmorel. GAUGE-COMPATIBLE TENSORS IN STATISTICAL MANIFOLDS. 2025. [hal-05435661]

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Distributed under a Creative Commons Attribution 4.0 International License
# GAUGE-COMPATIBLE TENSORS IN STATISTICAL MANIFOLDS

MIRJANA MILIJEVIC AND STÉPHANE PUECHMOREL

ABSTRACT. We investigate gauge-compatible (1, 1)-tensor fields on statistical manifolds and their impact on the cohomology of Koszul–Vinberg (pre-Lie) algebras naturally associated with torsion-free affine connections. Given a pair of dual connections ($\nabla, \nabla^*$), we study solutions of the gauge equation

$$ \nabla\theta = \theta\nabla^* $$

and analyze how such tensors induce algebraic structures and double complexes whose associated spectral sequences encode geometric information.

Our first contribution is the introduction of a new product based on the co-variant derivative of a gauge equation solution $\theta$. Provided $\theta$ is a weak Nijen- huis tensor, a curvature-like tensor can be defined and when it is vanishing, the product turns out to define a Koszul–Vinberg algebra on the sections of the tangent bundle. Our second main result establishes that, under suitable gauge-compatibility conditions adapted to invariant distributions, the Koszul– Vinberg spectral sequence degenerates at the $E_2$-page.

These general results are applied to statistical coKähler manifolds, where the structure tensor $\varphi$ provides a canonical, nontrivial solution of the gauge equation. In this setting, we obtain explicit geometric criteria ensuring spectral degeneracy and show that, for CR-submanifolds, the vanishing of two natural obstruction tensors is equivalent to both gauge flatness and degeneration of the associated spectral sequence. Our results unify gauge-theoretic compatibility, pre-Lie algebra cohomology, and geometric structure, providing explicit models where homological simplifications reflect intrinsic geometric properties.

## 1. INTRODUCTION

Koszul–Vinberg (KV) algebras, more commonly known as pre-Lie algebras, arise naturally in differential geometry through torsion-free affine connections. Given a smooth manifold $M$ equipped with a Koszul connection $\nabla$, the space of smooth vector fields $\Gamma(TM)$ carries a nonassociative product

$$ X \circ Y := \nabla_X Y, $$

whose commutator recovers the Lie bracket of vector fields. The failure of associativity is measured by the curvature of $\nabla$, making pre-Lie algebras a natural algebraic encoding of affine geometry. Their cohomology, introduced by Michel Boyom [4], plays a fundamental role in deformation theory, affine structures, and representation theory. Through this paper, pre-Lie algebra will be the preferred term, but Koszul–Vinberg is used for cohomology.

In parallel, statistical geometry provides a rich class of geometric structures involving pairs of dual torsion-free connections ($\nabla, \nabla^*$) compatible with a Riemannian

<footnote>Date: June 2025.</footnote>
metric. Such structures appear naturally in information geometry and affine differential geometry, and offer a fertile setting for studying compatibility conditions between geometric tensors and affine connections.

The central object of this paper is the gauge equation

$$ \nabla \theta = \theta \nabla^*, $$

where $\theta$ is a (1,1)-tensor field on $M$. This equation expresses the compatibility of $\theta$ with a pair of dual connections and generalizes the classical relation arising from conjugation of affine connections by invertible bundle morphisms. Even when $\theta$ is not invertible, the gauge equation has strong geometric consequences: it implies the integrability of $\ker \theta$ and $\im \theta$, induces chain maps on the de Rham complex, and governs the behavior of Nijenhuis-type torsion tensors.

The first aim of this work is to develop a systematic framework for the study of gauge-compatible tensors and the algebraic structures they induce. We introduce a categorical viewpoint in which gauge structures are objects and solutions of the gauge equation are morphisms, and we analyze how symmetric and skew-symmetric solutions give rise to Lie-admissible and pre-Lie products on spaces of vector fields. Special attention is paid to curvature-like tensors and weak Nijenhuis conditions controlling when these products define genuine pre-Lie algebra.

Our second, and main, objective is to understand how gauge compatibility influences Koszul–Vinberg cohomology. Given a gauge-compatible tensor $\theta$, one can construct double complexes combining the KV-complexes associated with $\nabla$ and $\nabla^*$, as well as geometric decompositions induced by $\ker \theta$ and $\im \theta$. The associated spectral sequences provide a powerful homological tool. We prove that, under natural geometric hypotheses, these spectral sequences degenerate at the $E_2$-page, revealing a strong simplification of the underlying cohomology.

A particularly rich and explicit class of examples is provided by statistical coKähler manifolds. On such manifolds, the almost contact structure tensor $\varphi$ is parallel with respect to the Levi–Civita connection and, under a natural compatibility condition on the difference tensor of a statistical connection, satisfies the gauge equation with respect to the dual connection. We show that $\varphi$ is a canonical gauge-compatible tensor whose associated Nijenhuis-type obstructions vanish identically. This leads to explicit constructions of Koszul–Vinberg double complexes and guarantees early spectral degeneracy.

Finally, we study CR-submanifolds of statistical coKähler manifolds. We identify two natural obstruction tensors—one tangential and one normal—that measure the failure of the ambient gauge equation to restrict to the submanifold. Our main result in this setting shows that the vanishing of these obstructions is equivalent to both gauge compatibility and degeneration of the associated spectral sequence. This provides a clear geometric interpretation of homological degeneracy in terms of submanifold geometry.

Overall, this work establishes a direct link between gauge equations, pre-Lie algebra cohomology, and spectral sequence degeneracy, showing how geometric compatibility conditions translate into algebraic simplifications. The results place statistical coKähler geometry alongside holomorphic statistical manifolds as a natural arena where gauge-theoretic and homological phenomena can be studied explicitly.

The paper is organized as follows. Section 2 recalls the necessary background on pre-Lie algebras and their cohomology. In Section 3 we review gauge equations for tensor fields and introduce the associated mixed connection. Section 4
discusses statistical coKähler manifolds and provides canonical examples of gauge-compatible tensors. In Section 5 we develop the theory of θ-CR submanifolds, establish the equivalence between gauge flatness and vanishing obstructions, and prove the degeneracy of the Koszul-Vinberg spectral sequence associated to the θ-product.

## 2. PRE-LIE ALGEBRAS

Pre-Lie algebras, arise naturally in differential geometry. Let $(M, \nabla)$ be a gauge structure with $\nabla$ a torsionless Koszul connection on $M$. A product is defined on $\chi(M) = \Gamma(TM)$ by:

$$ (1) \qquad X \circ Y = \nabla_X Y, X, Y \in \chi(M). $$

The commutator $X \circ Y - Y \circ X$ is the usual Lie bracket, but the Jacobi identity is not satisfied by the product $\circ$. The associator is given by the relation:

$$ (2) \qquad (X, Y, Z) = \nabla_{\nabla_X Y} Z - \nabla_X \nabla_Y Z = -\nabla_{\nabla_X Y}^2 Z, X, Y, Z \in \chi(M). $$

From the above property, it is easy to derive the next proposition:

**Proposition 2.1.** For any $X, Y, Z \in \chi(M)$:

$$ (3) \qquad (X, Y, Z) - (Y, X, Z) = -R^{\nabla} (X, Y) Z, \\ \text{with } R^{\nabla} \text{ the curvature of } \nabla. $$

*Proof.* Obvious from the expression: $R^{\nabla} (X, Y) Z = \nabla_{\nabla_X Y}^2 Z - \nabla_{\nabla_Y X}^2 Z$. $\square$

If $\nabla$ is flat, then: $(X, Y, Z) = (Y, X, Z)$, that is the defining property of a pre-Lie algebra.

### 2.1. Definitions.
Let $k$ be a field, that will always be $\mathbb{R}$ or $\mathbb{C}$ in the following, and let $A$ be a $k$-vector space.

**Definition 2.1.** $A$ is said to be a pre-Lie (or a KV-algebra) if equipped with a product $\circ$: $A \times A \to A$ such that, for any $a, b, c$ in $A$:

$$ (4) \qquad (a, b, c) = (b, a, c), (a, b, c) = (a \circ b) \circ c - a \circ (b \circ c). $$

**Proposition 2.2.** Let $A$ be a KV algebra. The bracket:

$$ (5) \qquad [a, b] = a \circ b - b \circ a. $$

provides $A$ with a Lie algebra structure, denoted by $A^L$.

**Definition 2.2.** Let $a \in A$. The left product endomorphism $L_a$ is defined by:

$$ (6) \qquad L_a b = a \circ b, b \in A. $$

**Proposition 2.3.** For any $a, b, \in A$:

$$ (7) \qquad [L_a, L_b] = L_{[a, b]}. $$

*Proof.* Let $w \in A$. Then:

$$ (8) \qquad [L_a, L_b] w = a \circ (b \circ w) - b \circ (a \circ w). $$

Using the associator, the right hand term can be rewritten as:

$$ (9) \qquad (a \circ b) \circ w - (a, b, w) - (b \circ a) \circ w + (b, a, w) = ([a, b]) \circ w = L_{[a, b]} w, $$

which concludes the proof. $\square$
The endomorphism $L_a$ extends to a derivation of the symmetric algebra $S(A)$ by application of the Liebniz rule:

$$ (10) \qquad L_a (a_1 \otimes \cdots \otimes a_q) = \sum_{i=1}^{q} a_1 \otimes \cdots aa_i \cdots \otimes a_q, $$

hence there is a Lie algebra morphism $L: a \in A \mapsto \text{Der}S(A)$ by equation 7.

**Definition 2.3 ([2]).** Let $A$ be a KV algebra. A representation of $A$ on a vector space $V$ is a pair of linear maps $L, R$ from $A$ to $\text{gl}(V)$ such that $L$ is a Lie algebra morphism from $A^L$ to $\text{gl}(V)$ and $R$ satisfies:

$$ (11) \qquad [R(b), L(a)] = R(b)R(a) - R(a \circ b). $$

A vector space $V$ equipped with a representation $L, R$ is said to be a KV-module (or pre-Lie module).

**Example 2.1.** The action of $L$ (resp. $R$) on an element $v$ of $V$ is conveniently denoted as a left (resp. right) product:

$$ L(a)v = a \triangleleft v, R(a)v = v \triangleright a. $$

**Remark 2.1.** Equation 11 is a weak form of associativity. In fact, the requirements on $L, R$ can be rewritten with associators as:

$$ (12) \qquad (a, b, v) = (b, a, v), (a, v, b) = (v, a, b), a, b \in A, v \in V. $$

with:

$$ (a, b, v) = (a \circ b) \triangleleft v - a \triangleleft (b \triangleleft v), $$

$$ (13) \qquad (a, v, b) = (a \triangleleft v) \triangleright b - a \triangleleft (v \triangleright b) $$

$$ (v, a, b) = (v \triangleright a) \triangleright b - v \triangleright (a \circ b). $$

**2.2. Cohomology.** The references for this section are [4, 10].

**Definition 2.4.** Let $V$ be a KV-module. For any positive integer $q$, let $C^q(A, V) = \text{hom}_{\mathbb{R}}(\otimes^q A, V)$, the vector space of $q$-multilinear $V$-valued maps. It has a natural KV-module structure with actions:

$$ (14) \qquad \begin{aligned} & (a \triangleleft f)(a_1 \otimes \cdots \otimes a_q) = a \triangleleft f (a_1 \otimes \cdots \otimes a_q) - \sum_{i=1}^{q} f (a_1 \otimes \cdots \otimes (a \circ a_i) \otimes \cdots a_q) \\ & (f \triangleright a)(a_1 \otimes \cdots \otimes a_q) = f (a_1 \otimes \cdots \otimes a_q) \triangleright a. \end{aligned} $$

**Definition 2.5.** Let $q$ be a positive integer. The face (resp. insertion) operator is defined by:

$$ (15) \qquad \partial_i : a_1 \otimes \cdots \otimes a_q = a_1 \otimes \cdots \otimes \hat{a}_i \otimes \cdots \otimes a_q $$

resp.

$$ (16) \qquad \mathcal{I}_i(a) : a_1 \otimes \cdots \otimes a_q = \begin{cases} a_1 \otimes \cdots a_{i-1} \otimes a \otimes a_i \otimes \cdots \otimes a_q, & a \in A, i = 2 \cdots q \\ a \otimes a_1 \otimes \cdots \otimes a_q, & a \in A, i = 1 \\ a_1 \otimes \cdots \otimes a_q \otimes a, & a \in A, i = q+1 \end{cases} $$

**Remark 2.2.** The notation $\hat{a}_i$ means, as usual, that the factor $i$ is omitted.
The composition $\partial_i \mathcal{I}_i(a)$ is the identity while $\mathcal{I}_i(a) \partial_i$ replaces the $i$-th element of $a_1 \otimes \cdots \otimes a_q$ by $a$. Other commutations rules are:

$$ (17) \qquad \begin{aligned} \partial_i \mathcal{I}_j(a) &= \mathcal{I}_{j-1}(a) \partial_i, \quad j > i \\ \partial_i \mathcal{I}_j(a) &= \mathcal{I}_j(a) \partial_{i-1}, \quad j < i. \end{aligned} $$

If $f \in C^q(A, V)$, then $f \circ \partial_i \in C^{q+1}(A, V)$ and $f \circ \mathcal{I}_i(a) \in C^{q-1}(A, V)$.

**Definition 2.6.** The coboundary operator $\delta^q: C^q(A, V) \to C^{q+1}(A, V)$ is defined by [4]:

$$ (18) \qquad \delta^q f = \sum_{j=1 \dots q} (-1)^j [a_j \triangleleft (f \partial_j) + f (\mathcal{I}_q(a_j) \partial_q \partial_j) \triangleright a_{q+1}] $$

**Proposition 2.4.** For $q \ge 0$:

$$ (19) \qquad \delta^{q+1} \delta^q = 0. $$

The proof of proposition 2.4 can be done by brute force, as in [4], but a simpler approach through semi-simplicial modules can be found in [5] and in [10]. It is also a consequence of the next proposition.

**Definition 2.7.** Let $1 \le i < q$ be integers. $\theta_i: \otimes^q A \to \otimes^{q-1} A$ is defined by:

$$ (20) \qquad \theta_i (a_1 \otimes \cdots \otimes a_q) = L_{a_i} \partial_i (a_1 \otimes \cdots \otimes a_q). $$

**Proposition 2.5.** Let $1 < i < j < q$ be three integers. Then, for $\xi \in \otimes^q A$:

$$ (21) \qquad \theta_j \theta_i \xi = (L_{(a_i a_{j+1})} + L_{a_{j+1} L_{a_i}}) \partial_i \partial_{j+1} \xi. $$

Furthermore, the following semi-simplicial identity holds:

$$ (22) \qquad \theta_j \theta_i \xi = \theta_i \theta_{j+1} \xi. $$

*Proof.* The first part of the proposition is obtained by writing the action of $\theta_j \theta_i$ on an elementary tensor $a_1 \otimes \cdots \otimes a_q$:

$$ (23) \qquad \theta_j \theta_i (a_1 \otimes \cdots \otimes a_q) = \sum_{k \neq i} \theta_j (a_1 \otimes \cdots \otimes a_i a_k \otimes \cdots \otimes \hat{a}_i \otimes \cdots \otimes a_q) $$

For $k = j + 1$:

$$ (24) \qquad \theta_j (a_1 \otimes \cdots \otimes a_i a_k \otimes \cdots \otimes \hat{a}_i \otimes \cdots \otimes a_q) = L_{(a_i a_{j+1})} \partial_i \partial_{j+1} (a_1 \otimes \cdots \otimes a_q), $$
and for $k \neq j + 1$:

$$ (25) \qquad \theta_j (a_1 \otimes \cdots \otimes a_i a_k \otimes \cdots \otimes \hat{a}_i \otimes \cdots \otimes a_q) = L_{a_{j+1}} L_{a_i} \partial_i \partial_{j+1} (a_1 \otimes \cdots \otimes a_q). $$

Summing the two terms yields the first claim. Now, again by direct computation:

$$ (26) \qquad \theta_i \theta_{j+1} \xi = (L_{(a_{j+1} a_i)} + L_{a_i L_{a_{j+1}}}) \partial_i \partial_{j+1} \xi. $$

The difference $(\theta_j \theta_i - \theta_i \theta_{j+1}) \xi$ computes as:

$$ (27) \qquad (L_{[a_i, a_{j+1}]} - [L_{a_i}, L_{a_{j+1}}]) \partial_i \partial_{j+1} \xi = 0. $$

□

The same scheme can be applied to the operator:

$$ (28) \qquad \eta_i: f \in C^q(A, V) \to a_i \triangleleft (f \partial_i) = L_{a_i} (f \partial_i). $$

to obtain the next proposition:
**Proposition 2.6.** Let $1 < i < j < q$ be three integers and $f \in C^q(A, V)$. Then:

$$ (29) \qquad \eta_j \eta_i f = (L_{(a_i a_{j+1})} + L_{a_{j+1}} L_{a_i}) (f \partial_i \partial_{j+1}). $$

Furthermore, the following semi-simplicial identity holds:

$$ (30) \qquad \eta_j \eta_i f = \eta_i \eta_{j+1} f. $$

*Remark 2.3.* It is quite interesting to note that the semi-simplicial identities for $\theta_i$ (resp. $\eta_i$) are due to the fact that the operator $L_a, a \in A$ acting on $\otimes^q A$ (resp. $C^q(A, V)$) is a Lie algebra representation.

Finally, the second term in the expression of $\delta^q$ can be written as an operator:

$$ (31) \qquad \rho_i: f \in C^q(A, V) \mapsto R_{a_{q+1}} f (\mathcal{I}_q(a_i) \partial_q \partial_i) $$

also satisfies a semi-simplicial property:

$$ (32) \qquad \rho_j \rho_i + \rho_j \eta_i + \eta_j \rho_i - (\rho_i \rho_{j+1} + \rho_i \eta_{j+1} + \eta_i \rho_{j+1}) = \\ \qquad [R_{a_{q+1}}, L_{a_i}] - R_{a_{q+1}} R_{a_i} - R_{a_{q+1}} a_i \\ \qquad - [R_{a_{q+1}}, L_{a_{j+1}}] - R_{a_{q+1}} R_{a_{j+1}} + R_{a_{q+1} a_{j+1}} = 0. $$

*Remark 2.4.* Here again, the semi-simplicial identity depends on the defining relations of a KV module. In the expression of $\delta^q$, both terms have a clear interpretation, a fact that is often overlooked.

Gathering things together:

$$ (33) \qquad \delta^{q+1} \delta^q = 0. $$

**Definition 2.8.** The KV-cohomology of $A$ with coefficient in $V$ is the cohomology of the complex $C^*(A, V), \delta^*$.

### 3. THE GAUGE EQUATION

Let $E$ be a vector bundle on a smooth manifold $M$. As usual, $\Gamma_E(U)$ denotes the $C^\infty(U)$-module of its smooth sections over an open set $U \subset M$. Given a pair $V \subset U \subset M$ of open sets, the inclusion map $i_{VU}: V \hookrightarrow U$ gives rise to a map of modules $\Gamma_E(i_{VU}): \Gamma_E(U) \to \Gamma_E(V)$. The functor $\Gamma_E$ defined that way is a sheaf, called the sheaf of smooth sections of $E$. It is well known [12] that if $\nabla$ is a Koszul connection and $\theta$ is an invertible (1,1)-tensor, then $\tilde{\nabla} = \theta \nabla \theta^{-1}$ is a Koszul connection. Rewriting the above relation gives:

$$ (34) \qquad \theta \nabla = \tilde{\nabla} \theta. $$

The form (34) is valid even if $\theta$ is not invertible, thus it makes sense to introduce the general gauge equation.

**Definition 3.1.** Let $(\nabla^1, \nabla^2)$ a pair of Koszul connections. A (1,1)-tensor is said to satisfy the gauge equation for $(\nabla^1, \nabla^2)$ if:

$$ (35) \qquad \nabla^2 \theta = \theta \nabla^1 \theta. $$

The gauge equation is compatible with the sheaf $\Gamma_E$. Let $U \subset M$ be an open set, $f \in C^\infty(U)$, and $s \in \Gamma_E(U)$ such that $\nabla^2 \theta_s = \theta \nabla^1 s$. Then:

$$ (36) \qquad \nabla^2 \theta f s = \nabla^2 f \theta s = \nabla^2(f) \theta s + f \nabla^2 \theta s = \nabla^1(f) \theta s + f \theta \nabla^1 s = \theta \nabla^1 f s. $$

It thus make sense to speak about the sheaf of solutions of the gauge equation for $(\nabla^1, \nabla^2)$. Finally, let $(E, \nabla^1), (F, \nabla^2)$ be gauge structures with $E, F$ bundles on
M. Koszul connections $\nabla^1, \nabla^2$ can be interpreted as exterior derivatives on (sheaves of) 0-forms. Given a bundle morphism $\theta: E \to F$, $\theta$ is said to be a solution of the gauge equation if the next diagram commutes:

$$ (37) \qquad \begin{array}{ccc} \Gamma_E & \xrightarrow{\nabla^1} & \Gamma_{T^*M \otimes E} \\ & \Big\downarrow_{\theta} & \Big\downarrow_{\mathrm{Id} \otimes \theta} \\ \Gamma_F & \xrightarrow{\nabla^2} & \Gamma_{T^*M \otimes F} \end{array} $$

This general definition obviously agrees with the previous one. The identity morphism $\mathrm{Id}_E: E \to E$ satisfies the gauge equation $\mathrm{Id} \nabla = \nabla \mathrm{Id}$ for any gauge structure $(E, \nabla)$ and the composition of two bundle morphisms satisfying the gauge equation satisfies again the gauge equation for if $\nabla^2\theta = \theta\nabla^1$ and $\nabla^3\eta = \eta\nabla^2$, then:

$$ \eta\theta\nabla^1 = \eta\nabla^2\theta = \nabla^3\eta\theta. $$

There is thus a category $\mathcal{GC}(M)$ whose objects are gauge structures and morphisms given by solutions of the gauge equation.

**Definition 3.2.** Let $\theta$ be a (1, 1)-tensor. Its adjoint, denoted by $\theta^\dagger$, is the graded operator of degree 0 acting on $p$-forms, $p = 1...n$, by:

$$ (38) \qquad (\theta^\dagger \omega)(X_1, \dots, X_p) = \omega(\theta X_1, \dots, \theta X_p). $$

**Proposition 3.1.** Let $(\nabla_1, \nabla_2, \theta)$ be a solution of the gauge equation $\nabla_1\theta = \theta\nabla_2$. Then, for any $p$-form $\omega$, $p = 1...n$:

$$ (39) \qquad \nabla_2 \theta^\dagger \omega = \theta^\dagger \nabla_1 \omega. $$

*Proof.* Let $\omega$ be a $p$-form. Then:

$$ (40) \qquad \begin{aligned} & (\nabla_2 \theta^\dagger \omega)(X_1, \dots, X_p) = \\ & \qquad \nabla_2 (\omega(\theta X_1, \dots, \theta X_p)) - \sum_{i=1}^p \omega(\theta X_1, \dots, \nabla_2 \theta X_i, X_{i+1}, \dots, \theta X_p) = \\ & \qquad \nabla_1 (\omega(\theta X_1, \dots, \theta X_p)) - \sum_{i=1}^p \omega(\theta X_1, \dots, \theta \nabla_1 X_i, X_{i+1}, \dots, \theta X_p) = \theta^t (\nabla_1 \omega). \end{aligned} $$

□

Since, for any torsion-free connection $\nabla$ and any $p$-form $\omega$:

$$ (41) \qquad (d\omega)(X_0, \dots, X_p) = \sum_{i=0}^{p} (\nabla_{X_i} \omega) (X_0, \dots, X_{i-1}, \hat{X}_i, X_{i+1}, \dots, X_p), $$

one deduce that:

$$ (42) \qquad \begin{aligned} & (d\theta^\dagger \omega)(X_0, \dots, X_p) = \sum_{i=0}^{p} (\nabla_{2X_i} \theta^\dagger \omega) (X_0, \dots, X_{i-1}, \hat{X}_i, X_{i+1}, \dots, X_p) \\ & = \sum_{i=0}^{p} (\theta^\dagger \nabla_{1X_i} \omega) (X_0, \dots, X_{i-1}, \hat{X}_i, X_{i+1}, \dots, X_p) = \theta^\dagger d\omega. \end{aligned} $$

Hence, $\theta^\dagger$ is a chain transformation of the de Rham complex. The next proposition is merely an application of definition 35.
**Proposition 3.2.** Let $(\nabla_1, \nabla_2, \theta)$ be a solution of the gauge equation $\nabla_1\theta = \theta\nabla_2$. Then $\text{im } \theta$ and $\text{ker } \theta$ are integrable.

*Proof.* If $X \in \text{ker } \theta$:

$$ (43) \qquad 0 = \nabla_1 \theta X = \theta \nabla_2 X, $$

hence $\nabla_2 \text{ker } \theta \subset \text{ker } \theta$ and the claim for $\text{ker } \theta$ is proved. Now, let $\mathcal{I}(\text{im } \theta)$ be the annihilator of $\text{im } \theta$, that is the ideal of forms vanishing on $\text{im } \theta$. It is clear that $\mathcal{I}(\text{im } \theta) = \text{ker } \theta^\dagger$, thus, using the gauge equation on forms given by proposition 3.1, one has:

$$ (44) \qquad d(\mathcal{I}(\text{im } \theta)) \subset \mathcal{I}(\text{im } \theta). $$

By Frobenius theorem, this implies that $\text{im } \theta$ is integrable.

<figure><img src="image_2.png" /></figure>

When the manifold $M$ is equipped with a Riemannian metric, there exists a dual connection associated to an arbitrary Koszul connection.

**Definition 3.3.** Let $\nabla$ be an affine connection. Its dual with respect to $g$ is the connection $\nabla^*$ defined by the relation:

$$ (45) \qquad \forall Z \in TM, \forall r, s \in \Gamma(TM), Z(g(r, s)) = g(\nabla_Z r, s) + g(r, \nabla_Z^* s) $$

**Definition 3.4.** Let $\theta$ be a bundle morphism on $TM$. Its (Riemannian) conjugate, denoted $\theta^*$, is the bundle morphism defined by:

$$ (46) \qquad \forall X, Y \in TM, g(\theta X, Y) = g(X, \theta^* Y) $$

**Proposition 3.3.** Let the triple $(\nabla, \nabla^*, \theta)$ satisfy the gauge equation $\nabla\theta = \theta\nabla^*$. Then the tensor:

$$ (47) \qquad g_\theta: (X, Y) \mapsto g(\theta X, Y) $$

is $\nabla$ parallel.

Proposition 3.3 implies that $\theta$ has constant rank.

It is clear from proposition 3.3 that if $\theta$ is symmetric, that is $\theta = \theta^*$, the tensor:

$$ (48) \qquad (X, Y) \mapsto \frac{1}{2} (g(\theta X, Y) + g(X, \theta Y)) $$

is $\nabla$-parallel. When $\theta$ is skew symmetric $\theta = -\theta^*$, the same is true for:

$$ (49) \qquad (X, Y) \mapsto \frac{1}{2} (g(\theta X, Y) - g(X, \theta Y)) $$

The next two examples are extracted from [13].

**Example 3.1.** Take $M = \mathbb{R}^2$ and consider the symplectic 2-form:

$$ (50) \qquad \omega: (x, y) \mapsto \exp xdx \wedge dy. $$

Let $\nabla$ be a Koszul connection such that $\nabla\omega = 0$, $g$ an arbitrary Riemannian metric on $\mathbb{R}^2$ and $\nabla^*$ the dual of $\nabla$ with respect to $g$. Finally, let $\theta$ be the unique $(1, 1)$-tensor such that, for all vector fields $X, Y$:

$$ (51) \qquad \omega(X, Y) = g(\theta X, Y) $$

Then:

$$ (52) \qquad \nabla_X^* \theta Y = \theta \nabla_X Y. $$
Example 3.2. Take $M = \mathbb{S}^3$, $\nabla$ a torsion-less connection and $g$ a Riemannian metric.
Any solution $\theta$ to the gauge equation:

$$ (53) \qquad \nabla_X \theta Y = \theta \nabla_X^* Y. $$

is either 0 or invertible.

**Proposition 3.4.** Let $\theta$ be normal and satisfying the gauge equation. Let $D = \nabla^* - \nabla$. Then the tensor:

$$ (54) \qquad T_{\theta} : (X, Y) \mapsto [\theta X, \theta^* Y] - \theta^* [\theta X, Y] - \theta [X, \theta^* Y] + \theta^* \theta [X, Y] \\ \qquad \qquad + \theta^* D(\theta X, Y) - \theta D(X, \theta^* Y) $$

vanishes identically.

*Proof.* We must first prove the tensoriality. Let $f: M \to \mathbb{R}$ be a smooth function. Since $D$ is known to be a tensor, there is no need to take the last two terms into account. A direct computation shows that:

$$ (55) \qquad T_{\theta}(fX, Y) = fT_{\theta}(X, Y) + (-\theta^*Y(f) + \theta^*Y(f) + \theta\theta^*Y(f) - \theta^*\theta Y(f)) T_{\theta}(X, Y) $$

Since $\theta$ is normal, $T_{\theta}(fX, Y) = fT_{\theta}(X, Y)$. Now, a similar computation yields:

$$ (56) \qquad T_{\theta}(X, fY) = fT_{\theta}(X, Y) + (\theta X(f) - \theta^* \theta X(f) - \theta X(f) + \theta^* \theta X(f)) T_{\theta}(X, Y) = 0 $$

proving that $T_{\theta}$ is a tensor.

$\nabla$ being without torsion:

$$ (57) \qquad \begin{aligned} [\theta X, \theta^* Y] &= \nabla_{\theta X} \theta^* Y - \nabla_{\theta^* Y} \theta X \\ &= \theta^* \nabla_{\theta X}^* Y - \theta \nabla_{\theta^* Y}^* X \\ &= \theta^* (\nabla_Y^* \theta X + [\theta X, Y]) - \theta (\nabla_X^* \theta^* Y + [\theta^* Y, X]) \end{aligned} $$

Introducing the difference tensor $D: (X, Y) \mapsto \nabla_X^* Y - \nabla_Y X$:

$$ (58) \qquad [\theta X, \theta^* Y] = \theta^* \theta \nabla_Y X - \theta \theta^* \nabla_X Y + \theta^* D(\theta X, Y) - \theta D(X, \theta^* Y) \\ \qquad \qquad + \theta^* [\theta X, Y] + \theta [X, \theta^* Y] $$

Since $\theta$ is normal, $\theta^*\theta = \theta\theta^*$, thus:

$$ (59) \qquad [\theta X, \theta^* Y] = \theta^* \theta [Y, X] + \theta^* [\theta X, Y] + \theta [X, \theta^* Y] + \theta^* D(\theta X, Y) - \theta D(X, \theta^* Y) $$

and the claim follows. $\square$

<figure><img src="image_3.png" /></figure>

*Remark 3.1.* The tensoriality of $T_{\theta}$ depends critically on the normality of $\theta$. This was already pointed out by Nijenhuis.

*Remark 3.2.* $T_{\theta}$ is the usual Nijenhuis torsion when $\theta$ is skew-symmetric and $\nabla = \nabla^*$. When $\theta$ is normal and satisfies the gauge equation, the above results shows that $\theta^* D(\theta X, Y) - \theta D(X, \theta^* Y)$ is the Nijenhuis torsion on $\theta, \theta^*$.

**Proposition 3.5.** $\theta$ satisfies the gauge equation for $\nabla, \nabla^*$ if and only if one of the next equivalent properties hold for any vector fields $X, Y$.

$$ (60) \qquad \begin{aligned} (\nabla_X^* \theta) Y &= D(X, \theta Y) \\ (\nabla_X \theta) Y &= \theta D(X, Y). \end{aligned} $$

*Proof.* This is only a matter of writing the definition $(\nabla_X \theta) Y = \nabla_X \theta Y - \theta \nabla_X Y$ and applying the gauge equation. $\square$

<figure><img src="image_4.png" /></figure>
**Proposition 3.6.** If $\theta$ is normal and satisfies the gauge equation for $\nabla, \nabla^*$, then the Nijenhuis torsion $N_{\theta,\theta^*}$ is such that:

$$ (61) \qquad N_{\theta,\theta^*}(X,Y) = (\nabla_{\theta X} \theta^*) Y - (\nabla_{\theta^* Y} \theta) X = \theta^* (\nabla_X^* \theta) Y - \theta (\nabla_Y^* \theta^*) X. $$

An interesting example is given by a symmetric solution of the gauge equation taking a diagonal form:

$$ (62) \qquad \theta(x) = \lambda_i(x) \frac{\partial}{\partial x_i} \otimes dx^i $$

in some local orthonormal chart $x_1, \dots, x_n$. By Eq. 61:

$$ (63) \qquad \begin{aligned} N_{\theta}(X_i, X_j) &= (\nabla_{\theta X_i} \theta) X_j - (\nabla_{\theta X_j} \theta) X_i = \lambda_i \theta D(X_i, X_j) - \lambda_j \theta D(X_j, X_i) \\ &= (\lambda_i - \lambda_j) (\nabla_{X_i} \theta) X_j. \end{aligned} $$

From Eq. 63, the Nijenhuis torsion vanishes if there is a single eigenvalue of multiplicity $n$ or if $\theta$ is $\nabla$-parallel. For other cases, $(\nabla_{X_i} \theta) X_j$ must vanish if $\lambda_i \neq \lambda_j$. This condition is similar to the one given in [15] (Lemma 9).

**Definition 3.5.** Let $\theta$ be a symmetric or skew-symmetric solution of the gauge equation $\nabla\theta = \theta\nabla^*$ for a pair of dual connections $\nabla, \nabla^*$. The product $\cdot_{\theta}$ is defined on $\Gamma(TM)$ by:

$$ (64) \qquad X \circ_{\theta} Y = (\nabla_{\theta X} \theta) Y. $$

By proposition 3.4, the commutator $X \circ_{\theta} Y - Y \circ_{\theta} X$ is $N_{\theta}(X, Y)$.

**Proposition 3.7.** If $\circ_{\theta}$ defines a Lie-admissible algebra structure on $\Gamma(TM)$, then the tensor:

$$ (65) \qquad R_{\theta}: (X, Y, Z) \mapsto (\nabla_{\theta X} \theta) (\nabla_{\theta Y} \theta) Z - (\nabla_{\theta Y} \theta) (\nabla_{\theta X} \theta) Z - (\nabla_{\theta N_{\theta}(X, Y)} \theta) Z \qquad \text{is curvature-like in the sense that it is skew-symmetric in the first two variables and:} $$

$$ (66) \qquad R_{\theta}(X, Y, Z) + R_{\theta}(Z, X, Y) + R_{\theta}(Y, Z, X) = 0. $$

If it vanishes identically, then $\circ_{\theta}$ defines a pre-Lie algebra structure on $\Gamma(TM)$. In such a case, $\theta$ is said to be gauge-flat.

*Proof.* By a straightforward computation and repeated use of the commutator:

$$ (67) \qquad \begin{aligned} R_{\theta}(X, Y, Z) + R_{\theta}(Z, X, Y) + R_{\theta}(Y, Z, X) = \\ N_{\theta}(X, N_{\theta}(Y, Z)) + N_{\theta}(Z, N_{\theta}(X, Y)) + N_{\theta}(Y, N_{\theta}(Z, X)). \end{aligned} $$

The vanishing of expression 67 is equivalent to $\circ_{\theta}$ defines a Lie-admissible algebra. Now, if $R_{\theta} = 0$:

$$ (68) \qquad \begin{aligned} R_{\theta}(X, Y, Z) = X \circ_{\theta} (Y \circ_{\theta} Z) - (X \circ_{\theta} Y) \circ_{\theta} Z \\ - Y \circ_{\theta} (X \circ_{\theta} Z) + (Y \circ_{\theta} X) \circ_{\theta} Z = 0, \end{aligned} $$

proving that $\circ_{\theta}$ defines a pre-Lie algebra on $\Gamma(TM)$. $\square$

<figure><img src="image_5.png" /></figure>

*Remark 3.3.* A vanishing Nijenhuis torsion implies of course that $\circ_{\theta}$ defines a Lie-admissible algebra.

**Definition 3.6.** Let

$$ (69) \qquad T_{\theta}(X, Y, Z) = N_{\theta}(X, N_{\theta}(Y, Z)) + N_{\theta}(Z, N_{\theta}(X, Y)) + N_{\theta}(Y, N_{\theta}(Z, X)) $$

If $T = 0$, then $\theta$ is said to be a weak Nijenhuis tensor [7].
Since $N_{\theta}$ is skew-symmetric, it is clear that $T_{\theta}$ is non-zero only when $X, Y, Z$ are all different. A simple expansion of equation 69 yields the next proposition:

**Proposition 3.8.** Let $\theta$ be of diagonal form:

$$ \theta(x) = \lambda_i(x) \frac{\partial}{\partial x_i} \otimes dx^i $$

in some local chart. Then:

$$ \begin{align} T(e_i, e_j, e_k) = & N_{\theta}(e_i, e_j) \left( (\lambda_k - \lambda_i) \frac{\partial \lambda_i}{\partial x_k} - (\lambda_k - \lambda_j) \frac{\partial \lambda_j}{\partial x_k} \right) \tag{70} \\ & + N_{\theta}(e_j, e_k) \left( (\lambda_i - \lambda_j) \frac{\partial \lambda_j}{\partial x_i} - (\lambda_i - \lambda_k) \frac{\partial \lambda_k}{\partial x_i} \right) \nonumber \\ & + N_{\theta}(e_k, e_i) \left( (\lambda_j - \lambda_k) \frac{\partial \lambda_k}{\partial x_j} - (\lambda_j - \lambda_i) \frac{\partial \lambda_i}{\partial x_j} \right) \nonumber \end{align} $$

From proposition 3.8, $T_{\theta}$ vanishes of course when $N_{\theta} = 0$, but also when the differential conditions:

$$ (\lambda_k - \lambda_i) \frac{\partial \lambda_i}{\partial x_k} - (\lambda_k - \lambda_j) \frac{\partial \lambda_j}{\partial x_k} = 0 $$

are satisfied.

*Remark 3.4.* An arbitrary solution $\theta$ of the gauge equation can be uniquely written as $\theta = \theta_s + \theta_a$ where $\theta_s$ is symmetric, $\theta_a$ is skew symmetric and both are solutions of the gauge equation. Two products $\circ_{\theta_s}, \circ_{\theta_a}$ can thus be defined.

## 4. SPECTRAL SEQUENCES ARISING FROM THE GAUGE EQUATION

All the material in this section can be found in [13] when the pre-lie algebra arises from dually flat connections. The product $\circ_{\theta}$ allows, mutatis mutandis, another spectral sequence to be defined when the flatness condition is satisfied in the sense of prop. 3.7.

**4.1. Dually flat structures.** Let $(M, g, \nabla, \nabla^*)$ be a statistical structure with $\nabla, \nabla^*$ dual, torsion-free and flat dual connections. Let $\theta$ be a solution of the gauge equation for $\nabla, \nabla^*$. Define $\Theta$ (resp. $\Theta^*$) by:

$$ \begin{align} \theta_s &= \frac{1}{2} (\theta + \theta^*) \tag{71} \\ \theta_a &= \frac{1}{2} (\theta - \theta^*) \nonumber \end{align} $$

Then, since $\ker \theta_s, \im \theta_s$ are respectively $\nabla, \nabla^*$ invariant, there is a 2-web $TM = \ker \theta_s \oplus \im \theta_s$. In the same manner, there exists a 2-web $TM = \ker \theta_a \oplus \im \theta_a$. Any of the pairs $(\ker \theta_s, \im \theta_s), (\ker \theta_a, \im \theta_a)$ gives rise to a de Rham double complex:

$$ (72) \qquad \Omega(K, I) = \oplus_{q,p} \Omega^p(K) \otimes \Omega^q(I) $$

where $K = \ker \theta_{s,a}$, $I = \im \theta_{s,a}$ and $\Omega(M)$ denotes the de Rham complex of $M$. There is also a double $KV$-complex:

$$ (73) \qquad C_{KV}(A, A^*; V) = \oplus_{p,q} C_{KV}^p(A; V) \otimes C_{KV}^q(A^*; V) $$

where $A$ (resp. $A^*$) is the pre-Lie algebra associated to $\nabla$ (resp. $\nabla^*$). Then, considering the spectral sequences associated with these double complexes, they are converging to the cohomology of the total complex.
## 4.2. Flat solutions of the gauge equation.
Assuming that $T_{\theta} = 0$, $R_{\theta} = 0$ for a solution of the gauge equation, two pre-Lie algebras can be defined from $\circ_{\theta_s}$, $\circ_{\theta_a}$, thus yielding a double KV-complex with associated spectral sequence.

# 5. COKÄHLER MANIFOLDS

CoKähler manifolds provide a natural setting for Koszul–Vinberg structures, since the Levi–Civita connection is torsion-free and preserves the structural tensors $(\varphi, \xi, \eta, g)$. This makes them ideal for studying gauge-equation compatibility and spectral degeneracy. They form a class of almost contact metric manifolds closely related to both Kähler and cosymplectic geometry. Let $(M^{2n+1}, \varphi, \xi, \eta, g)$ be an almost contact metric manifold, where:

(1) $\varphi$ is a (1,1)-tensor field,

(2) $\xi$ is the Reeb (characteristic) vector field,

(3) $\eta$ is the contact 1-form,

(4) $g$ is a Riemannian metric satisfying:

$$ \eta(\xi) = 1, \quad \varphi^2 = -\mathrm{Id} + \eta \otimes \xi, \quad g(\varphi X, \varphi Y) = g(X, Y) - \eta(X)\eta(Y). $$

A coKähler structure is defined by the following conditions:

(1) $\nabla\varphi = 0$,

(2) $\nabla\eta = 0$, and

(3) $d\eta = 0$, $d\Phi = 0$, where $\Phi(X, Y) = g(X, \varphi Y)$.

These conditions imply that $\varphi$, $\eta$, and $g$ are all parallel with respect to the Levi-Civita connection $\nabla$, and that $(M, \varphi, \xi, \eta, g)$ is both normal and cosymplectic ([6, 3]). Importantly, the almost contact structure induces a foliation by leaves orthogonal to $\xi$, each of which inherits a Kähler structure from $\varphi$ and $g$.

This leads to a canonical gauge structure on coKähler manifolds, making them suitable for the application of KV-algebra techniques. Specifically, the Levi-Civita connection $\nabla$ on a coKähler manifold is torsion-free and preserves the underlying structure tensors, allowing one to define pre-Lie products via

$$ X \circ Y := \nabla_X Y $$

for vector fields $X, Y \in \Gamma(TM)$. The curvature conditions further imply symmetries in the associator that reflect pre-Lie algebra identities.

In particular, the canonical splitting $TM = \mathcal{D} \oplus \langle\xi\rangle$, where $\mathcal{D} = \ker(\eta)$, allows one to study the restriction of the Koszul-Vinberg algebra to the Kähler leaves and analyze the corresponding spectral sequence. These foliated structures are flat in transverse directions and lead naturally to vanishing torsion tensors $T_{\theta}$ when considering natural normal fields such as $\varphi$. Thus, coKähler manifolds provide an explicit class of geometric models where the degeneracy of the KV spectral sequence can be investigated under controlled curvature and holonomy assumptions.

Let $M = S^1 \times \mathbb{R}^{2n}$ with coordinates $(t, x^1, y^1, \dots, x^n, y^n)$. Define the flat product metric:

$$ g = dt^2 + \sum_{j=1}^{n} ((dx^j)^2 + (dy^j)^2) $$

Define a coKähler structure:
• $\xi = \partial_t$;

• $\eta = dt$;

• $\varphi(\partial_{x^j}) = \partial_{y^j}$, $\varphi(\partial_{y^j}) = -\partial_{x^j}$, $\varphi(\partial_t) = 0$.

Let $\nabla$ be the flat connection. Define the dual connection $\nabla^*$ as ([1]):

$$ \nabla_X^* Y = \nabla_X Y + D(X, Y) $$

where $D$ is a symmetric (1,2)-tensor field such that:

$$ g(D(X, Y), Z) = -g(Y, D(X, Z)). $$

Define $\theta$ diagonally:

$$ \theta = \lambda_0 \partial_t \otimes dt + \sum_{j=1}^{n} (\lambda_j \partial_{x^j} \otimes dx^j + \mu_j \partial_{y^j} \otimes dy^j) $$

This $\theta$ is symmetric by construction.

Given $\nabla\theta = 0$, the gauge equation reduces to:

$$ 0 = \theta D(X, \cdot). $$

*Example 5.1.* In dimension 2, let $A(\partial_x, \partial_x) = \partial_y$. Then $\theta(\partial_y) = 0$ is required. Thus, $\mu = 0$ is forced.

*Example 5.2.* Let $(M, \varphi, \xi, \eta, g)$ be a flat coKähler manifold with local orthonormal frame $\{\xi, E_2, \dots, E_n\}$ satisfying $\eta(\xi) = 1$, $\eta(E_j) = 0$, and let $\nabla$ denote the flat, torsion-free Levi-Civita connection. Define the symmetric (1,1)-tensor

$$ \theta = a(t) \eta \otimes \xi + \sum_{j=2}^{n} b_j(t) E^j \otimes E_j, $$

where $E^j$ are the dual 1-forms of $E_j$, and $\theta$ is time-dependent only. Assume $E_j$ are parallel along $\xi$, i.e., $\nabla_\xi E_j = 0$.

We verify the gauge-type condition

$$ (\nabla_X \theta)(Y) = \eta(Y)\theta(X) + \eta(X)\theta(Y), $$

for all vector fields $X, Y$ on $M$.

• For $X = Y = \xi$:

$$ (\nabla_{\xi}\theta)(\xi) = \nabla_{\xi}(\theta(\xi)) - \theta(\nabla_{\xi}\xi) = \nabla_{\xi}(a(t)\xi) = a'(t)\xi. $$

On the other hand, the right-hand side gives $2a(t)\xi$. Therefore,

$$ a'(t) = 2a(t) \Rightarrow a(t) = Ae^{2t}. $$

• For $X = \xi$, $Y = E_j$:

$$ (\nabla_{\xi}\theta)(E_j) = \nabla_{\xi}(b_j(t)E_j) = b'_j(t)E_j, $$

$$ \eta(E_j) = 0, \quad \eta(\xi) = 1 \Rightarrow \eta(Y)\theta(\xi) + \eta(X)\theta(Y) = \theta(E_j). $$

Thus,

$$ b'_j(t) = b_j(t) \Rightarrow b_j(t) = B_j e^t. $$
• For $X = E_k$, $Y$ arbitrary: The gauge condition becomes:

$$ (\nabla_{E_k} \theta)(Y) = \eta(Y)\theta(E_k). $$

Compute the left-hand side:

$$ (\nabla_{E_k} \theta)(Y) = \nabla_{E_k}(\theta(Y)) - \theta(\nabla_{E_k} Y). $$

For $Y = \xi$:

$$ (\nabla_{E_k} \theta)(\xi) = E_k(a)\xi + a\nabla_{E_k}\xi - \theta(\nabla_{E_k}\xi). $$

Equating both sides:

$$ E_k(a)\xi + (a - \theta)\nabla_{E_k}\xi = \theta(E_k). $$

Thus, either $\nabla_{E_k}\xi = 0$ and $E_k(a) = 0$ (implying $\theta(E_k) = 0$), or $\theta$ must be chosen to satisfy:

$$ \theta(E_k) = E_k(a)\xi + (a - \theta)(\nabla_{E_k}\xi). $$

This example demonstrates how exponential time-dependent components of $\theta$ can satisfy the gauge-type condition on flat coKähler manifolds under appropriate geometric constraints.

**Example 5.3.** We now present a nontrivial example where the tensor field $\theta$ depends polynomially on the coordinate $t$ and satisfies the gauge equation $\nabla\theta = \theta\nabla^*$, but only after modifying the background geometry. This illustrates the necessity of adjusting the difference tensor $D$ when seeking polynomial solutions in $\theta$.

Let $M = S^1 \times \mathbb{R}^{2n}$ with coordinates $(t, x_1, y_1, \dots, x_n, y_n)$ and flat coKähler structure as in previous examples. Define the symmetric $(1, 1)$-tensor field

$$ \theta = a(t) \eta \otimes \xi + \sum_{j=1}^{n} (b_j(t) dx_j \otimes \partial_{x_j} + c_j(t) dy_j \otimes \partial_{y_j}), $$

where the coefficient functions are taken to be polynomials in $t$, e.g., $a(t) = At^k$, $b_j(t) = B_j t^{l_j}$, $c_j(t) = C_j t^{m_j}$ for constants $A, B_j, C_j$ and non-negative integers $k, l_j, m_j$.

Assume the connection $\nabla$ is flat and torsion-free, so that $\nabla\theta = \partial_t\theta$. In the previous examples, a constant $\theta$ sufficed to make $\nabla\theta = 0$ and the gauge equation reduced to $\theta D = 0$. However, for this variable $\theta$, the equation $\nabla\theta = \theta D$ becomes

$$ \frac{d}{dt}\theta = \theta D(\partial_t, \cdot), $$

which imposes strong constraints on $D$.

To satisfy this equation, we define a non-flat difference tensor:

$$ D(X, Y) := \eta(X) \varphi(Y) + t \cdot \varphi(X) \eta(Y). $$

We compute:

$$ \partial_t b_j(t) = t \cdot b_j(t) \quad \Rightarrow \quad b_j(t) = B_j \exp\left(\frac{t^2}{2}\right), $$

so true polynomial solutions do not arise. However, for $b_j(t) = B_j t$, we find that:

$$ \partial_t b_j(t) = B_j = t \cdot b_j(t) \quad \Leftrightarrow \quad t = 1, $$

which is only valid on a slice of $M$. Thus, a polynomial solution is only compatible with the gauge equation if $D$ is choosen.
In the statistical coKähler setting we define, the structure tensor $\varphi$ is shown to satisfy the gauge equation $\nabla\varphi = \varphi\nabla^*$. This is a nontrivial and highly constrained equation in general, but here it emerges naturally from the compatibility condition on the difference tensor $D$. The result provides a canonical geometric solution to the gauge equation, allowing us to build Koszul–Vinberg double complexes and spectral sequences with intrinsic geometric meaning. This places the coKähler structure alongside holomorphic statistical geometry as a setting where deep gauge-theoretic phenomena can be studied naturally.

**Definition 5.1.** A coKähler statistical structure on a smooth manifold $M^{2n+1}$ is a quadruple $(\nabla, g, \varphi, \xi)$ such that:

(1) $(g, \varphi, \xi, \eta)$ defines a coKähler structure on $M$, with $\eta = g(\cdot, \xi)$,

(2) $\nabla$ is a torsion-free affine connection, and the pair $(\nabla, g)$ defines a statistical structure ([1]) i.e., there exists a dual torsion-free connection $\nabla^*$ such that

$$X \cdot g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X^* Z),$$

(3) The difference tensor $D = \nabla - \nabla^g$ satisfies the compatibility condition:

$$D(X, \varphi Y) + \varphi D(X, Y) = 0,$$

for all vector fields $X, Y$.

**Proposition 5.1.** Let $(\nabla, g, \varphi, \xi)$ be a statistical coKähler structure on a manifold $M$. Then the structure tensor $\varphi$ satisfies the gauge equation:

$$\nabla_X \varphi = \varphi \nabla_X^* \quad \text{for all } X \in TM.$$

*Proof.* Let $D = \nabla - \nabla^g$ be the difference tensor between the affine connection $\nabla$ and the Levi-Civita connection $\nabla^g$, and let $\nabla^* = 2\nabla^g - \nabla$ be the dual connection with respect to $g$ ([1]). Then for any vector fields $X, Y$, we compute:

$$\begin{aligned} \nabla_X(\varphi Y) - \varphi(\nabla_X^* Y) &= [\nabla_X^g(\varphi Y) + D(X, \varphi Y)] - \varphi[\nabla_X^g Y - D(X, Y)] \\ &= (\nabla_X^g \varphi)Y + D(X, \varphi Y) + \varphi D(X, Y). \end{aligned}$$

Since the coKähler structure satisfies $\nabla^g \varphi = 0$, and by assumption $D(X, \varphi Y) + \varphi D(X, Y) = 0$, we conclude:

$$\nabla_X(\varphi Y) = \varphi(\nabla_X^* Y),$$

which is the gauge equation. $\square$

**Proposition 5.2.** Let $(\nabla, g, \varphi, \xi)$ be a statistical coKähler structure on a manifold $M$. Then, the tensor expression

$$-\varphi^* D(\varphi X, Y) + \varphi D(X, \varphi^* Y)$$

vanishes identically for all vector fields $X, Y$, where $\varphi^* = -\varphi$ is the $g$-adjoint of $\varphi$. In particular, the corresponding Nijenhuis-type tensor from Remark 3.4 vanishes identically.

**Remark 5.1.** This result shows that the coKähler structure tensor $\varphi$ is a canonical gauge-compatible tensor in the sense of Section 3. Its normality and skew-symmetry, together with the statistical compatibility condition on $D$, ensure the vanishing of the associated Nijenhuis-type tensor. This places $\varphi$ on the same footing as complex structures in holomorphic statistical manifolds, providing a natural source of geometric gauge solutions and structured double complexes.
On a (2n+1)-dimensional coKähler manifold $(M, \varphi, \xi, \eta, g)$, there is the canonical orthogonal splitting

$$TM = \mathcal{D} \oplus \langle \xi \rangle, \quad \mathcal{D} = \ker \eta,$$

where $\mathcal{D}$ is the 2n-dimensional distribution orthogonal to the Reeb vector field $\xi$. In this adapted decomposition, the structure tensor $\varphi$ satisfies

$$\varphi(\xi) = 0, \quad \varphi^2|_{\mathcal{D}} = -\mathrm{Id}_{\mathcal{D}},$$

so that $\varphi$ induces an almost complex structure on $\mathcal{D}$.

We use a local $g$-orthonormal frame adapted to the splitting $TM = \mathcal{D} \oplus \langle \xi \rangle$:

$$\{\xi, E_1, \dots, E_{2n}\}, \quad E_{2k} = J E_{2k-1}, \quad k = 1, \dots, n,$$

so that any vector $v \in T_pM$ decomposes as

$$v = (v_{\mathcal{D}}, v_{\xi}),$$

with $v_{\mathcal{D}} \in \mathcal{D}_p$ and $v_{\xi} \in \langle \xi_p \rangle$.

In block form we may write

$$\varphi = \begin{pmatrix} J & 0 \\ 0 & 0 \end{pmatrix}, \quad \text{acting on} \quad \begin{pmatrix} v_{\mathcal{D}} \\ v_{\xi} \end{pmatrix},$$

where $J^2 = -\mathrm{Id}_{\mathcal{D}}$ and $v_{\mathcal{D}} \in \mathcal{D}, v_{\xi} \in \langle \xi \rangle$.

We can write a general (1, 1)-tensor $\theta$ (candidate gauge solution) in block form with respect to the orthogonal splitting $TM = \mathcal{D} \oplus \langle \xi \rangle$ as

$$\theta = \begin{pmatrix} B & u \\ v^{\top} & a \end{pmatrix},$$

where:

* $B: \mathcal{D} \to \mathcal{D}$ is the restriction of $\theta$ to $\mathcal{D}$ (a $2n \times 2n$ matrix in a chosen basis),

* $u: \langle \xi \rangle \to \mathcal{D}$ is given by $u = \pi_{\mathcal{D}}\theta(\xi)$, i.e. a column vector in $\mathcal{D}$ representing the $\mathcal{D}$-component of $\theta(\xi)$,

* $v^{\top}: \mathcal{D} \to \langle \xi \rangle$ is the linear functional $v(Y) = g(\theta(Y), \xi)$, i.e. a row vector representing the $\langle \xi \rangle$-component of $\theta$ applied to $\mathcal{D}$,

* $a \in C^{\infty}(M)$ is the scalar component of $\theta(\xi)$ along $\xi$.

The commutator of $\theta$ and $\varphi$ is

$$[\theta, \varphi] = \theta\varphi - \varphi\theta = \begin{pmatrix} BJ - JB & -Ju \\ v^{\top}J & 0 \end{pmatrix}.$$

Thus $[\theta, \varphi] \neq 0$ may arise in three distinct ways:

(1) The $\mathcal{D}$-component $B$ does not commute with $J$, i.e. $[B, J] \neq 0$.

(2) The vector $u$ is nonzero (mixing $\xi \to \mathcal{D}$): the upper-right block $-Ju$ is nonzero.

(3) The covector $v$ is nonzero (mixing $\mathcal{D} \to \xi$): the lower-left block $v^{\top}J$ is nonzero.

For each fixed $X \in T_pM$, the map

$$Y \mapsto D(X, Y)$$
is a linear endomorphism of $T_pM$. Thus, for each $X$ we can represent $D(X)$ as a block matrix

$$D(X) \equiv \begin{pmatrix} D_{11}(X) & D_{12}(X) \\ D_{21}(X) & D_{22}(X) \end{pmatrix},$$

where the blocks correspond to the orthogonal splitting $T_pM = D_p \oplus \langle \xi_p \rangle$.

On the other hand, the gauge equation is equivalent to

$$(\star) \qquad (\nabla_X^g \theta)(Y) + D(X, \theta Y) + \theta(D(X, Y)) = 0, \quad \forall X, Y \in TM.$$

The equation $(\star)$ becomes, for each $X \in TM$,

$$(1) \qquad (\nabla_X^g \theta) + D(X)\theta + \theta D(X) = 0.$$

Expanding (1) in block form gives the following system of equations:

$$(1a) : \quad (\nabla_X^g B) + D_{11}(X)B + D_{12}(X)v^\top + BD_{11}(X) + uD_{21}(X) = 0,$$

$$(1b) : \quad (\nabla_X^g u) + D_{11}(X)u + D_{12}(X)a + BD_{12}(X) + uD_{22}(X) = 0,$$

$$(1c) : \quad (\nabla_X^g v^\top) + D_{21}(X)B + D_{22}(X)v^\top + v^\top D_{11}(X) + aD_{21}(X) = 0,$$

$$(1d) : \quad X(a) + D_{21}(X)u + v^\top D_{12}(X) + 2a D_{22}(X) = 0.$$

**Lemma 5.3.** Let $(M, \varphi, \xi, \eta, g)$ be a statistical coKähler manifold. With respect to the orthogonal splitting $TM = D \oplus \langle \xi \rangle$ and the block form

$$\phi = \begin{pmatrix} J & 0 \\ 0 & 0 \end{pmatrix},$$

write $D(X)$ as

$$D(X) = \begin{pmatrix} D_{11}(X) & D_{12}(X) \\ D_{21}(X) & D_{22}(X) \end{pmatrix}.$$

Then, for every $X \in TM$,

$$D_{12}(X) = 0, \quad D_{21}(X) = 0, \quad D_{11}(X)J + J D_{11}(X) = 0,$$

and $D_{22}(X)$ is a scalar multiple of the identity on $\langle \xi \rangle$.

*Proof.* First take $Y = \xi$. Since $\varphi\xi = 0$, the relation $D(X, \varphi Y) + \varphi D(X, Y) = 0$ reduces to $\varphi D(X, \xi) = 0$. This means $D(X, \xi) \in \langle \xi \rangle$, hence $D_{12}(X) = 0$.

Next, take $Y \in D$. Then $\varphi Y = JY \in D$. The $\langle \xi \rangle$-component of $D(X, \varphi Y) + \varphi D(X, Y) = 0$ is

$$D_{21}(X)(JY) + 0 = 0.$$

Since $J$ is invertible on $D$, this implies $D_{21}(X) \equiv 0$. The $D$-component of the same relation reads

$$D_{11}(X)JY + J D_{11}(X)Y = 0,$$

for all $Y \in D$, i.e.

$$D_{11}(X)J + J D_{11}(X) = 0.$$

Finally, since $\langle \xi \rangle$ is one-dimensional, $D_{22}(X)$ is necessarily a scalar. $\square$
With the above simplifications, the gauge equation $\nabla\theta = \theta\nabla^*$ in block form

$$ \theta = \begin{pmatrix} B & u \\ v^{\top} & a \end{pmatrix} $$

reduces to the following system: for all $X \in TM$,

$$ (1a') \qquad (\nabla_X^g B) + D_{11}(X)B + B D_{11}(X) = 0, $$

$$ (1b') \qquad (\nabla_X^g u) + D_{11}(X)u + u D_{22}(X) = 0, $$

$$ (1c') \qquad (\nabla_X^g v^\top) + v^\top D_{11}(X) + D_{22}(X) v^\top = 0, $$

$$ (1d') \qquad X(a) + 2a D_{22}(X) = 0. $$

Fix a vector field $X$ and an integral curve $\gamma(s)$ with $\dot{\gamma}(s) = X(\gamma(s))$. Let $\nabla_{\dot{\gamma}}^g$ denote the covariant derivative along $\gamma$. Define

$$ \delta(s) := \int_{0}^{s} D_{22}(X)(\gamma(\tau)) d\tau, \qquad \mathcal{P}_{11}(s) := \mathcal{P} \exp\left(-\int_{0}^{s} D_{11}(X)(\gamma(\tau)) d\tau\right), $$

where $\mathcal{P} \exp$ is the path-ordered exponential.

If $B_0, u_0, v_0^\top, a_0$ denote the initial values of the components of $\theta$ at $s = 0$, the unique solution of $(1a')-(1d')$ along $\gamma$ is given by

$$ (A) \qquad B(s) = \mathcal{P}_{11}(s) B_0 \mathcal{P}_{11}(s), $$

$$ (B) \qquad u(s) = e^{-\delta(s)} \mathcal{P}_{11}(s) u_0, $$

$$ (C) \qquad v^\top(s) = e^{-\delta(s)} v_0^\top \mathcal{P}_{11}(s), $$

$$ (D) \qquad a(s) = a_0 e^{-2\delta(s)}. $$

**Lemma 5.4.** Let $(M, \varphi, \xi, \eta, g, \nabla)$ be a statistical coKähler manifold and assume the hypotheses of Lemma 4.3. Along an integral curve $\gamma(s)$ of a vector field $X = \dot{\gamma}(s)$ the block equation

$$ (1a') \qquad \nabla_X^g B + D_{11}(X)B + B D_{11}(X) = 0 $$

implies the following.

(1) The endomorphism $BJ$ satisfies the linear ODE

$$ \nabla_X^g (BJ) + [D_{11}(X), BJ] = 0, $$

hence, if we set

$$ U(s) = \mathcal{P} \exp\left(-\int_{0}^{s} D_{11}(\dot{\gamma}(\tau)) d\tau\right), \quad U(0) = \text{Id}, $$

we have the conjugation formula

$$ BJ(s) = U(s) (BJ)(0) U(s)^{-1}. $$

(2) The endomorphism $JB$ satisfies the linear ODE

$$ \nabla_X^g (JB) + [JB, D_{11}(X)] = 0, $$

and with $U(s)$ as above one obtains

$$ JB(s) = U(s)^{-1} (JB)(0) U(s). $$
(3) Let $C(s) := [B(s), J] = B(s)J - JB(s)$. Suppose that at $s = 0$ the pair $(BJ(0), JB(0))$ is not conjugate, i.e. there exists no invertible $Q \in \text{GL}(D)$ with

$$BJ(0) = Q(JB(0))Q^{-1}.$$

Then $C(s) \neq 0$ for all $s$ along $\gamma$; equivalently, if $[B(0), J] \neq 0$ and $BJ(0)$ is not conjugate to $JB(0)$, the commutator $[B(s), J]$ never vanishes along $\gamma$.

*Proof.* Multiply (1a') on the right by $J$. Using $\nabla^g J = 0$ and the anti-commutation $D_{11}J = -JD_{11}$ (Lemma 4.3) we get

$$\nabla_X^g (BJ) + D_{11}(X)BJ - BJ D_{11}(X) = 0,$$

which is exactly

$$\nabla_X^g (BJ) + [D_{11}(X), BJ] = 0.$$

Along the curve $\gamma$ this is an ordinary linear matrix ODE for the time-dependent endomorphism $A(s) := BJ(s)$. The standard solution of $\dot{A} + [D_{11}(s), A] = 0$ is conjugation by the fundamental solution $U(s)$ of $\dot{U} = -D_{11}(s)U$, $U(0) = \text{Id}$; writing the solution via the path-ordered exponential gives the formula $BJ(s) = U(s)(BJ)(0)U(s)^{-1}$.

Similarly, multiply (1a') on the left by $J$ and use $JD_{11} = -D_{11}J$:

$$\nabla_X^g (JB) - D_{11}(X)JB + JB D_{11}(X) = 0,$$

i.e.

$$\nabla_X^g (JB) + [JB, D_{11}(X)] = 0.$$

If $U(s)$ is as above then $U(s)^{-1}$ satisfies $\dot{W} = D_{11}(s)W$, $W(0) = \text{Id}$, and one checks that $JB(s) = U(s)^{-1}(JB)(0)U(s)$.

Finally, assume by contradiction that there exists $s_0$ with $C(s_0) = 0$. Then $BJ(s_0) = JB(s_0)$. Using the conjugation formulas we obtain

$$U(s_0)(BJ)(0)U(s_0)^{-1} = U(s_0)^{-1}(JB)(0)U(s_0).$$

Multiply on the left by $U(s_0)^{-1}$ and on the right by $U(s_0)^{-1}$ to get

$$(BJ)(0) = U(s_0)^{-2}(JB)(0)U(s_0)^2,$$

so $(BJ)(0)$ and $(JB)(0)$ are conjugate (by $Q = U(s_0)^{-2}$). This contradicts the hypothesis that they are not conjugate. Hence $C(s) \neq 0$ for all $s$, which proves persistence of non-commutativity along $\gamma$. $\square$

## 6. CR SUBMANIFOLDS

**Definition 6.1.** Let $(\overline{M}^{2n+1}, \varphi, \xi, \eta, g)$ be a coKähler manifold, i.e. an almost contact metric manifold which is normal and satisfies $d\eta = 0$ and $d\Phi = 0$, where $\Phi(X, Y) = g(X, \varphi Y)$.

A submanifold $M \subset \overline{M}$ is called a CR submanifold of the coKähler manifold if there exists an orthogonal decomposition of the tangent bundle

$$TM = D \oplus D^{\perp}$$

such that

(1) $\varphi(D_p) = D_p$ for all $p \in M$ (the distribution $D$ is $\varphi$-invariant),

(2) $\varphi(D_p^{\perp}) \subset T_p^\perp M$ for all $p \in M$ (the distribution $D^{\perp}$ is totally real, or anti-invariant).
Depending on the situation, the Reeb vector field $\xi$ may be either tangent to $M$ (and contained in $D$ or $D^{\perp}$) or normal to $M$.

Let $(\overline{M}, \varphi, \xi, \eta, g)$ be a statistical coKähler manifold with an affine connection $\overline{\nabla}$ and its dual $\overline{\nabla}^*$ (so that $(\overline{\nabla}, g)$ is a statistical pair). Then the ambient gauge equation holds:

$$ (74) \qquad \overline{\nabla}_X(\varphi Y) = \varphi(\overline{\nabla}_X^* Y), \quad \forall X, Y \in \Gamma(\overline{T\overline{M}}). $$

Let $M \subset \overline{M}$ be a CR-submanifold. Denote by $TM$ the tangent bundle and by $T^{\perp}M$ the normal bundle. For tangent vector fields $X, Y \in \Gamma(T\overline{M})$ we use the Gauss-Weingarten decompositions:

$$ (75) \qquad \overline{\nabla}_X Y = \nabla_X Y + h(X, Y), $$

$$ (76) \qquad \overline{\nabla}_X \nu = -A_\nu X + \nabla_X^\perp \nu, \quad \nu \in \Gamma(T^{\perp}M), $$

where $\nabla$ is the induced connection on $TM$, $h$ is the second fundamental form, $A_\nu$ is the shape operator, and $\nabla^\perp$ the normal connection.

For any $Y \in \Gamma(TM)$, decompose

$$ (77) \qquad \varphi Y = PY + FY, \quad PY \in \Gamma(TM), FY \in \Gamma(T^{\perp}M). $$

Here $P : \Gamma(TM) \to \Gamma(TM)$ is the tangential part of $\varphi|_{TM}$ and $F : \Gamma(TM) \to \Gamma(T^{\perp}M)$ the normal part.

In the case of a CR-submanifold $M$, we have the tangent splitting $TM = D \oplus D^{\perp}$, where $D$ is $\varphi$-invariant $(\varphi(D) \subseteq D)$ and $D^{\perp}$ is totally real $(\varphi(D^{\perp}) \subseteq T^{\perp}M)$. Thus:

$$ Y \in D \Rightarrow FY = 0, \quad Y \in D^{\perp} \Rightarrow PY = 0, \quad \varphi(Y) \in T^{\perp}M. $$

Let $X, Y \in \Gamma(TM)$. Applying the ambient gauge equation (74) to $\varphi Y = PY + FY$, and using the Gauss-Weingarten formulas (75)-(76), we obtain

$$ \begin{aligned} \overline{\nabla}_X(\varphi Y) &= \overline{\nabla}_X(PY + FY) \\ &= \nabla_X(PY) + h(X, PY) - A_{FY}X + \nabla_X^\perp(FY). \end{aligned} $$

Thus, the tangent and normal projections of the left-hand side are

$$ (78) \qquad (\overline{\nabla}_X(\varphi Y))^T = \nabla_X(PY) - A_{FY}X, $$

$$ (79) \qquad (\overline{\nabla}_X(\varphi Y))^{\perp} = \nabla_X^\perp(FY) + h(X, PY). $$

On the other hand, for the right-hand side of (74), we decompose ([11])

$$ \overline{\nabla}_X^* Y = \nabla_X^* Y + h^*(X, Y), $$

so that

$$ \varphi(\overline{\nabla}_X^* Y) = \varphi(\nabla_X^* Y) + \varphi(h^*(X, Y)). $$

Splitting into tangent and normal parts, we obtain two compatibility conditions.

$$ (80) \qquad \nabla_X(PY) - A_{FY}X = P(\nabla_X^* Y) + P(h^*(X, Y)). $$

The difference

$$ O^T(X, Y) := \nabla_X(PY) - P(\nabla_X^* Y) = A_{FY}X + P(h^*(X, Y)) $$

measures the tangential obstruction to the gauge equation on $M$.

$$ (81) \qquad \nabla_X^\perp(FY) + h(X, PY) = F(\nabla_X^* Y) + F(h^*(X, Y)). $$
The difference

$$O^{\perp}(X, Y) := \nabla_{X}^{\perp}(FY) - F(h^{*}(X, Y)) = -h(X, PY) + F(\nabla_{X}^{*}Y)$$

measures the normal obstruction.

**Theorem 6.1.** Let $M$ be a CR-submanifold of a statistical coKähler manifold $(\bar{M}, \varphi, \xi, \eta, g, \bar{\nabla})$. Define the obstruction tensors

$$O^{T}(X, Y) := \nabla_{X}(PY) - P(\nabla_{X}^{*}Y) = A_{FY}X + P(h^{*}(X, Y)),$$

and

$$O^{\perp}(X, Y) := \nabla_{X}^{\perp}(FY) - F(h^{*}(X, Y)) = -h(X, PY) + F(\nabla_{X}^{*}Y).$$

Then the following are equivalent:

(1) The restriction of the ambient gauge equation

$$\nabla_X(\varphi Y) = \varphi(\nabla_X^* Y), \quad X, Y \in \Gamma(TM),$$

to $M$ holds identically.

(2) Both obstruction tensors vanish:

$$O^T(X, Y) = 0, \quad O^{\perp}(X, Y) = 0, \quad \forall X, Y \in \Gamma(TM).$$

(3) The following explicit conditions are satisfied:

$$A_{FY}X + P(h^*(X, Y)) = 0, \quad F(\nabla_X^*Y) = h(X, PY), \quad \forall X, Y \in \Gamma(TM).$$

**Proposition 6.2** (Geometric consequences of the obstructions). Let $M$ be a CR-submanifold of a statistical coKähler manifold. Then the obstruction tensors $O^T$ and $O^{\perp}$ admit the following geometric interpretations:

(1) **Tangential obstruction.** The condition $O^T = 0$ is equivalent to

$$A_{FY}X + P(h^*(X, Y)) = 0, \quad \forall X, Y \in \Gamma(TM).$$

In this case the CR distribution $\mathcal{D} = \ker \eta \cap TM$ is preserved by the statistical structure $(\nabla, \nabla^*)$, and $(\mathcal{D}, P)$ carries a natural statistical holomorphic structure.

(2) **Normal obstruction.** The condition $O^{\perp} = 0$ is equivalent to

$$F(\nabla_X^*Y) = h(X, PY), \quad \forall X, Y \in \Gamma(TM).$$

Thus the normal bundle is gauge-compatible with respect to the induced normal connection. In particular, if $M$ is totally geodesic ($h = 0$), then $O^{\perp} = 0$ reduces to $F(\nabla_X^*Y) = 0$.

**Corollary 6.3** (Special cases). Let $M$ be a CR-submanifold of a statistical coKähler manifold. Then the obstruction tensors simplify in the following situations:

(1) **Invariant CR-submanifold ($F = 0$).** In this case,

$$O^{\perp}(X, Y) = -h(X, PY), \quad O^{T}(X, Y) = P(h^{*}(X, Y)).$$

Thus gauge compatibility holds if and only if

$$h(X, PY) = 0, \quad P(h^*(X, Y)) = 0.$$

(2) **Anti-invariant CR-submanifold ($P = 0$).** In this case,

$$O^T(X, Y) = A_{FY}X, \quad O^{\perp}(X, Y) = F(\nabla_X^*Y).$$

Hence gauge compatibility requires vanishing of the shape operator $A_{FY}$ together with $F(\nabla_X^*Y) = 0$.
(3) Dually totally geodesic submanifold ($h = 0, h^* = 0$). In this case,

$$O^T(X, Y) = A_{FY}X, \quad O^\perp(X, Y) = F(\nabla_X^*Y).$$

**Theorem 6.4** (Degeneracy of the KV spectral sequence on CR-submanifolds). Let $M \subset \bar{M}$ be a CR-submanifold of a statistical coKähler manifold. Then the Koszul-Vinberg spectral sequence associated to the pre-Lie algebra of vector fields on $M$ degenerates at the second page $E_2$ if and only if both obstruction tensors vanish:

$$O^T(X, Y) = 0, \quad O^\perp(X, Y) = 0, \quad \forall X, Y \in \Gamma(TM).$$

Equivalently, degeneracy occurs precisely when

$$A_{FY}X + P(h^*(X, Y)) = 0, \quad F(\nabla_X^*Y) = h(X, PY), \quad \forall X, Y \in \Gamma(TM).$$

**Corollary 6.5.** In each of the special cases below, degeneracy of the KV spectral sequence is characterized as follows:

(1) **Invariant CR-submanifold** ($F = 0$). Degeneracy holds if and only if

$$h(X, PY) = 0, \quad P(h^*(X, Y)) = 0.$$

(2) **Anti-invariant CR-submanifold** ($P = 0$). Degeneracy holds if and only if

$$A_{FY}X = 0, \quad F(\nabla_X^*Y) = 0.$$

(3) **Dually totally geodesic submanifold** ($h = 0, h^* = 0$). Degeneracy holds if and only if

$$A_{FY}X = 0, \quad F(\nabla_X^*Y) = 0.$$

**Proposition 6.6.** Let $M$ be a CR-submanifold of a statistical coKähler manifold $(\bar{M}, \phi, \xi, \eta, g, \nabla)$. Assume $M$ is invariant, i.e. $\phi(TM) \subset TM$ (equivalently $F \equiv 0$ in the splitting $\phi Y = PY + FY$), and that the normal bundle is $\phi$-invariant, i.e. $\phi(T^\perp M) \subset T^\perp M$. Then

$$P(h^*(X, Y)) = 0 \quad \text{for all } X, Y \in \Gamma(TM).$$

Consequently, in the invariant case the obstruction tensors reduce to

$$O^\perp(X, Y) = -h(X, PY), \quad O^T(X, Y) = 0,$$

so the gauge-compatibility condition (hence $E_2$-degeneracy of the KV spectral sequence) is equivalent to

$$h(X, PY) = 0 \quad \text{for all } X, Y \in \Gamma(TM).$$

**Definition 6.2** (Mixed connection on $\operatorname{Hom}(TM, TM)$). Let $(M, \nabla, \nabla^*)$ be a statistical manifold. The mixed connection on the endomorphism bundle $\operatorname{Hom}(TM, TM)$ is defined by

$$(82) \qquad (\nabla_X^{\text{mix}} T)(Y) = \nabla_X(TY) - T(\nabla_X^*Y),$$

for all $T \in \Gamma(\operatorname{Hom}(TM, TM))$ and $X, Y \in \Gamma(TM)$.

This connection acts by $\nabla$ on the target argument and by the dual connection $\nabla^*$ (with opposite sign) on the source argument. It naturally extends to the ambient bundle $TM$ and to any subbundle preserved by both $\nabla$ and $\nabla^*$.
**Definition 6.3** (Mixed curvature). The curvature of the mixed connection $\nabla^{\text{mix}}$ is the (1, 3)-tensor

$$ (83) \qquad \Omega^{\text{mix}}(X, Y)T := [\nabla_X^{\text{mix}}, \nabla_Y^{\text{mix}}]T - \nabla_{[X,Y]}^{\text{mix}}T, \quad X, Y \in \Gamma(\text{TM}). $$

A direct computation shows that, for every $Z \in \Gamma(\text{TM})$,

$$ (84) \qquad (\Omega^{\text{mix}}(X, Y)T)(Z) = R(X, Y)(TZ) - T(R^*(X, Y)Z). $$

where $R$ and $R^*$ denote the curvature tensors of $\nabla$ and $\nabla^*$, respectively.

**Lemma 6.7.** Let $M$ be a CR submanifold of a statistical coKähler manifold $(\bar{M}, \varphi, \xi, \eta, g, \bar{\nabla})$. Denote

$$ P = \text{pr}_{\text{TM}} \circ \varphi|_{\text{TM}}, \quad F = \text{pr}_{T^{\perp}M} \circ \varphi|_{\text{TM}}. $$

Then, for all $X, Y \in \Gamma(\text{TM})$,

$$ (85) \qquad (\tilde{\nabla}_X^{\text{mix}} \varphi)Y = O_T(X, Y) + O_{\perp}(X, Y), $$

where

$$ (86) \qquad O_T(X, Y) = \nabla_X(PY) - P(\nabla_X^*Y) = A_{FY}X + P(h^*(X, Y)), $$

$$ (87) \qquad O_{\perp}(X, Y) = \nabla_X^\perp(FY) - F(\nabla_X^\perp Y) = -h(X, PY) + F(h^*(X, Y)). $$

The tensors $O_T$ and $O_{\perp}$ represent the tangential and normal components of the failure of the gauge compatibility condition $\nabla_X(\varphi Y) = \varphi(\nabla_X^*Y)$.

**Proposition 6.8** (Geometric meaning of the mixed curvature). For the structural endomorphism $P$ on a CR submanifold $M$, the curvature of the mixed connection satisfies

$$ (88) \qquad \Omega^{\text{mix}}(X, Y)P = R(X, Y) \circ P - P \circ R^*(X, Y). $$

Its vanishing is equivalent to the gauge-flatness condition

$$ O_T = 0 \quad \text{and} \quad O_{\perp} = 0. $$

Hence the CR submanifold is gauge-flat precisely when $(\tilde{\nabla}^{\text{mix}}\varphi)|_{\text{TM}} = 0$. In this case the geometric flatness of the induced gauge field implies the $E_2$-degeneracy of the Koszul-Vinberg spectral sequence, thus establishing the correspondence between geometric gauge flatness and homological degeneracy.

## 7. $\theta$-CR SUBMANIFOLDS

**Definition 7.1.** Let $(\bar{M}, \varphi, \xi, \eta, g, \bar{\nabla}, \nabla^*)$ be a statistical coKähler manifold and let $\theta: \bar{TM} \to TM$ be a (1, 1)-tensor field satisfying the ambient gauge equation

$$ \bar{\nabla}_X \Theta = \Theta \bar{\nabla}_X^*, \quad X \in \Gamma(\bar{TM}). $$

A submanifold $M \subset \bar{M}$ is called a $\theta$-CR submanifold if the following conditions hold:

(i) There exists an orthogonal decomposition of the tangent bundle

$$ TM = D_{\Theta} \oplus D_{\Theta}^{\perp}, $$

such that

$$ \Theta(D_{\Theta}) \subset D_{\Theta}, \quad \Theta(D_{\Theta}^{\perp}) \subset T^{\perp}M. $$
(ii) Writing, for $Y \in \Gamma(TM)$,

$$ \Theta Y = P_{\Theta}Y + F_{\Theta}Y, \quad P_{\Theta}Y \in \Gamma(TM), F_{\Theta}Y \in \Gamma(T^{\perp}M), $$

the $\theta$-obstruction tensors

$$ O_{T}^{\Theta}(X, Y) = \nabla_{X}(P_{\Theta}Y) - P_{\Theta}(\bar{\nabla}_{X}^{*}Y), \quad O_{\perp}^{\Theta}(X, Y) = \nabla_{\bar{X}}^{\perp}(F_{\Theta}Y) - F_{\Theta}(\bar{\nabla}_{\bar{X}}^{*}Y) $$ 
vanish identically for all $X, Y \in \Gamma(TM)$.

**Remark 7.1.** (1) Any classical CR submanifold of a statistical coKähler manifold is a $\theta$-CR submanifold for $\theta = \varphi$. In this case, $P_{\theta} = P$, $F_{\theta} = F$, and the $\theta$-obstruction tensors coincide with the classical CR obstruction tensors.

(2) In general, $\theta$ need not satisfy $\theta^2 = -\mathrm{Id}$ on any distribution. Thus $\theta$-CR submanifolds form a strictly larger class than classical CR submanifolds.

(3) The vanishing of $O_{T}^{\theta}$ and $O_{\perp}^{\theta}$ is equivalent to the restriction of the ambient gauge equation

$$ \bar{\nabla}_{X}(\theta Y) = \theta(\bar{\nabla}_{X}^{*}Y) $$

to the tangent bundle $TM$, hence $\theta$-CR submanifolds are precisely those submanifolds on which $\theta$ is gauge-flat.

**Definition 7.2.** Let $M$ be a $\theta$-CR submanifold of a statistical coKähler manifold $(\bar{M}, \varphi, \xi, \eta, g, \bar{\nabla}, \bar{\nabla}^*)$. We define a bilinear product on $\Gamma(TM)$ by

$$ X \circ_{\theta} Y := (\bar{\nabla}_{\theta X} \theta) Y^T, \quad X, Y \in \Gamma(TM), $$

where $(\cdot)^T$ denotes the tangential component.

**Theorem 7.1.** Let $M$ be a $\theta$-CR submanifold of a statistical coKähler manifold, and assume that $\theta$ is normal. Then:

(i) The product $\circ_{\theta}$ defines a Koszul–Vinberg (pre–Lie) algebra structure on $\Gamma(TM)$.

(ii) The curvature-type tensor associated to $\circ_{\theta}$ vanishes identically.

(iii) The Koszul–Vinberg spectral sequence associated to $(\Gamma(TM), \circ_{\theta})$ degenerates at the second page:

$$ E_2 \cong E_{\infty}. $$

**Proof.** By the $\theta$-version of Theorem 5.4, the vanishing of the $\Theta$-obstruction tensors $O_{T}^{\theta}$ and $O_{\perp}^{\theta}$ is equivalent to the flatness of the mixed connection acting on $\theta$, that is,

$$ \nabla^{\mathrm{mix}} \theta = 0 \quad \text{on } TM. $$

Consequently, the mixed curvature tensor satisfies

$$ \Omega^{\mathrm{mix}}(X, Y)\theta = 0, \quad \forall X, Y \in \Gamma(TM). $$

This flatness condition implies that the curvature–like tensor controlling the failure of the pre–Lie identity for the product

$$ X \circ_{\theta} Y := (\bar{\nabla}_{\theta X} \theta) Y^T $$

vanishes identically. In particular, the associator of $\circ_{\theta}$ is symmetric in the first two arguments.

Moreover, since $\theta$ is assumed to be normal, the associated $\theta$-Nijenhuis tensor vanishes identically. Therefore, the product $\circ_{\theta}$ satisfies the Koszul–Vinberg (pre–Lie) identity and defines a pre–Lie algebra structure on $\Gamma(TM)$.
Finally, the degeneration of the Koszul–Vinberg spectral sequence associated to $(\Gamma(TM), \circ_{\theta})$ follows from the same homological argument used in the case of $\varphi$–CR submanifolds. Indeed, the flatness of the mixed connection implies that all higher obstruction classes vanish, so that all differentials $d_r$ for $r \ge 3$ are trivial and

$$E_2 \cong E_{\infty}.$$

This concludes the proof.

□

*Example 7.1 (Classical CR submanifolds).* Let $M$ be a CR submanifold of a statistical coKähler manifold and take $\theta = \varphi$. Then $M$ is a $\theta$–CR submanifold, and the induced product $\circ_{\theta}$ coincides with the $\varphi$–product studied in Section 5.

*Example 7.2 (Diagonal $\theta$ on flat statistical coKähler manifolds).* Let $\bar{M} = S^1 \times \mathbb{R}^{2n}$ with its flat coKähler statistical structure and let $\theta$ be a diagonal $(1, 1)$–tensor satisfying the gauge equation $\bar{\nabla}\theta = \theta\bar{\nabla}^*$. Any submanifold $M$ invariant under the eigenspace decomposition of $\theta$ is a $\theta$–CR submanifold. In this case the induced $\theta$–product is explicitly computable and defines a finite-dimensional family of pre-Lie algebras parameterized by the eigenvalues of $\theta$.

*Example 7.3 (Totally geodesic $\theta$–CR submanifolds).* If $M$ is totally geodesic with respect to both $\bar{\nabla}$ and $\bar{\nabla}^*$, then $h = h^* = 0$ and the $\theta$–obstruction tensors vanish automatically. Hence any submanifold invariant under $\theta$ is $\theta$–CR, and the $\theta$–product reduces to

$$X \circ_{\theta} Y = \nabla_{\theta X}(\theta Y) - \theta(\nabla_{\theta X} Y),$$

giving a particularly simple pre-Lie structure.

*Remark 7.2.* $\theta$–CR submanifolds provide a new geometric setting in which gauge theory, extrinsic geometry, and homological algebra interact. They generalize classical CR submanifolds and are characterized equivalently by:

* gauge-flatness of $\theta$ along $M$,

* vanishing of $\theta$–obstruction tensors,

* degeneration of the Koszul–Vinberg spectral sequence.

This establishes $\theta$–CR geometry as a natural framework for studying homological degeneracy phenomena in statistical and gauge-compatible geometries.

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FACULTY OF ECONOMICS, UNIVERSITY OF BANJA LUKA

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