1 Introduction

Although \(A\)-structures were originally intended to provide a general setting for many important \(G\)-structures, questions on \(A\)-analyticity and \(A\)-differentiability (cf. [7, 23]) have posed interesting problems in analysis of hypercomplex variables, notably in the quaternionic case. The development of \(A\)-manifolds has been also influenced by the theory of supermanifolds, graded manifolds, gauge theories (e.g., see [6, 11, 12, 17]), and, in general, by the structures on manifolds modeled over a finite-dimensional ground \(\mathbb R \)-algebra—instead of the real or complex fields—as introduced in the general survey [21] (also see [9]).

Below, a common definition of a connection for the class of \(G\)-structures satisfying: i) \(\mathfrak g ^{(1)}=0\) and ii) \(\mathfrak g \) is invariant under transposition (\(\mathfrak g \) being the Lie algebra of \(G\)), is proposed and then, such a connection is compared with the connections appearing in the literature for the classical geometric \(G\)-structures. In the case of \(A\)-structures, the algebras \(A\) for which \(\mathfrak gl (r,A)\) satisfies the previous conditions i) and ii) are classified.

2 \(A\)-manifolds

Let \(A\) be a (not necessarily commutative) \(n\)-dimensional \(\mathbb R \)-algebra, let \(\mathfrak gl (r,A)\) be the \(\mathbb R \)-algebra of \(r\times r\) matrices with entries in \(A\), and let \(GL(r,A)\) be the group of invertible matrices in \(\mathfrak gl (r,A)\) endowed with its Lie-group structure of dimension \(\dim GL(r,A)=r^2n\), whose Lie algebra can be identified to \(\mathfrak gl (r,A)\) itself with the standard matrix Lie bracket. As \(\mathfrak gl (r,A)\) can be identified to right \(A\)-linear mappings \(\Lambda :A^r\rightarrow A^r\), there exists a linear representation \(\lambda ^r:GL(r,A)\rightarrow GL(nr,\mathbb R )\).

Let \(M\) be an \(m\)-dimensional \(C^\infty \) manifold, \(m=nr\). An almost \(A\) -structure (resp. an \(A\)-structure) on \(M\) is a \(GL(r,A)\)-structure (resp. an integrable \(GL(r,A)\)-structure) on \(M\). This definition covers a large class of interesting geometric structures on manifolds, such as integrable almost tangent structures (\(A=\mathbb R [t]/(t^2)\)), complex and paracomplex structures (\(A=\mathbb R [t]/(t^2\pm 1)\)), hypercomplex and quaternionic-like structures (\(A=\mathbb R [i,j,k]\)), Yano’s structures (\(A=\mathbb R [t]/(t^3+t)\)).

Proposition 2.1

Let \(M\) be an \(m\)-dimensional \(C^\infty \) manifold, \(m=nr\). There is a natural bijection between almost \(A\)-structures on \(M\) and free right \(A\)-module structures on the tangent spaces \(T_xM\), \(\forall x\in M\). The manifold \(M\) admits an \(A\)-structure if and only if, with the natural identification \(T_zA^r\cong A^r\), \(z\in A^r\), there exists an atlas \(\{ \varphi _i:U_i\rightarrow A^r\} _{i\in I}\) such that the mapping

$$\begin{aligned} \left( \varphi _i\circ \varphi _j^{-1} \right) _*:T_{\varphi _j(x)}A^r \rightarrow T_{\varphi _i(x)}A^r \end{aligned}$$

is right \(A\)-linear for every \(x\in U_i\cap U_j\).

Proof

Assume \(M\) admits a \(GL(r,A)\)-structure \(\pi :P\rightarrow M\), i.e., \(P\) is a subbundle of the bundle of linear frames \(\pi :FM\rightarrow M\). A linear frame of \(M\) at \(x\in M\) can be identified to a \(\mathbb R \)-linear isomorphism \(u:A^r\rightarrow T_x(M)\). Once a frame \(u\in P_x=\pi ^{-1}(x)\) has been fixed, we can define a right free \(A\)-module structure on \(T_x(M)\) by setting \(X\cdot a=u(u^{-1}(X)\cdot a)\), \(\forall a\in A\), \(\forall X\in T_x(M)\), and the definition does not depend on the chosen frame, as any other linear frame \(v\in P_x\) can uniquely be written as \(v=u\circ \Lambda \), \(\Lambda \in GL(r,A)\), and we thus have

$$\begin{aligned} v\left( v^{-1} \left( X \right) \cdot a \right)&= \left( u\circ \Lambda \right) \left( \Lambda ^{-1} \left( u^{-1} \left( X\right) \cdot a\right) \right) \\&= u\left( u^{-1} \left( X\right) \cdot a\right) . \end{aligned}$$

Conversely, if the fibers of the tangent bundle \(TM\) are endowed with a structure of right free \(A\)-module, then we can define a \(GL(r,A)\)-structure \(\pi :P\rightarrow M\) as the linear frames \(u:A^r\rightarrow T_xM\) which are right \(A\)-linear with respect to that structure. Moreover, if an atlas exists satisfying the condition in the statement, then we can define a right \(A\)-module structure on \(T_xM\) by simply choosing an index \(i\) such that \(x\in U_i\) and transporting the right \(A\)-module structure of \(A^r\) via \((\varphi _i)_*\). According to the first part of the proof, we have thus defined a \(GL(r,A)\)-structure on \(M\), which is integrable as it follows from its very definition that this structure is locally isomorphic to the standard \(GL(r,A)\)-structure on \(A^r\). The converse is similar. \(\square \)

3 Functorial connections

Let \(G\) be a closed subgroup of \(GL(m,\mathbb R )\), \(m=\dim M\). The \(G\)-structures over \(M\) are in bijection with the sections of the quotient bundle \(\bar{\pi }:FM/G\rightarrow M\); e.g., see [15], I, Propositions 5.5 and 5.6]. The bijection \(s\leftrightarrow P_s\) between sections and \(G\)-structures is given by \(P_s=\left\{ u\in FM:u\cdot G =s(\pi (u)) \right\} \). For each diffeomorphism \(\varphi :M\rightarrow M^\prime \), let \(\tilde{\varphi }:FM\rightarrow FM^\prime \) be the \(GL(m,\mathbb R )\)-principal bundle isomorphism given by (cf. [15], VI, p. 226]): \(\tilde{\varphi }( X_1,\ldots ,X_m) =(\varphi _*X_1,\ldots ,\varphi _*X_m)\), for every linear frame \((X_1,\ldots ,X_m)\in F_xM\). By passing to the quotient, \(\tilde{\varphi }\) induces a diffeomorphism \(\bar{\varphi }:FM/G\rightarrow FM^\prime /G\), such that \(\bar{\pi }^\prime \circ \bar{\varphi } =\varphi \circ \bar{\pi }\), and for every \(G\)-structure \(s:M\rightarrow FM/G\) over \(M\), we can define a \(G\)-structure over \(M^\prime \)—the \(G\)-structure obtained by transporting \(s\) to \(M^\prime \) via \(\varphi \)—by setting \(\varphi \cdot s =\bar{\varphi }\circ s\circ \varphi ^{-1}\).

Let \(CM\rightarrow M\) be the bundle of linear connections on \(M\), which is an affine bundle modeled over the vector bundle \(\otimes ^2T^*M\otimes TM\). The global sections of \(CM\) are identified to the linear connections on \(M\).

Let \(p^k:J^kE\rightarrow M\) be the bundle of \(k\)-jets of local sections of a fibered manifold \(p:E\rightarrow M\) with projections \(p^l_k:J^lE\rightarrow J^kE\) for \(l\ge k\).

A precise definition of a functorial connection is as follows (cf. [19]):

A functorial connection on \(G\)-structures over \(M\) is a presheaf morphism \(s\mapsto \Gamma (s)\), from the presheaf of sections of \(FM/G\rightarrow M\) into the presheaf of sections of \(CM\rightarrow M\), such that,

  1. 1.

    If \(s:U\rightarrow FM\) is a section on an open subset \(U\subseteq M\), then \(\Gamma (s)\) is a linear connection on \(U\) adapted to the \(G\)-structure defined by \(s\); i.e., \(\Gamma (s)\) is reducible to the subbundle \(P_s\).

  2. 2.

    Let \(U^\prime \subseteq M^\prime \) be an open subset in another manifold \(M^\prime \). If \(\varphi :U\rightarrow U^\prime \) is a diffeomorphism, then \(\Gamma (\varphi \cdot s)=\varphi \cdot \Gamma (s)\), where the right-hand side stands for the direct image of \(\Gamma (s)\) by \(\varphi \) (cf. [15], II.Proposition 6.1]).

  3. 3.

    \(\Gamma \) factors smoothly through \(J^k(FM/G) \) for some integer \(k\), i.e., the value of the section \(\Gamma (s)\) of \(C(U)\rightarrow U\) at a point \(x\in U\) depends only on \(j_x^ks\), and the induced mapping \(\Gamma ^k:J^k(FM/G)\rightarrow CM\), \(\Gamma ^k(j_x^ks)=\Gamma (s)(x)\) is smooth.

We recall that the first prolongation \(\mathfrak g ^{(1)}\) of a Lie subalgebra \(\mathfrak g \subseteq \mathfrak gl (m,\mathbb R )\) is the kernel of the antisymmetrization (or Spencer) operator \(\mathcal A \); namely

$$\begin{aligned}&\mathcal A (L)(u,v)=L(u)v-L(v)u, \\&\forall u,v\in \mathbb R ^m,\;\forall L\in (\mathbb R ^m)^*\otimes \mathfrak g \cong \mathrm{* }{Hom}(\mathbb R ^m,\mathfrak g ). \\ \end{aligned}$$

Hence, the following exact sequence holds:

$$\begin{aligned} 0\rightarrow \mathfrak g ^{(1)}\rightarrow \left( \mathbb R ^m \right) ^*\otimes \mathfrak g \overset{\mathcal{A }}{\longrightarrow }\wedge ^2 \left( \mathbb R ^m \right) ^*\otimes \mathbb R ^m. \end{aligned}$$

Let \(G\subseteq GL(m,\mathbb R )\) be a Lie subgroup with Lie algebra \(\mathfrak g \subseteq \mathfrak gl (m,\mathbb R )\). In [22, Theorems 1.1 and 1.2], the following criterion is proved: If the first prolongation of \(\mathfrak g \) vanishes, i.e., \(\mathfrak g ^{(1)}=0\), and \(\mathfrak g \) is further invariant under transposition, then any \(G\)-structure admits a functorial connection.

As is well known, the adjoint bundle of an arbitrary principal \(G\)-bundle \(\pi :P\rightarrow M\) is the bundle associated with \(P\) under the adjoint representation of \(G\) on its Lie algebra \(\mathfrak g \), i.e., \(\mathrm{ad }P=(P\times \mathfrak g )/G\), where the action of \(G\) on \(P\times \mathfrak g \) is given by \((u,v)\cdot g =(u\cdot g,\mathrm{Ad }_{g^{-1}}(v))\), for all \(g\in G\), \(u\in P\), and \(v\in \mathfrak g \). In particular, if \(G\) is a closed subgroup on \(GL(m,\mathbb R )\), then \(\mathfrak g \) is a subalgebra of \(\mathfrak gl (m,\mathbb R ) =(\mathbb R ^m)^*\otimes \mathbb R ^m\), and for every \(G\)-structure \(P\subset FM\), the adjoint bundle is a subbundle of the bundle of endomorphisms of the tangent bundle, \(T^*M\otimes TM=\mathrm{End }(TM)\).

Bearing this in mind, the connection \(\gamma \) functorially associated with \(P\) is the only principal connection on \(P\) such that

$$\begin{aligned} \mathrm{trace } \left( \sigma \circ i_X\mathrm{Tor }_{\nabla ^\gamma } \right) =0, \quad \forall \sigma \in \Gamma \left( M,\mathrm{ad }P \right) , \; \forall X\in \mathfrak X (M), \end{aligned}$$
(1)

where \(\nabla ^\gamma \) denotes the covariant derivative attached to \(\gamma \) (cf. [22, Theorem 1.1]).

Theorem 3.1

In the classical settings, the connection defined by (1) can be identified as follows:

  1. (i)

    Let \(\pi :F_gM\rightarrow M\) be the bundle of orthonormal frames over an \(m\)-dimensional Riemannian manifold \((M,g)\). The Levi–Civita connection of \(g\) is the only connection on \(F_gM\) satisfying the condition (1).

  2. (ii)

    Let \(\pi :F_{g,J}M\rightarrow M\) be the bundle of unitary frames over a \(2m\)-dimensional almost Hermitian manifold \((M,g,J)\). The connection \(\gamma \) on \(F_{g,J}M\) satisfying the condition (1) is completely determined by the following properties:

    $$\begin{aligned} \left. \begin{array}{l} g\left( \mathrm{Tor }_{\nabla ^\gamma } (Z,V),\bar{U} \right) =g\left( \mathrm{Tor }_{\nabla ^\gamma } (Z,\bar{U}),V \right) \\ g\left( \mathrm{Tor }_{\nabla ^\gamma } (\bar{Z},V),\bar{U} \right) =g\left( \mathrm{Tor }_{\nabla ^\gamma } (\bar{Z},\bar{U}),V \right) \end{array} \right\} \quad \forall Z,U,V\in T_x^{1,0}M. \end{aligned}$$
    (2)

    This connection coincides with the canonical connection corresponding to \(t=\frac{1}{3}\) in the sense of [8] (also see [5, 18]).

  3. (iii)

    Let \(\pi :F_{I,J}M\rightarrow M\) be the bundle of linear frames adapted to an almost hypercomplex structure \((I,J,K=I\circ J)\) on a \(4m\)-dimensional manifold \(M\). The connection \(\gamma \) on \(F_{I,J}M\) satisfying the condition (1) coincides with the Obata connection (cf. [2, 4]).

proof

(i) As the Levi–Civita connection is the unique symmetric connection on \(F_gM\) (e.g., see [15], IV, Theorem 2.2]), it will suffice to prove that, in the present case, the condition (1) implies \(\mathrm{Tor }_{\nabla ^\gamma }=0\). An endomorphism \(\sigma :TM\rightarrow TM\) defines a section of the adjoint bundle if and only if, \(g(\sigma (x)X,Y)+g(X,\sigma (x)Y)=0\) for all \(x\in M\) and all \(X,Y\in T_xM\). Let \((U;x^1,\ldots ,x^m)\) be a coordinate system centered at \(x\) such that \((X_1,\ldots ,X_m)\), \(X_i=(\partial /\partial x^i)_x\), \(1\le i\le m\), is an orthonormal basis for \(T_xM\), i.e., \(g(X_i,X_j)=\delta _{ij}\), with dual coframe \((w^1,\ldots ,w^m)\), \(w^i=(X_i)^\flat \). Evaluating (1) at \(x\) and letting \(\sigma (x)=w^b\otimes X_a-w^a\otimes X_b\), \(X=X_c\), one obtains,

$$\begin{aligned} \Gamma ^b_{ca}(x)-\Gamma ^b_{ac}(x) =\Gamma ^a_{cb}(x)-\Gamma ^a_{bc}(x), \quad \forall a,b,c=1,\ldots ,m, \end{aligned}$$
(3)

where \(\Gamma ^a_{bc}\) are the components of \(\gamma \) with respect to \((x^1,\ldots ,x^m)\). If \(a=c\), then from (3), one deduces \(\Gamma ^a_{ab}(x)=\Gamma ^a_{ba}(x)\), if \(a\ne c\) but \(b\in \{ a,c\} \), then from (3) and the previous formulas one also deduce \(\Gamma ^b_{ac}(x)=\Gamma ^b_{ca}(x)\) for \(a=b\) or \(a=c\). Finally, if the indices \(a,b,c\) are pairwise distinct, then letting \(b\leftrightarrow c\), \(a\mapsto b\mapsto c\mapsto a\) in (3) one obtains \(\Gamma ^c_{ba}(x)-\Gamma ^c_{ab}(x) =\Gamma ^a_{bc}(x)-\Gamma ^a_{cb}(x)\) and \(\Gamma ^c_{ab}(x)-\Gamma ^c_{ba}(x) =\Gamma ^b_{ac}(x)-\Gamma ^b_{ca}(x)\), respectively, thus proving that \(\Gamma ^a_{bc}(x) =\Gamma ^a_{cb}(x)\).

(ii) Let \((U;x^1,\ldots ,x^m,y^1,\ldots ,y^m)\) be a coordinate system centered at \(x\in M\) such that \((X_1,\ldots ,X_m,Y_1,\ldots ,Y_m)\), \(X_j=(\partial /\partial x^j)_x\), \(Y_j=(\partial /\partial y^j)_x\), \(1\le j\le m\), is a unitary basis for \(T_xM\); i.e., \(JX_j=Y_j\), \(JY_j=-X_j\), \(g(X_j,X_k)=\delta _{jk}\), \(g(X_j,Y_k)=0\), \(g(Y_j,Y_k)=\delta _{jk}\).

We also write \(Z_j=(\partial /\partial z^j)_x =\frac{1}{2}(X_j-\mathbf i Y_j)\), \(\bar{Z}_j=(\partial /\partial \bar{z}^j)_x =\frac{1}{2}(X_j+\mathbf i Y_j)\).

An endomorphism \(\sigma :T^cM\rightarrow T^cM\) defines a section of the adjoint bundle if and only if \(\sigma \) is anti-Hermitian, i.e.,

$$\begin{aligned} \sigma (x)&= c_j^h \left( \mathrm{d}z^j\right) _x \otimes Z_h+\bar{c}_j^h \left( \mathrm{d}\bar{z}^j \right) _x \otimes \bar{Z}_h \\&= a_j^h \left\{ \left( \mathrm{d}z^j \right) _x \otimes Z_h +\left( \mathrm{d}\bar{z}^j \right) _x \otimes \bar{Z}_h \right\} \\&+\mathbf i b_j^h \left\{ \left( \mathrm{d}z^j \right) _x \otimes Z_h -\left( \mathrm{d}\bar{z}^j \right) _x \otimes \bar{Z}_h \right\} , \end{aligned}$$

where \(c_j^h=a_j^h+\mathbf i b_j^h\in \mathbb C \), and \((a_j^h)_{h,j=1}^m\) (resp. \((b_j^h)_{h,j=1}^m\)) is a skew–symetric (resp. symmetric) matrix. Hence, we can confine ourselves to impose the condition (1) at the point \(x\) for the standard basis

$$\begin{aligned}&\sigma _j^{\prime h}(x) =\left( \mathrm{d}z^j\right) _x \otimes Z_h +\left( \mathrm{d}\bar{z}^j \right) _x \otimes \bar{Z}_h -\left( \mathrm{d}z^h \right) _x \otimes Z_j -\left( \mathrm{d}\bar{z}^h \right) _x \otimes \bar{Z}_j,\quad h<j, \\&\sigma _j^{\prime \prime h}(x) =\left( \mathrm{d}z^j \right) _x \otimes Z_h -\left( \mathrm{d}\bar{z}^j \right) _x \otimes \bar{Z}_h +\left( \mathrm{d}z^h \right) _x \otimes Z_j -\left( \mathrm{d}\bar{z}^h \right) _x \otimes \bar{Z}_j, \quad h\le j, \end{aligned}$$

and for \(X=Z_a\) or \(X=\bar{Z}_a\). By using the standard notations (e.g., see [15], IX, §5, p. 155]), in imposing such conditions, the following equations are obtained:

$$\begin{aligned} \Gamma _{ah}^j(x)-\Gamma _{ha}^j(x)&= \Gamma _{a\bar{\jmath }}^{\bar{h}}(x) -\Gamma _{\bar{\jmath }a}^{\bar{h}}(x),\end{aligned}$$
(4)
$$\begin{aligned} \Gamma _{\bar{a}\bar{\jmath }}^{\bar{h}}(x) -\Gamma _{\bar{\jmath }\bar{a}}^{\bar{h}}(x)&= \Gamma _{\bar{a}h}^j(x) -\Gamma _{h\bar{a}}^j(x), \end{aligned}$$
(5)

for \(a,h,j=1,\ldots ,m\), which are equivalent to the Eq. (2).

Furthermore, by imposing \(\nabla J(x)=0\) (e.g., see [15], IX, §5, (9)]), one has \(\Gamma _{b\bar{c}}^a(x)=0\), \(\Gamma _{\bar{b}\bar{c}}^a(x)=0\), \(\Gamma _{bc}^{\bar{a}}(x)=0\), and \(\Gamma _{\bar{b}c}^{\bar{a}}(x)=0\). Therefore, the Eqs. (4) and (5) transform respectively into the following:

$$\begin{aligned} \Gamma _{a\bar{c}}^{\bar{b}}(x)&= \Gamma _{ab}^c(x)-\Gamma _{ba}^c(x),\end{aligned}$$
(6)
$$\begin{aligned} \Gamma _{\bar{a}b}^c(x)&= \Gamma _{\bar{a}\bar{c}}^{\bar{b}}(x) -\Gamma _{\bar{c}\bar{a}}^{\bar{b}}(x). \end{aligned}$$
(7)

Hence, \(\Gamma ^{\bar{b}}_{a\bar{c}}(x) =-\Gamma ^{\bar{a}}_{b\bar{c}}(x)\). Moreover, from (6) and (7) with the same notations as in [8], the following formulas hold:

$$\begin{aligned} \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \left( Z_i,Z_j,Z_k \right)&= 0,\\ \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \left( Z_i,Z_j,\bar{Z}_k \right)&= \tfrac{1}{3} \Gamma _{j\bar{k}}^{\bar{\imath }}(x),\\ \mathfrak b \left( \left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^{1,1} \right) \left( Z_i,Z_j,Z_k \right)&= 0,\\ \mathfrak b \left( \left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^{1,1} \right) \left( Z_i,Z_j,\bar{Z}_k \right)&= \tfrac{2}{3} \Gamma _{j\bar{k}}^{\bar{\imath }}(x), \end{aligned}$$

thus proving that \(\mathfrak b ((\mathrm{Tor }_{\nabla ^\gamma })^{1,1}) =2\mathfrak b (\mathrm{Tor }_{\nabla ^\gamma })\).

In addition, denoting as in [8], Lemme 3] by \((\mathrm{Tor }_{\nabla ^\gamma })_s^{1,1}\) the projection of \((\mathrm{Tor }_{\nabla ^\gamma })^{1,1}\) onto the subspace of elements satisfying the zero cyclic sum property and by \((\mathrm{Tor }_{\nabla ^\gamma })_a^{1,1}\) the component in its orthogonal subspace, the formula [8], (1.4.3)] leads one to the following equation:

$$\begin{aligned} (\mathrm{Tor }_{\nabla ^\gamma })_a^{1,1} =\tfrac{3}{2} \left( \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) +\mathfrak M \left( \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \right) \right) , \end{aligned}$$

and then,

$$\begin{aligned} \left( \mathrm{Tor }_{\nabla ^{\gamma }} \right) _{a}^{1,1} \left( Z_{i},Z_{j},Z_{k} \right) =&\!\! 0&\!\! =\left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^{1,1} \left( Z_i,Z_j,Z_k\right) , \\ \left( \mathrm{Tor }_{\nabla ^\gamma } \right) _a^{1,1} \left( Z_i,Z_j,\bar{Z}_k \right) =&\!\! \Gamma _{j\bar{k}}^{\bar{\imath }}(x)&\!\! =\left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^{1,1} \left( Z_{i},Z_{j},\bar{Z}_{k}\right) , \\ \left( \mathrm{Tor }_{\nabla ^\gamma } \right) _a^{1,1} \left( Z_i,\bar{Z}_j,Z_k \right) =&\!\! -\Gamma _{k\bar{\jmath }}^{\bar{\imath }}(x)&\!\! =\left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^{1,1} \left( Z_i,\bar{Z}_j,Z_k\right) , \\ \left( \mathrm{Tor }_{\nabla ^\gamma } \right) _a^{1,1} \left( \bar{Z}_i,Z_j,Z_k \right) =&\!\! 0&\!\! =\left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^{1,1} \left( \bar{Z}_i,Z_j,Z_k \right) . \end{aligned}$$

Hence, \((\mathrm{Tor }_{\nabla ^\gamma })_s^{1,1} =0\).

Finally, in order to obtain the value of the parameter \(t\), we use the second formula in [8], (2.3.5)], from which it follows:

$$\begin{aligned} (d^cF)^+ =3\left( \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^+ -2\mathfrak b \left( (\mathrm{Tor }_{\nabla ^\gamma })^{1,1}_a \right) \right) . \end{aligned}$$

As the components \((3,0)\) and \((0,3)\) of \(\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \) both vanish, one obtains

$$\begin{aligned} \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) ^+ =\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) . \end{aligned}$$

Furthermore, from the following equalities:

$$\begin{aligned} \mathfrak b \left( (\mathrm{Tor }_{\nabla ^\gamma })^{1,1}_a \right) =\mathfrak b \left( (\mathrm{Tor }_{\nabla ^\gamma })^{1,1} \right) =2\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) , \end{aligned}$$

we obtain \((d^cF)^+ =3\left( \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) -4\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \right) =-9\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \). Hence, the second formula in [8], (2.5.1)] yields

$$\begin{aligned} \mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) =\frac{2t-1}{3} \left( -9\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) \right) =3(1-2t)\mathfrak b \left( \mathrm{Tor }_{\nabla ^\gamma } \right) . \end{aligned}$$

(iii) Here, we follow [1–4]. Let \(\le \) be the ordering in the set of indices \(\tilde{\imath }\in \{ i,i^\prime ,i^{\prime \prime }, i^{\prime \prime \prime }:1\le i\le m\} \) given by the following rules: \(i<j^\prime <k^{\prime \prime }<l^{\prime \prime \prime }\), \(a^\prime <b^\prime \Longleftrightarrow a^{\prime \prime }<b^{\prime \prime } \Longleftrightarrow a^{\prime \prime \prime } <b^{\prime \prime \prime } \Longleftrightarrow a<b\). Let \((x^{\tilde{a}})\) be a coordinate system centered at \(x\in M\) such that the basis \(v_{\tilde{a}}=(\partial /\partial x^{\tilde{a}})_x\) is adapted to \(I\), \(J\), \(K\), i.e., \(v_{a^\prime }=I(v_a)\), \(v_{a^{\prime \prime }}=J(v_a)\), \(v_{a^{\prime \prime \prime }}=K(v_a)\). Hence, the matricial expression of the operators \(I\),\(J\) and \(K\) at \(x\) (writing \(I(v_{\tilde{a}})=I^{\tilde{b}}_{\tilde{a}}v_{\tilde{b}}\), and similarly for \(J\), \(K\)) is

$$\begin{aligned} \text{ Matrix } \text{ of } I_x =\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} O &{} -\mathrm{id}_m &{} O &{} O \\ \mathrm{id}_m &{} O &{} O &{} O \\ O &{} O &{} O &{} -\mathrm{id}_m \\ O &{} O &{} \mathrm{id}_m &{} O \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \text{ Matrix } \text{ of } J_x =\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} O &{} O &{} -\mathrm{id}_m &{} O \\ O &{} O &{} O &{} \mathrm{id}_m \\ \mathrm{id}_m &{} O &{} O &{} O \\ O &{} -\mathrm{id}_m &{} O &{} O \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \text{ Matrix } \text{ of } K_x =\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} O &{} O &{} O &{} -\mathrm{id}_m \\ O &{} O &{} -\mathrm{id}_m &{} O \\ O &{} \mathrm{id}_m &{} O &{} O \\ \mathrm{id}_m &{} O &{} O &{} O \end{array} \right) . \end{aligned}$$

Let

$$\begin{aligned} \left( \mathrm{Tor }_{\nabla ^{\gamma _{\mathrm{Ob}}}} \right) _x&= \sum \limits _{\tilde{a}<\tilde{b}} \tau _{\tilde{a}\tilde{b}}^{\tilde{c}} v_{\tilde{c}}\otimes v^{\tilde{a}}\wedge v^{\tilde{b}}\\&= \tfrac{1}{12} \left\{ N(I)_x+N(J)_x+N(K)_x \right\} , \end{aligned}$$

be the torsion of the Obata connection with \(\tau _{\tilde{a}\tilde{b}}^{\tilde{c}} =\left( \Gamma ^{\gamma _{\mathrm{Ob}}} \right) _{\tilde{a}\tilde{b}}^{\tilde{c}}(x) -\left( \Gamma ^{\gamma _{\mathrm{Ob}}} \right) _{\tilde{b}\tilde{a}}^{\tilde{c}}(x)\), \(N(I),N(J),N(K)\) being the Nijenhuis tensor field of \(I,J,K\), respectively. The local expression of \(N(I)\) is (cf. [15], IX, p. 124]),

$$\begin{aligned} N_{\tilde{a}\tilde{b}}^{\tilde{c}}(I)(x) =2\left\{ I_{\tilde{a}}^{\tilde{h}}(x) \tfrac{\partial I_{\tilde{b}}^{\tilde{c}}}{\partial x^{\tilde{h}}}(x) -I_{\tilde{b}}^{\tilde{h}}(x) \tfrac{\partial I_{\tilde{a}}^{\tilde{c}}}{\partial x^{\tilde{h}}}(x) -I_{\tilde{h}}^{\tilde{c}}(x) \tfrac{\partial I_{\tilde{b}}^{\tilde{h}}}{\partial x^{\tilde{a}}}(x) +I_{\tilde{h}}^{\tilde{c}}(x) \tfrac{\partial I_{\tilde{a}}^{\tilde{h}} }{\partial x^{\tilde{b}}}(x) \right\} , \end{aligned}$$

and similarly for \(J\) and \(K\). Differentiating the equations \(I^2=-\mathrm{id}\), \(J^2=-\mathrm{id}\), \(IJ=K\), and \(IJ=-JI\) at \(x\in M\) yields the following set of constraints on the derivatives of \(I\), \(J\), and \(K\):

$$\begin{aligned} 0&= \frac{\partial I^{\tilde{b}}_{\tilde{h}}}{\partial x^{\tilde{c}}}(x) I^{\tilde{h}}_{\tilde{a}}(x) +\frac{\partial I^{\tilde{h}}_{\tilde{a}}}{\partial x^{\tilde{c}}}(x) I^{\tilde{b}}_{\tilde{h}}(x), \\ 0&= \frac{\partial J^{\tilde{b}}_{\tilde{h}}}{\partial x^{\tilde{c}}}(x) J^{\tilde{h}}_{\tilde{a}}(x) +\frac{\partial J^{\tilde{h}}_{\tilde{a}}}{\partial x^{\tilde{c}}}(x) J^{\tilde{b}}_{\tilde{h}}(x), \\ \frac{\partial K^{\tilde{b}}_{\tilde{a}}}{\partial x^{\tilde{c}}}(x)&= \frac{\partial J^{\tilde{h}}_{\tilde{a}}}{\partial x^{\tilde{c}}}(x) I^{\tilde{b}}_{\tilde{h}}(x) +\frac{\partial I^{\tilde{b}}_{\tilde{h}}}{\partial x^{\tilde{c}}}(x)J^{\tilde{h}}_{\tilde{a}}(x),\\ 0&= \frac{\partial I^{\tilde{b}}_{\tilde{h}}}{\partial x^{\tilde{c}}}(x) J^{\tilde{h}}_{\tilde{a}}(x) +\frac{\partial J^{\tilde{h}}_{\tilde{a}}}{\partial x^{\tilde{c}}}(x) I^{\tilde{b}}_{\tilde{h}}(x) +\frac{\partial J^{\tilde{b}}_{\tilde{h}}}{\partial x^{\tilde{c}}}(x) I^{\tilde{h}}_{\tilde{a}}(x) +\frac{\partial I^{\tilde{h}}_{\tilde{a}}}{\partial x^{\tilde{c}}}(x) J^{\tilde{b}}_{\tilde{h}}(x). \end{aligned}$$

Taking these constraints into account, it follows that the components \(\tau _{\tilde{a}\tilde{b}}^{\tilde{c}}\) of the torsion of the Obata connection fulfill the condition (1), which are written down as follows in the present case:

$$\begin{aligned}&0=\tau _{\tilde{l}q}^p +\tau _{\tilde{l}q^\prime }^{p^\prime } +\tau _{\tilde{l}q^{\prime \prime }}^{p^{\prime \prime }} +\tau _{\tilde{l}q^{\prime \prime \prime }}^{p^{\prime \prime \prime }}, \quad 0=\tau _{\tilde{l}q^\prime }^p -\tau _{\tilde{l}q}^{p^\prime } -\tau _{\tilde{l}q^{\prime \prime \prime }}^{p^{\prime \prime }} +\tau _{\tilde{l}q^{\prime \prime }}^{p^{\prime \prime \prime }}, \\&0=\tau _{\tilde{l}q^{\prime \prime }}^p +\tau _{\tilde{l}q^{\prime \prime \prime }}^{p^\prime } -\tau _{\tilde{l}q}^{p^{\prime \prime }} -\tau _{\tilde{l}q^\prime }^{p^{\prime \prime \prime }}, \quad 0=\tau _{\tilde{l}q^{\prime \prime \prime }}^p -\tau _{\tilde{l}q^{\prime \prime }}^{p^\prime } +\tau _{\tilde{l}q^\prime }^{p^{\prime \prime }} -\tau _{\tilde{l}q}^{p^{\prime \prime \prime }}. \end{aligned}$$

\(\square \)

Remark 3.2

The conclusion of item (i) in Theorem 3.1 also holds for a pseudo-Riemannian metric, and item (ii) holds for a para-Hermitian manifold with the same value of the parameter \(t=\frac{1}{3}\). Both proofs are similar; see [13] for the para-Hermitian case.

As far as we know, the canonical connections in the Hermitian as well as in the para-Hermitian case do not appear as distinguished canonical connections in the literature.

4 A criterion for the existence of a functorial connection

Below, we characterize the algebras \(\mathfrak gl (r,A)\) satisfying the criterion given in [22, Theorems 1.1 and 1.2].

Theorem 4.1

With the previous hypotheses and notations, we have

  1. (a)

    The first prolongation of the algebra \(\mathfrak gl (r,A)\) can be identified with the set \(\mathcal S ^r\) of systems \((v_{ij})\), \(1\le i\le j\le r\), \(v_{ij}\in A^r\), such that,

    $$\begin{aligned} v_{ij}\cdot [a,b]=0, \quad \forall a,b\in A,\; 1\le i\le j\le r. \end{aligned}$$
    (8)

    Accordingly, \(\mathfrak gl (r,A)^{(1)}\) vanishes if and only if the condition \(x\cdot [a,b]=0\), \(\forall a,b\in A\), implies \(x=0\).

  2. (b)

    The subalgebra \(\mathfrak gl (r,A) \subset \mathfrak gl (nr,\mathbb R )\) is invariant under transposition if and only if \(A\) is a \(C^\star \)-algebra.

Hence, the unique subalgebras \(\mathfrak gl (r,A)\) invariant under transposition and further satisfying \(\mathfrak gl (r,A)^{(1)}=0\), correspond to the algebras \(A\) that are isomorphic to a finite direct sum of matrix algebras of the form

$$\begin{aligned} \mathrm{Mat}_k(\mathbb R ) \, (k\ge 2), \qquad \mathrm{Mat}_k(\mathbb C ) \, (k\ge 2), \qquad \mathrm{Mat}_k(\mathbb H ) \, (k\ge 1). \end{aligned}$$

Proof

The elements of \(\mathfrak gl (r,A)^{(1)}\) can be viewed as the symmetric \(\mathbb R \)-bilinear mappings \(B:A^r\times A^r\rightarrow A^r\) such that for every \(v\in A^r\), \(B(v,-) \in \mathfrak gl (r,A)\), or equivalently \(B(v,-):A^r\rightarrow A^r\) is a right \(A\)-linear mapping. Let \((v_1,\ldots ,v_r)\) be the standard basis of \(A^r\). For every \(1\le i\le j\le r\), we set \(B(v_i,v_j)=v_{ij}\). Hence, for all \(a,b\in A\), we have

$$\begin{aligned} B\left( v_{i}\cdot a,v_{j}\cdot b\right)&= B\left( v_{i}\cdot a,v_{j}\right) \cdot b=B\left( v_{j},v_{i}\cdot a\right) \cdot b=v_{ij}\cdot ab,\\ B\left( v_{j}\cdot b,v_{i}\cdot a\right)&= B\left( v_{j}\cdot b,v_{i}\right) \cdot a=B\left( v_{i},v_{j}\cdot b\right) \cdot a=v_{ij}\cdot ba. \end{aligned}$$

Taking the symmetry of \(B\) into account, we conclude that the Eq. (8) holds. Accordingly, we can define an \(\mathbb R \)-linear mapping \(\varphi :\mathfrak gl (r,A)^{(1)} \rightarrow \mathcal S ^r\) by setting \(\varphi (B)=(v_{ij})\), \(1\le i\le j\le r\), which is injective since if \((e_1,\ldots ,e_n)\) is a basis of \(A\) as an \(\mathbb R \)-vector space, then \((v_i\cdot e_h)\), \(1\le i\le r\), \(1\le j\le n\), is a basis of \(A^r\) over \(\mathbb R \), and from the equation above, we obtain \(B(v_i\cdot e_h,v_k\cdot e_j) =v_{ik}\cdot e_he_j\), for \(i,k=1,\ldots ,r\), \(1\le h,j\le n\), with \(v_{ik}=v_{ki}\) whenever \(i>k\).

Conversely, given a system \((v_{ij})\in \mathcal S ^r\), we can define a symmetric \(\mathbb R \)-bilinear mapping \(B:A^r\times A^r\rightarrow A^r\) by the previous formula and \(B(v_i\cdot e_h,-)\) is right \(A\)-linear as for every \(1\le l\le n\), we have

$$\begin{aligned} B\left( v_i\cdot e_h, \left( v_k\cdot e_j \right) \cdot e_l \right)&= \sum \limits _{p=1}^n\lambda _{jl}^pB \left( v_i\cdot e_h,v_k\cdot e_p \right) \\&= \sum \limits _{p=1}^n \lambda _{jl}^pv_{ik}\cdot e_he_p\\&= B\left( v_i\cdot e_h,v_k\cdot e_j \right) \cdot e_l, \end{aligned}$$

where the scalars \(\lambda _{jl}^p\in \mathbb R \) are defined by \(e_j\cdot e_l=\sum _{p=1}^n\lambda _{jl}^pe_p\), thus proving that \(\varphi \) is surjective.

Let \(R^r_a:A^r\rightarrow A^r\) be the right translation by \(a\in A\), i.e.,

$$\begin{aligned} R^r_a(x_1,\ldots ,x_r)=(x_1a,\ldots ,x_ra). \end{aligned}$$

Hence, \(R^r_a=\oplus ^rR^1_a\) and \((R^r_a)^\mathrm{T}=\oplus ^r(R^1_a)^\mathrm{T}\), where \(\Lambda ^\mathrm{T}\) denotes the transpose of the matrix \(\Lambda \in \mathfrak gl (nr,\mathbb R )\). Therefore, \((R^r_a)^\mathrm{T}\) is \(A\)-linear if and only if \((R^1_a)^\mathrm{T}\) is. Accordingly, if \(\mathfrak gl (r,A)\) is invariant by transposition, then for every \(a\in A\), there exists a unique element \(a^\star \in A\) such that \((R_a)^\mathrm{T}=R_{a^\star }\). The algebra \(A\) endowed with \(a\mapsto a^\star \) and the norm \(a\mapsto \Vert a\Vert =\Vert R_a\Vert \) inherited from the standard norm in \(\mathfrak gl (nr,\mathbb R )\) determines a \(C^\star \)-algebra structure on \(A\); e.g., see [10, Chapter 8].

We can thus deduce the final conclusion of the statement by simply applying the Wedderburn theorem for real \(C^\star \)-algebras, e.g., see [10, Theorem 8.4].

In order to prove the converse of (b), we can confine ourselves to consider only the cases in which \(A\) is equal to \(\mathrm{Mat}_k(\mathbb R )_{k\ge 2}\), \(\mathrm{Mat}_k(\mathbb C )_{k\ge 2}\), or \(\mathrm{Mat}_k(\mathbb H )_{k\ge 1}\), as transposition commutes with direct sums.

If \(A=\mathrm{Mat}_k(\mathbb F )\), where \(\mathbb F \) denotes either \(\mathbb R \), or \(\mathbb C \), or \(\mathbb H \) and \(\Lambda \in \mathfrak gl (r,A)\), then we set \(\Lambda (v_i)=\sum _{h=1}^rv_h\cdot \Lambda ^h_i\), where \(\Lambda ^h_i\in A\), \(h,i=1,\ldots ,r\).

Assume \(\mathbb F =\mathbb R \) (hence \(n=k^2\)) and let \((E^a_b)_{a,b=1}^k\) be the standard basis for \(A=\mathrm{Mat}_k(\mathbb R )\) as a vector \(\mathbb R \)-space, i.e., \(E^a_b\) is the square matrix such that the entry at the \(a\)th row and \(b\)-th column is 1 and all the other entries are zero.

The lexicographic order is considered on the basis \((v_h\cdot E^a_b)\), \(1\le h\le r\), \(a,b=1,\ldots ,k\), for \(\mathfrak gl (nr,\mathbb R )\) as a vector \(\mathbb R \)-space; namely

$$\begin{aligned}&v_1\cdot E_1^1<v_1\cdot E_2^1<\cdots <v_1\cdot E_k^1<\cdots \\&<v_1\cdot E_1^k<v_1\cdot E_2^k<\cdots <v_1\cdot E_k^k<\cdots \\&\qquad \qquad \qquad \quad \vdots \\&<v_r\cdot E_1^1<v_r\cdot E_2^1<\cdots <v_r\cdot E_k^1<\cdots \\&<v_r\cdot E_1^k<v_r\cdot E_2^k<\cdots <v_r\cdot E_k^k. \end{aligned}$$

Let \(\tilde{\Lambda }\) be the matrix representing \(\Lambda \) in the embedding \(\mathfrak gl (r,A) \subset \mathfrak gl (nr,\mathbb R )\) with respect to the basis above. As the straightforward—but rather long—computation of the products \(\Lambda ^h_iE^a_b\) shows, the matrix \(\tilde{\Lambda }\) is the \(r\times r\) block matrix \(\tilde{\Lambda }=(\lambda ^{h,a}_{i,b}I_r)\), \(h,i=1,\ldots ,r\), \(a,b=1,\ldots ,k\), where \(I_r\) denotes the identity matrix of order \(r\) and \(\lambda ^{h,a}_{i,b}\) are the entries of \(\Lambda ^h_i\). Hence, \(\tilde{\Lambda }^\mathrm{T} =\widetilde{\Lambda ^\mathrm{T}}\), the \(A\)-linear mapping \(\Lambda ^\mathrm{T}\) in \(\mathfrak gl (r,A)\) being defined by \(\Lambda ^\mathrm{T}(v_i) =\sum _{h=1}^rv_h\cdot (\Lambda ^h_i)^\mathrm{T}\), \(1\le i\le r\).

Next, assume \(\mathbb F =\mathbb C \) (hence \(n=2k^2\)) and let us put the following order on the basis \((v_h\cdot E^a_b, v_h\cdot E^a_b\mathbf i )\), \(1\le h\le r\), \(a,b=1,\ldots ,k\), for \(\mathfrak gl (nr,\mathbb R )\):

$$\begin{aligned}&v_1\cdot E_1^1<v_1\cdot E_1^1 \mathbf i <v_1\cdot E_2^1<v_1\cdot E_2^1\mathbf i <\cdots <v_1\cdot E_k^1<v_1\cdot E_k^1\mathbf i <\cdots \\&<v_1\cdot E_1^k<v_1\cdot E_1^k\mathbf i <v_1\cdot E_2^k<v_1\cdot E_2^k\mathbf i <\cdots <v_1\cdot E_k^k<v_1\cdot E_k^k\mathbf i <\cdots \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \vdots \\&<v_r\cdot E_1^1<v_r\cdot E_1^1\mathbf i <v_r\cdot E_2^1<v_r\cdot E_2^1\mathbf i <\cdots <v_r\cdot E_k^1<v_r\cdot E_k^1\mathbf i <\cdots \\&<v_r\cdot E_1^k<v_r\cdot E_1^k\mathbf i <v_r\cdot E_2^k<v_r\cdot E_2^k\mathbf i <\cdots <v_r\cdot E_k^k<v_r\cdot E_k^k\mathbf i . \end{aligned}$$

If \(\lambda ^{h,a}_{i,b}=\lambda ^{h,a}_{0,i,b} +\lambda ^{h,a}_{1,i,b}\mathbf i \), \(\lambda ^{h,a}_{0,i,b},\lambda ^{h,a}_{1,i,b} \in \mathbb R \), are the entries of \(\Lambda ^h_i\), and we set

$$\begin{aligned} \tilde{\lambda }_{i,b}^{h,a} =\left( \begin{array}{c@{\quad }c} \lambda _{0,i,b}^{h,a} &{} -\lambda _{1,i,b}^{h,a}\\ \lambda _{1,i,b}^{h,a} &{} \lambda _{0,i,b}^{h,a} \end{array} \right) , \qquad \tilde{\Lambda }_{i,b}^{h,a} =\left( \begin{array}{c@{\quad }c} \tilde{\lambda }_{i,b}^{h,a} &{} O_2\\ O_2 &{} \tilde{\lambda }_{i,b}^{h,a} \end{array} \right) , \end{aligned}$$

where \(O_2\) denotes the zero matrix of size \(2\times 2\), then \(\tilde{\Lambda }\) is the \(r\times r\) block matrix \(\tilde{\Lambda } =(\tilde{\Lambda }^{h,a}_{i,b})\). Hence, \(\tilde{\Lambda }^\mathrm{T} =\widetilde{\Lambda ^\star }\), where the \(A\)-linear mapping \(\Lambda ^\star \) in \(\mathfrak gl (r,A)\) is given by,

$$\begin{aligned} \Lambda ^\star (v_i) =\sum \limits _{h=1}^rv_h \cdot (\overline{\Lambda ^h_i})^\mathrm{T}, \quad 1\le i\le r, \end{aligned}$$
(9)

the bar denoting complex conjugation.

Finally, if \(\mathbb F =\mathbb H \) (hence \(n=4k^2\)), then the following order is put on the basis \((v_h\cdot E^a_b, v_h\cdot E^a_b\mathbf i , v_h\cdot E^a_b\mathbf j ,v_h\cdot E^a_b\mathbf k )\), \(1\le h\le r\), \(a,b=1,\ldots ,k\), for \(\mathfrak gl (nr,\mathbb R )\):

$$\begin{aligned} \cdots <v_h\cdot E^a_b<v_h\cdot E^a_b \mathbf i <v_h\cdot E^a_b \mathbf j <v_h\cdot E^a_b \mathbf k <\cdots \end{aligned}$$

If \(\lambda ^{h,a}_{i,b} =\lambda ^{h,a}_{0,i,b} +\lambda ^{h,a}_{1,i,b}\mathbf i +\lambda ^{h,a}_{2,i,b}\, \mathbf j +\lambda ^{h,a}_{3,i,b}\mathbf k \), \(\lambda ^{h,a}_{c,i,b}\in \mathbb R \), \(0\le c\le 3\), are the entries of \(\Lambda ^h_i\), and we set

$$\begin{aligned} \tilde{\lambda }_{i,b}^{h,a} = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \lambda _{0,i,b}^{h,a}&{} -\lambda _{1,i,b}^{h,a} &{} -\lambda _{2,i,b}^{h,a} &{} -\lambda _{3,i,b}^{h,a}\\ \lambda _{1,i,b}^{h,a} &{} \lambda _{0,i,b}^{h,a} &{} -\lambda _{3,i,b}^{h,a} &{} \lambda _{2,i,b}^{h,a}\\ \lambda _{2,i,b}^{h,a} &{} \lambda _{3,i,b}^{h,a} &{} \lambda _{0,i,b}^{h,a} &{}-\lambda _{1,i,b}^{h,a}\\ \lambda _{3,i,b}^{h,a}&{} -\lambda _{2,i,b}^{h,a} &{} \lambda _{1,i,b}^{h,a} &{}\lambda _{0,i,b}^{h,a} \end{array} \right) , \qquad \tilde{\Lambda }_{i,b}^{h,a} =\left( \begin{array}{c@{\quad }c} \tilde{\lambda }_{i,b}^{h,a} &{} O_{4}\\ O_{4} &{} \tilde{\lambda }_{i,b}^{h,a} \end{array} \right) , \end{aligned}$$

where \(O_4\) denotes the zero matrix of size \(4\times 4\), then \(\tilde{\Lambda }\) is the \(r\times r\) block matrix \(\tilde{\Lambda } =(\tilde{\Lambda }^{h,a}_{i,b})\), and as in the previous case, we obtain, \(\tilde{\Lambda }^\mathrm{T} =\widetilde{\Lambda ^\star }\), where the \(A\)-linear mapping \(\Lambda ^\star \in \mathfrak gl (r,A)\) is again defined by the formula (9) and the bar now denotes quaternionic conjugation.\(\square \)

Corollary 4.2

Although every almost complex manifold admits almost complex linear connections with special properties (cf. [15], IX, Theorem 3.4]), the category of such manifolds does not admit any functorial connection as its associated algebra, i.e., \(A=\mathbb C \), is not included in the list of Theorem 4.1.

Remark 4.3

As a consequence of the property (a) in the previous theorem, it follows that the property (b) implies \(\mathfrak gl (r,A)^{(1)}=0\).

Remark 4.4

We should finally like to mention the weak connection of the results above with nonassociative algebras. If \(A\) is a nonassociative \(F\)-algebra, then there exists a natural one-to-one correspondence between left \(A\)-modules and left \(\bar{A}\)-modules, \(\bar{A}\) being the associative algebra attached to \(A\) (i.e., \(\bar{A}\) is the quotient algebra of \(A\) by the bilateral ideal generated by the associators \((a,b,c)=(ab)c-a(bc)\) of any three elements in \(A\), cf. [20, p. 13]). In fact, if \(N\) is a left \(A\)-module, then every associator \((a,b,c)\) belongs to the kernel of the homomorphism \(A\rightarrow \mathrm{* }{End}_FN\), \(a\mapsto L_a\). Hence, \(N\) inherits a structure of left \(\bar{A}\)-module. In practice, however, this result is not interesting, as the associative algebras attached to classical nonassociative algebras are the ground field or alike.