Study on the Concreate Model for the Development of Supergeometry as a New Category of Supermanifolds

Study on the Concreate Model for the Development of Supergeometry as a New Category of Supermanifolds

Khondokar M. Ahmed

Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

DOI: https://doi.org/10.51244/IJRSI.2024.1108031

Received: 06 July 2024; Revised: 24 July 2024; Accepted: 27 July 2024; Published: 02 September 2024

ABSTRACT

In this present paper sheaves notion of \(GH^\infty\) supermanifolds graded manifolds morphism \(Z\)-expansion functions of variables in \(BL^{m,n}\) sheaves of graded commutative \(BL\)-algebra \(G\)-supermanifolds topologies of rings of \(G\)-functions are studied. We establish a theorem demonstrating that the \(Z\)-expansion on an isometry onto its image and prove a related metric isomorphism.

Keywords: Supermanifolds, Sheaf, Morphism, Graded \(G\)-module, Seminorms \(p_{K,I,\mu}\)

INTRODUCTION

The category of \(G\)-supermanifolds [3] [4] provides a consistent and concrete model for the development of supergeometry. In order to supply exact motivation for the development these objects and also for historical argument we started with a brief representation graded manifolds these are basically introduced by Berzin and Lei∧tes [8] [14]; although the most widespread treatment can be found in Kostant [12] and Manin [13]. In this way \(G\)-supermanifolds could be expanded more consistent or concrete compared to traditional model. Graded manifolds [2] play a direct role in the theory developed in this paper in that some results holding in that category can be applied as they are in the context of \(G\)-supermanifolds. The discussion of the relationship between \(G\)-supermanifolds and the axiomatics for supermanifolds proposed by Rathstein [19]. The classes of \(G^\infty\), \(GH^\infty\), and \(H^\infty\) supersmooth functions are used which allow us to define supermanifolds in the sense of Rogers [15] [16] [17] [18]; the discussion of their shortcomings leads us to introduce the notion of \(G\)-supermanifolds and \(Z\)-expansion. In the present work a theorem on an isometry and its image of \(Z\)-expansion and on a metric isomorphism is established.

PRELIMINARIES

The original idea of geometric approach to supermanifolds [10] is to patch open sets in \(BL^{m,n}\) by means of transition functions which fulfill a suitable ‘smoothness’ condition. We call generically supersmooth functions such as \(G^\infty\), \(GH^\infty\), and \(H^\infty\) functions. These functions are introduced in a unified manner in terms of a morphism called \(Z\)-expansion which maps functions of real variables into functions of variables in \(BL^{m,n}\). Unless otherwise stated whenever referring explicitly or implicitly to a topology on \(BL^{m,n}\) we mean its \(\mathbb{N}\)-vector space topology. Throughout this paper we assume to choose integers \(L\), \(m\), and \(n\) with \(L > 0\) and \(m n \geq 0\) subject to the condition \(L \geq n\). For every integer \(L’\) such that \(0 \leq L’ \leq L\) the exterior algebra is regarded as a subalgebra of \(BL\) so that \(BL\) acquires a structure of a graded -module which is not free unless \(L’ = 0\) or \(L’ = L\). We recall that the graded vector space associated with \(BL^{m|n}\) according to the procedure is simply \(\mathbb{N}^m \oplus \mathbb{N}^n\). We denote by \(\sigma^{m,n}: BL^{m,n} \rightarrow \mathbb{N}^m\) the restriction of the augmentation map to \(BL^{m,n}\).

For any \(C^\infty\) differentiable manifold \(X\) let us denote by \(C^{L’^\infty}(W)\) the graded algebra of -valued \(C^\infty\) functions on the open set \(W \subset X\). For each integer \(L’ \leq L\) and any \(U \subset \mathbb{N}^m\) the \(Z\)-expansion is the morphism of graded algebras

\[
Z_{L’}: C^{L’^\infty}(U) \rightarrow C^\infty((\sigma^{m,0})^{-1}(U))
\]

defined by the formula (cf. [18])

\[
Z_{L’}(h)(x) = h(\sigma^{m,0}(x)) + \sum_{j=1}^{L’} \frac{1}{j!} D^{(j)}h\sigma^{m,0}(x)(s^{m,0}(x) \dots s^{m,0}(x)) \quad (2.1)
\]

for all \(h \in C^{L’^\infty}(U)\) and all \(x \in (\sigma^{m,0})^{-1}(U)\); here the \(j\)-th Fre’cht differential \(D^jh\sigma^{m,0}x\) at the point \(\sigma^{m,0}x\) acts on \(BL^{m,0} \times \cdots \times BL^{m,0}\) (\(j\)-times) simply by extending by \(BL^0\)-linearity its action on \(\mathbb{N}^m \times \cdots \times \mathbb{N}^m\). The mapping \(S^{m,0}: BL^{m,0} \rightarrow nL^{m,0}\) is the projection onto the second component of the direct sum \(BL^{m,0} = \mathbb{N}^m \oplus nL^{m,0}\).

For each open \(U \subset \mathbb{N}^m\) \(\sigma^{m,0}^{-1}(U) \subset BL^{m,0}\) is a subset of \(BL^{m,n}\) so that we can define on the open set \(\sigma^{m,n-1}(U) \subset BL^{m,n}\) the graded algebra \(S^{L’}\sigma^{m,n-1}(U)\) formed by the functions having the following expression:

\[
f(x_1 \dots x_m y_1 \dots y_n) = \sum_{\mu \in \Xi_n} f_\mu(x_1 \dots x_m) y_\mu \quad (2.2)
\]

where \(f_\mu \in Z_{L’}(C^{\sigma^{m,n-1}(U)})\) and \(y_\mu = y_{\mu_1} \dots y_{\mu_r}\) if \(f = \mu_1 \dots \mu_r\).

We can therefore introduce a sheaf [1] \(S^{L’}\) of graded-commutative \(BL’\)-algebras over \(BL^{m,n}\) by letting for each open \(V \subset BL^{m,n}\)

\[
S^{L’}(V) = S^{L’}(\sigma^{m,n-1}(\sigma^{m,n}(V))) \quad (2.3)
\]

The sections of the sheaf \(S^{L’}\) on an open set \(V\) are \(C^\infty\) functions which show a kind of holomorphic behavior in the nilpotent directions in that the coefficients of the various powers of the \(y\)’s in the equation (2.2) are determined at every point \(z\) of the fiber \(\sigma^{m,n-1}(x)\) of \(BL^{m,n}\) over \(x = \sigma^{m,n}(z) \in \mathbb{N}^m\) by their germs at \(x\).

We denote by \(S^{L’}\) the subsheaf of \(S^{L’}\) whose sections are functions not depending on the odd variables \(y_a\) namely they have only the first term in the sum (2.2). In other words, the sheaf \(S^{L’}\) on \(BL^{m,n}\) is the inverse image under the projection \(BL^{m,n} \rightarrow BL^{m,0}\) of the sheaf \(S^{L’}\) on \(BL^{m,0}\). Then equation (2.2) shows the existence for any open \(U \subset BL^{m,n}\) of a surjective morphism:

\[
\lambda: S^{L’}(U) \otimes_{\mathbb{N}} \wedge \mathbb{N}^n \rightarrow S^{L’}
\]

\[
\mu \in \Xi_n f_\mu \otimes y_\mu \mapsto \sum_{\mu \in \Xi_n} f_\mu y_\mu \quad (2.4)
\]

G-SUPERMANIFOLDS

We have seen that the classes of supersmooth functions which is free from inconsistencies and yields a theory applicable to supersymmetry [5] is not trivial. In particular, it seems rather difficult to combine the following requirements:

(a) the sheaf of derivations of the function sheaf under consideration should be locally free;

(b) the coefficients of the ‘superfield expansion’ (2.2) when restricted to real arguments should take values in a graded-commutative algebra \(B\);

(c) there should be a good theory of superbundles and in particular there is a sensible notion of graded tangent space.

These difficulties can be overcome by introducing a new category of supermanifolds [6] called \(G\)-supermanifolds characterized in terms of a sheaf \(G\) on \(BL^{m,n}\) which is in a sense a ‘completion’ of \(GH^{L’}\) (condition \(L – L’ \geq n\) is assumed to hold). More precisely, we define the sheaf of graded-commutative \(BL\)-algebras on \(BL^{m,n}\)

\[
G^{L’} \equiv GH^{L’} \otimes_{BL’} BL \quad (3.1)
\]

It is convenient to introduce an evaluation morphism \(\delta: G^{L’} \rightarrow C^L\) (we denote by \(C^L\) the sheaf of \(BL\)-valued continuous functions on \(BL^{m,n}\)) by extending by additivity the mapping

\[
\delta(f \otimes a) = f \cdot a \quad (3.2)
\]

Proposition 3.1: The image of \(\delta\) is isomorphic to the sheaf \(G^\infty\) of \(G^\infty\) functions on \(BL^{m,n}\). The morphism \(\delta\) is injective when restricted to the subsheaf \(G^{L’} = GH^{L’} \otimes_{BL’} BL\).

Proof: The first claim is evident in view of the definition of the sheaf of \(G^\infty\) functions. In order to prove that \(\delta: G^{L’} \rightarrow G^\infty\) is an isomorphism we exhibit the inverse morphism \(\lambda: G^\infty \rightarrow G^{L’}\). Given an open set \(U \subset BL^{m,n}\) every \(f \in G^\infty(U)\) can be written in accordance with equation (2.1) in the form

\[
f = \sum_{\mu \in \Xi_n} Z_0(f_\mu(U)) \cdot \beta_\mu \quad (3.3)
\]

where the \(f_\mu\)’s are suitable sections of \(C^\infty(\mathbb{N}^m, \sigma^{m,n}(U))\). After letting \(\lambda(f) = Z_0(f_\mu(U)) \otimes \beta_\mu\) we verify that \(\lambda \circ \delta = id = \delta \circ \lambda\).

Corollary 3.2: Given two integers \(L’\) and \(L”\) satisfying the condition \(L – L’ \geq n\) there is a canonical isomorphism of sheaves of graded commutative \(BL\)-algebras \(G^{L’} \rightarrow G^{L”}\).

Proof: Proposition 3.1 entails the isomorphism \(G^{L’} -\). On the other hand, for any open \(U \subset BL^{m,n}\) the surjective isomorphism gives

\[
G^{L’} – \otimes_{\mathbb{N}} \wedge \mathbb{N}^n \quad (3.4)
\]

so that our claim is proved.

Therefore it is possible to introduce on \(BL^{m,n}\) a canonical sheaf of graded commutative \(BL\)-algebras \(G\) formally defined as the isomorphism class of the sheaves \(G^{L’}\) while \(L’\) varies among the non-negative integers such that \(L – L’ \geq n\). Alternatively, one can assume \(L \geq 2n\) and take once for all \(L’ = \left[\frac{L}{2}\right]\) the biggest integer less than \(L/2\) (cf. [17]). A subsheaf \(G\) of germs of sections of \(G’\) not depending on the odd variables is defined in the same fashion and one obtains the isomorphism:

\[
G – G \otimes_{\mathbb{N}} \wedge \mathbb{N}^n \quad (3.5)
\]

Proposition 3.3: There is an isomorphism of sheaves of graded \(BL\)-modules \(DerG – DerGH \otimes_{BL’} BL\).

Proof: By virtue of the surjective isomorphism for any open \(U \subset BL^{m,n}\) it is enough to show that \(DG – DerGH \otimes_{BL’} BL\). By identifying \(G\) with \(G^\infty\) we define a morphism \(\eta: DG^\infty \rightarrow D \otimes_{BL’} BL\) given by

\[
\eta(Df) = \sum_{\mu \in \Xi_n} D \cdot Z_0(f_\mu) \otimes \beta_\mu
\]

where \(f\) has been factorized according to equation (3.3). It is easily verified that \(\eta\) is an isomorphism.

Proposition 3.4: \(DerG\) is a locally free graded \(G\)-module on \(BL^{m,n}\) of rank \((m + n)\). On every open set \(U \subset BL^{m,n}\) \(DerG(U)\) is generated over \(G(U)\) by the derivations:

\[
\frac{\partial}{\partial x^i}, \frac{\partial}{\partial y^\alpha} \quad i = 1 \dots m, \alpha = 1 \dots n
\]

defined as follows:

\[
\frac{\partial}{\partial x^i}(f \otimes a) = \frac{\partial f}{\partial x^i} \otimes a \quad i = 1 \dots m
\]

\[
\frac{\partial}{\partial y^\alpha}(f \otimes a) = \frac{\partial f}{\partial y^\alpha} \otimes a \quad \alpha = 1 \dots n \quad (3.6)
\]

Definition 3.5: An \((m + n)\)-dimensional \(G\)-supermanifold is a graded locally ringed \(BL\)-space \((M, A)\) satisfying the following conditions:

(a) \(M\) is a Hausdorff paracompact topological space;

(b) \((M, A)\) is locally isomorphic with \((BL^{m,n}, G)\);

(c) denoting by \(C^L(M)\) the sheaf of continuous \(BL\)-valued functions on \(M\) there exists a morphism of sheaves of \(BL\)-algebras \(\delta_M: A \rightarrow C^L(M)\) which is locally compatible with the evaluation morphism (3.2) and with the isomorphisms ensuing from condition (b).

Thus by assumptions any point \(z \in M\) has a neighborhood \(U\) such that:

(i) there is an isomorphism of graded locally ringed spaces

\[
\phi: (U, A(U)) \rightarrow \phi(U, G(\phi(U))) \quad (3.7)
\]

(ii) the following diagram commutes:

\[
G(\phi(U)) \xrightarrow{\phi} A(U)
\]

\[
\delta \downarrow \hspace{3cm} \delta_M \downarrow
\]

\[
C^L(\phi(U)) \xrightarrow{\phi^*} C^L(M)
\]

where \(\phi^*\) is the ordinary pull-back associated with the mapping \(\phi\).

If there is no confusion the evaluation morphism \(\delta_M\) will be denoted simply by \(\delta\). The image of the sheaf \(A\) through \(\delta\) is a sheaf on \(M\) of graded-commutative \(BL\)-algebras denoted by \(A^\infty\).

Proposition 3.6:

(a) The atlas \(U^\infty = \{(U_i, \phi_i) | i \in \mathbb{I}\}\) endows \(M\) with a structure of \(G^\infty\) supermanifold of the same dimension as \((M, A)\).

(b) The \(G^\infty\) structure sheaf of \(M\) coincides with \(A^\infty\).

It is clear that \(G\)-supermanifolds generalize the notion of \(GH^\infty\) supermanifolds; indeed if \((M, GH^\infty(M))\) is a \(GH^\infty\) supermanifold [7] the pair \((M, A = GH^\infty(M) \otimes_{BL’} BL)\) is a \(G\)-supermanifold. The resulting \(G\)-supermanifold will be called the trivial extension of the original \(GH^\infty\) supermanifold [19].

Graded Tangent Space

As a consequence of Proposition 3.3, the sheaf \(Der(A)\) of graded derivations on a \(G\)-supermanifold \((M, A)\) is locally free with local bases given by the derivations

\[
\frac{\partial}{\partial x^i}, \frac{\partial}{\partial y^\alpha} \quad i = 1 \dots m, \alpha = 1 \dots n
\]

associated with a local coordinate system \(x_1 \dots x_m, y_1 \dots y_n\).

Definition 3.7: The graded tangent space \(T_z(M, A)\) at a point \(z \in M\) is the graded \(BL\)-module whose elements are the graded derivations \(X: A_z \rightarrow BL\).

The graded tangent space \(T_z(M, A)\) is quite evidently free of rank \((m + n)\) and the elements

\[
\frac{\partial}{\partial x^i}, \frac{\partial}{\partial y^\alpha} \quad i = 1 \dots m, \alpha = 1 \dots n
\]

defined by

\[
\frac{\partial}{\partial x^i}(f) = \frac{\partial f}{\partial x^i}(z), \frac{\partial}{\partial y^\alpha}(f) = \frac{\partial f}{\partial y^\alpha}(z) \quad \text{for all } f \in A_z
\]

yield a graded basis for it. Furthermore, there is a canonical isomorphism of graded \(BL\)-modules

\[
T_z(M, A) \rightarrow (Der(A))_z/(L_z \cdot (Der(A))_z)
\]

where \(L_z\) is the ideal of germs in which vanish when evaluated i.e.

\[
L_z = \{f \in A | f(z) = 0\}.
\]

TOPOLOGIES OF RINGS OF G-FUNCTIONS.

In order to introduce the notions of morphisms and products of \(G\)-supermanifolds and to discuss Rothstein’s axiomatics we need to topologize in a suitable way the rings of sections of the structure sheaves of \(G\)-supermanifolds [9]. This will parallel the analogous study performed in the case of graded manifolds [2].

Let \((M, A)\) be a \(G\)-supermanifold and let \( \| \cdot \|_{l_1} \) denote the \(l_1\) norm in \(BL\); for every open subset \(U \subset M\) the rings \(A(U)\) of \(A\) can be topologized by means of the seminorms \(p_{L,K}: A(U) \rightarrow \mathbb{N}\) defined by

\[
p_{L,K}(f) = \max_{z \in K} \|L \cdot f(z)\|_{l_1}
\]

where \(L\) runs over the differential operators of \(A\) on \(U\) and \(K \subset U\) is compact. The above topology is also given by the family of seminorms

\[
p_{K,I}(f) = \max_{z \in K} \left|\frac{\partial^J}{\partial x^J}\frac{\partial^\mu}{\partial y^\mu}f(z)\right|_{l_1} \quad (4.1)
\]

where \(K\) runs over the compact subsets of a coordinate neighborhood \(W\) with coordinates \(x_1 \dots x_m, y_1 \dots y_n\). Under this form, it is clear that this topology makes \(A(U)\) into a locally convex metrizable graded algebra. The next results will allow proving that \(A(U)\) is complete so that it is in fact a graded Fre’chet algebra. Without loss of generality we may assume that \((M, A) = (BL^{m,n}, G)\). With reference to the isomorphism (3.5) we topologize the rings \(U\) by means of the seminorms

\[
p_{K,I}(f) = \max_{z \in K} \left|\frac{\partial^J}{\partial x^J}f(z)\right|_{l_1} \quad (4.2)
\]

The tensor product \(U \otimes_{\mathbb{N}} \wedge \mathbb{N}^n\) is in turn given its natural topology which is induced by the seminorms

\[
p_{K,I}^\mu(f) = p_{K,I}(f_\mu)
\]

having set \(f = \sum_{\mu \in \Xi_n} f_\mu \otimes y_\mu\).

Lemma 4.1: The isomorphism (3.5) \(G(U) \otimes_{\mathbb{N}} \wedge \mathbb{N}^n\) is a metric isomorphism.

Proof: A direct majoration argument shows that

\[
p_{K,I} \leq \sum_{\mu \in \Xi_n} c_\mu p_{K,I}^\mu
\]

where \(c_\mu = \max_{z \in K, \nu \in \Xi_n} \left|\frac{\partial^\nu}{\partial y^\nu}y_\mu(z)\right|\).

This shows the continuity of the inverse morphism. We now display the opposite majoration. The seminorm \(p_{K,I}\) is explicitly written as

\[
p_{K,I}(f) = \max_{z \in K, j \leq I, v \in \Xi_n, \mu \in \Xi_n} \epsilon_{\mu,v} \frac{\partial f_\mu}{\partial x^J}(z) \frac{\partial^\nu}{\partial y^\nu}y_\mu(z) \quad (4.3)
\]

with \(\epsilon_{\mu,v}\) a suitable sign. The seminorms \(p_{K,I}^\mu\) are majorated by descending recurrence starting from the last one i.e. from \(p_{K,I}^\omega\) where \(\omega\) is the sequence \(1,2,\dots,n\). Indeed, from (4.3) we obtain \(p_{K,I}^\mu \leq p_{K,I}\) since \(p_{K,I}^\mu\) is one of the terms over which the maximum (4.3) is taken. For the same reason, if we consider the seminorms \(p_{K,I}^{\omega_i}\) for \(i = 1 \dots n\) with \(\omega_i = 1,2,\dots,i,\dots,n\) we obtain

\[
p_{K,I}^{\omega_i}(f) = \max_{z \in K, j \leq I} \left|\frac{\partial f_{\omega_i}}{\partial x^J}(z) + \frac{\partial f_\omega}{\partial x^J}(z)\frac{\partial y_i}{\partial y^\nu}(z) – \frac{\partial f_\omega}{\partial x^J}(z)\frac{\partial y_i}{\partial y^\nu}(z)\right| \leq p_{K,I}(f) + \max_{z \in K, j \leq I} \left|\frac{\partial f_\omega}{\partial x^J}(z)\frac{\partial y_i}{\partial y^\nu}(z)\right|
\]

\[
\leq (1 + c_i(K)) p_{K,I}(f)
\]

where \(c_i(K) = \max_{z \in K} \left|\frac{\partial y_i}{\partial y^\nu}(z)\right|\). The remaining majorations are performed in the same way.

For any open \(W \subset \mathbb{N}^m\) the space \(C^\infty(W) \otimes_{\mathbb{N}} BL’\) is equipped with the usual topology of uniform convergence of derivatives of any order which is induced by the family of seminorms

\[
q_{K,I}(h) = \max_{z \in K, J \leq I} \left|\frac{\partial^J}{\partial x^J} h(z)\right|
\]

where \(K\) is a compact in \(W\) and the norm is taken in \(BL’\). Moreover, since \(\delta\) is injective when restricted to \(G\), we may identify the sheaves \(G\) and \(G^\infty\).

Theorem 4.2: For any open \(U \subset BL^{m,n}\) and all such that \(0 \leq L’ \leq L\) the \(Z\)-expansion

\[
Z_{L’}: C^\infty(U) \rightarrow C^\infty(\sigma^{m,n}(U)) \quad (4.4)
\]

is an isometry onto its image. In particular, when \(L’ = L\) we obtain a metric isomorphism \(C^\infty(\sigma^{m,n}(U)) \otimes_{BL} U\) while for \(L’ = 0\) we obtain a metric isomorphism \(C^\infty(\sigma^{m,n}(U)) \rightarrow U\).

Proof: One easily shows that the seminorms which define the topology on the right-hand side are majorated in terms of the relevant seminorms on the left-hand side. To show the converse, let \(K\) be a compact subset of an open \(W \subset \mathbb{N}^m\) and \(I\) be a nonnegative integer; for any \(h \in C^\infty(\mathbb{N}^n, W)\) we have

\[
q_{K,I}(h) \leq \max_{z \in K, J \leq I} \left|\frac{\partial^J}{\partial x^J} Z_{L’}(h)(z)\right| = p_{K,I}(Z_{L’}(h))
\]

where \(K\) is a compact in \(\sigma^{m,n-1}(W)\) containing \(K\). It is clear that the previous minoration implies the thesis.

Proposition 4.3:

(a) The functions \(p_{K,r}: A(W) \rightarrow \mathbb{N}\) are submultiplicative seminorms in that

\[
p_{K,r}(fg) \leq 2^n r p_{K,r}(f) p_{K,r}(g)
\]

(b) \(A(W)\) equipped with the topology induced by the seminorms \(p_{K,r}\) where \(r \geq 0\) and \(K\) is an arbitrary compact coordinate subset of \(W\) is a Fre’chet algebra.

Corollary 4.4: The spaces \(G(U)\), \(H^\infty(U) \otimes_{\mathbb{N}} BL\), and \(C^\infty(\sigma^{m,n}(U)) \otimes_{BL} \otimes_{\wedge \mathbb{N}} \mathbb{N}\) are isometrically isomorphic for any open \(U \subset BL^{m,n}\).

Proposition 4.5: Let \((M, A)\) be a \(G\)-supermanifold. For every open \(U \subset M\) the space \(A(U)\) endowed with the topology induced by the seminorms (4.1) is a graded Fre’chet algebra.

Reasoning as in Proposition 4.3 one proves that the topological algebra \(U\) is complete where using Lemma 4.1 and reasoning as in Proposition 4.3 again the algebra \(G(U)\) is complete as well. We eventually obtain the results which are Corollary 4.4 and Proposition 4.5.

Example 4.6:

The previous Lemma 4.1 and Theorem 4.2 also imply a further result that will be significant when dealing with morphisms of \(G\)-supermanifolds. For any open \(W \subset \mathbb{N}^m\) we topologize the space

\[
C^\infty(W) \otimes_{BL} \otimes_{\wedge \mathbb{N}} \rightarrow C^\infty(W) \otimes_{\wedge \mathbb{N}} \mathbb{L+n}
\]

CONCLUSIONS

The \(Z\)-expansion is the morphism of graded algebras \(Z_{L’}\) which is defined by (2.1). A theorem on an isometry onto its image of \(Z\)-expansion and on a metric isomorphism is derived. This theorem makes possible the definition of coordinate neighborhood and odd and even coordinate systems, and allows for the study of odd symplectic supermanifolds [11] as well as integration on supermanifolds such as integration on \(\mathbb{N}S^{m,n}\) and Rothstein’s theory of integration on non-compact supermanifolds. Thus, this theorem implies further research which will be useful when some author has to deal with morphisms of \(G\)-supermanifolds.

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