Riemannian geometry of a discretized circle and torus

We classify all linear connections for the minimal noncommutative differential calculus over a finite cyclic group and solve the torsion free and metric compatibility condition in general. We show that there are several classes of solutions, out of which only special ones are compatible with a metric that gives a Hilbert C*-module structure on the space of the one-forms. We compute the curvature and scalar curvature for these metrics and find its continuous limit.


Introduction
Noncommutative geometry [13,22] offers a new insight into spaces and their generalizations by allowing to skip the traditional assumptions of points and to use the methods of differential geometry that are adaptable to the algebraic language. The construction of differential calculi has been one of the first steps that allowed the extension of the formalism of gauge theory to the realm of noncommutative spaces. In particular, the spaces that consist of finite number of points or discrete lattices have appeared not only as discrete approximations of differentiable spaces but as manifolds in the generalized sense [2,16,8].
One of the crucial aspects of the differential geometry is, however, the link between the metric aspects, that is distances and the norm on the space of states with the relevant objects in the differential algebra. In the classical differential geometry this link is provided by the metric tensor and leads to the notion of metric compatible and torsion-free linear connections that provide relevant and physically significant constructions of the curvature and appropriate geometrical objects like Ricci tensor and the scalar curvature. The noncommutative geometry has been, so far, unable to retrace these steps in full generality despite many efforts. Apart from the usual problem of the choice of the differential calculus for the given algebra the main problem is the definition of the metric over the bimodule of differential forms and the linear connection [25,26]. The choice of the metric and the linear connection that are compatible with the bimodule structure of the differential forms lead necessarily to severe restrictions not only on the possible metrics [28] but also on connections [17] and curvature [18].
Recently an updated version of the approach to linear connections for a special types of differential calculi was studied in general and for particular examples of noncommutative spaces [9,10,11,12].
A systematic approach to the general issue of bimodule linear connections and bimodule metrics over the differential forms was started by Majid [6,7] and developed in [5]. The formalism used there allows to generalize in a systematic way almost all classical notions like torsion-freeness and Ricci and scalar curvature [23] (depending on certain choices).
It is worth mentioning that the Connes' spectral approach based on the Dirac operators [13] that was much successful in the reconstruction of the Einstein-Hilbert gravity action for the standard and modified almost-commutative geometries [14] can be applied to the studies of the generalized scalar of curvature for certain noncommutative manifolds, in particular the noncommutative tori (see [19,20] and [15] for the specific example of an asymmetric torus). Yet there is currently no method to recover all geometric objects like Ricci tensor or the torsion through the spectral methods. There exists a huge discrepancy between the usual methods of recovering the geometric objects like the scalar of curvature for the manifolds and their deformations and the attemt to use of spectral methods [3,4,21,27] in the finite-dimensional case.
In this paper we start systematic computations of linear connections for finite groups, starting with the easiest example of finite cyclic groups and their products. We provide a complete classification of linear connections that are torsion free and compatible with any nondegenerate metric, demonstrating that there are severe restrictions on possible metrics and only a certain class of metrics allows the existence of non-unique compatible linear connections.
The main result is that for the special case of left-right symmetric metric there exist still a choice of linear connections that are torsion-free and compatible with the metric a scalar curvature that vanishes only for the constant (equivariant) metric (for some canonical choices of the arbitrary constants in the theory). We demonstrate that the freedom is much larger in the case of the products of two discrete circles even in the case of the constant metric.

Preliminaries
Let us start by recalling basic definitions. In what follows for a group G with its neutral element e we denote G × = G \ {e}. For a subset H ⊂ G × by H ⊥ we shall always denote G × \ H. Furthermore, for X ⊂ G we denote by χ X the characteristic function of the set X, i.e. χ X (g) = 1, g ∈ X 0, g ∈ X.
Definition 2.1. The (first-order) differential calculus over an algebra A over a field k is a pair (Ω 1 (A), d), where Ω 1 (A) is a bimodule over A, d is a linear map d : A → Ω 1 (A), which satisfies the Leibniz rule, d(ab) = a db + (da) b, (2.1) and Ω 1 (A) is generated as a left module by the image of d. We say that (Ω 1 (A), d) is connected if ker d ∼ = k.
In case the algebra A is a Hopf algebra (with a coproduct ∆, counit ε and antipodal map S) we have the following. Definition 2.2. We say that the differential calculus over a Hopf algebra A is left-covariant if there exists a coassociative left coaction of A on Ω 1 (A), δ L : for every a, b ∈ A, ω ∈ Ω 1 (A), and In a similar manner we define the right-covariance and bicovariance (as simultaneous left and right covariance). The canonical example of a first-order differential calculus is given by the universal calculus, with Ω 1 u (A) = ker m ⊆ A ⊗ A, where m : A ⊗ A → A is the multiplication map for A, and the universal differential d u : A→ker m of the form d u a := a ⊗ 1 − 1 ⊗ a. The universal calculus over a Hopf algebra is bicovariant. The bicovariant calculi over an arbitrary Hopf algebra were classified by Woronowicz [30].
For a * -algebra A we can consider differential calculi that in addition possess a * -structure, that is we assume d is a derivation of a * -algebra, i.e. d(a * ) = (da) * for every a ∈ A.

Finite cyclic groups
As we consider finite cyclic groups Z N with N ≥ 2 and since the groups are abelian many results are much simpler. We skip the derivation of the results, which are based on [16,8]. By e g , g ∈ G, we denote a function that vanishes everywhere apart from g: e g (h) = δ g,h . By R g (f ) we denote the right translation R g (f )(h) = f (hg). In a similar manner we introduce the left translation L g . Note that the left covariance of the calculus is equivalent to the fact that for all ω ∈ Ω 1 (A) and all g ∈ G we have L g ω ∈ Ω 1 (A).
Each connected, star-compatible first-order bicovariant differential calculus over C(Z N ) is determined by a subset H ⊂ G × such that H = H −1 and H generates the entire group Z N . By |H| we denote the number of elements in H. There are |H| left invariant forms such that the star-and the bimodule structure over

and the calculus is inner
Moreover, there exists a unique minimal extension of the first order differential calculus (as defined by Woronowicz) so that: Proof. The first part follows directly from the Section 2 in [16]. From [30], Proposition 3.1, for any bicovariant differential calculus (Ω 1 H (Z N ), d) there exists a unique bimodule automorphism σ W of Then we introduce the symmetrization map as the extension of the σ W to the tensor algebra of Ω 1 (A). Its kernel is identified with the exterior algebra over A.
Theorem 2.2. Consider the cyclic group Z N , N > 2 with the generator p. Denote byp its inverse in Z N . Then there exists a minimal bicovariant, star-compatible connected differential calculus, generated by θ p , θp with the following structure:

4)
Proof. Since N = 2 we have p =p. The first order differential calculus generated by H = {p,p} is then bicovariant, connected and compatible with the star structure. By a direct computation we see that for any g ∈ H: Since H = {p,p} and if p ∈ H, then for N > 3 we get pp −1 ∈ H (otherwise p = e or is order 3), we immediately infer that dθ p = dθp = 0. For N = 3 in the above sum there is only one term: dθ g = θ g −1 ∧ θ g −1 for g = p,p, and as a result dθ p = dθp = 0 also for this case. Notice that since dθ p = dθp = 0, for ω = ω p θ p + ωpθp we get which is exactly θ ∧ ω + ω ∧ θ.

Bimodule linear connections
Following [5] for a first-order differential calculus (Ω 1 (A), d), we set and a bimodule map, called the generalized braiding, such that for all a ∈ A, ω ∈ Ω 1 (A).
Notice that (see e.g. [7] -Prop.2.1. 3) with such a definition, the linear connection can be naturally extended to the whole tensor algebra In our case of the algebra C(Z N ) since the calculus is inner, we can use for some bimodule maps σ, α. [24]-Thm.2.1.
As an immediete consequence of the above definition we get the following result: For a minimal bicovariant calculus over C(Z N ) with N = 3 a bimodule linear connection is determined by a bimodule map σ.
Proof. We use shortened notation Ω 1 (A) to denote Ω 1 H (C(Z N )) from Theorem 2.2. First of all, observe that there are no bimodule maps apart from the zero map between and Ω 1 (A) and Ω 1 (A) ⊗ A Ω 1 (A). Indeed, there are no objects in Ω 1 (A) ⊗ A Ω 1 (A) that have the same bimodule commutation rules as in Ω 1 (A). Otherwise p would be of order 3. Therefore, necessarily α ≡ 0. Hence, the bimodule connection ∇ and σ are mutually determined.

Torsion-free connection
Let us now concentrate on the notion of a torsion. We define torsion as a map T ∇ : (3.5) Following [24] we say that the connection is compatible with a torsion if Im(id + σ) ⊆ ker ∧. The connection is said to be torsion-free if T ∇ = 0. Observe first that we have the following result.
Proposition 3.2. For a minimal bicovariant calculus over C(Z N ) with the torsion-free connection, the map σ must satisfy, , which gives us the claimed formula.
Notice that it follows from the last proposition that the torsion-free connection is compatible with a torsion. This is a manifestation of the more general result: For a inner calculus (3.4) with the extension to Ω 2 (A) such that θ ∧ θ = 0, the connection is torsion-free if and only if is torsion-compatible and imα ⊆ ker ∧.
Remark 3.2. Notice that the similar result was stated in [24]-Theorem 2.1., part (3), but in their formulation α was enforced to be a zero bimodule map instead of satisfying imα ⊆ ker ∧. As one can easily see for the case of the Z 3 group such a formulation is not true in general, since it is possible to have a torsion-free connection with nontrivial α, mainly α(θ p ) = θp ⊗ A θp (because p =p 2 ).
From Proposition 3.1 it follows that for N = 3 the pair (∇, σ) is mutually unambiguously determined. The case with N = 3 has to be consider separatelly. Even the torsion-freeness does not guarantee vanishing of α.
Definition 3.2. We say that the connection is star-compatible, if: where (ω ⊗ A η) † = η * ⊗ A ω * , i.e. † is the induced * -structure on higher tensors. Proposition 3.4. The torsion-free bimodule connections over the minimal bicovariant calculi over C(Z N ) with N = 4 are determined by a family of functions A p , Ap, B p , Bp, so that σ is, Proof. If follows directly from the fact that σ is a bimodule map, p 2 =p 2 for N = 4, and the compatibility condition of σ with the ∧ (3.6).
The assumption for the connection to be compatible with the star structure imposes further restrictions on the functions A and B.
In our situation, we have: Lemma 4.1. A nondegenerate metric over the minimal bicovariant calculus over C(Z N ) is given by functions G p , Gp, which are everywhere different from 0, Proof. Since for arbitrary f ∈ C(G), then we can now analyse the conditions we have from the required properties of a metric g. First, we obviously have Since the right action is free it implies that j = i −1 whenever (θ i , θ j ) = 0. Therefore the bimodule map (·, ·) has to be of the following form where F a ∈ C(Z N ). We are now ready to explore conditions that follows from equation (4.3). Let us write g in the basis, here H = {p,p}, and consider the condition ω = g (1) (g (2) , ω) with ω = θ c . We have, The equality holds if an only if: Taking a = c we immediately obtain the claimed result.
Corollary 4.1. If the metric in the Lemma 4.1 is also compatible with the higher-order differential calculus (i.e. ∧g = 0), then it can be described by the only one function G := G p = Gp.
It follows immediately that g is a central element in Ω 1 H (A) ⊗ A Ω 1 H (A) and we can compute both contractions of the metric, that is not only (g (1) , g (2) ) but also (g (2) , g (1) ) make sense. We have, (4.10) where we have used the unique extension of right (resp. left) translations to the whole differential algebra, so that .
The metric g is left-invariant if and only if for every g ∈ {p,p}, G p = const. A nondegenerate metric is * -compatible iff for the metric coefficients are real, G g = G * g .
Finally let us see when a *-compatible metric defines a norm on the module of one-forms.
If all G g are real and negative then Ω 1 (A) equipped with ·, · is a Hilbert C * -module over A.

Metric compatibility condition
Let us now concentrate on the metric compatibility condition for a bimodule linear connection over the minimal bicovariant calculus on C(Z N ). Although we shall later concentrate on the solutions that correspond to the real-valued metrics that provide nondegenerate scalar products over Ω 1 , we solve the metric compatibility problem in all generality.
Before we proceed with the conditions for the general Z N case, N > 4, let us consider a much simpler case of N = 2.
Example 5.1 (Levi-Civita bimodule connections for Z 2 ). In the case of Z 2 , we have p =p and therefore the entire connection is determined by one function S: the metric is given by G θ p ⊗ A θ p and the metric compatibility then reads: Using notation G 0 = G(e), G 1 = G(p) and S 0 , S 1 for the respective values of S we have and The above system of equations is equivalent to the following two Observe that even in the case of constant metric we can have a one-parameter family of torsion-free, metric compatible connections given by: Theorem 5.1. For the torsion-free bimodule connection for the minimal bicovariant calculus over Z N with N > 4 the metric compatibility conditions takes the following form: for g = p,p.
Proof. First, notice that On the other hand: Taking the sum of these two expressions we get the final result.
Let us now solve the system of equations (5.2). To start we substitute A g = a g + 1 and B g = b g + 1, then the equations read: Introducing X g = RgGg G g −1 and combining the first and the third equation we obtain As the left-hand side is unchanged when we replace g by g −1 and act on the result with R g , we obtain Since X g satisfies: we obtain: which leads to: and as a result Notice that the above relation is, effectively equivalent to (g (1) , g (2) ) = c, which means that in this case both contractions as computed in (4.10) are constant.
Writing explicitly X p , Xp as functions over Z N , the relation (5.6) can be reformulated in the form of the following recurrence system, here for simplicity we denote the function X p as f and choose p = 1 for a function f : N → C. Note that we can equivalently choose the equation for Xp (denote this function as F ) but this corresponds to the choice of −1 as the generator of Z N and therefore give the equations, which is equivalent to (5.7) since

Solving the recurrence relation
We begin with solving the following recurrence equation (5.7). First, let us choose γ such that (c − γ)γ = 1. There are two possible solutions of this equation, which may be, in general, complex numbers and are mutual inverses, that is γ − = (γ + ) −1 . Fixing one root γ we define f (n) = k(n) + γ, so that the equation we have to solve reduces to an equivalent one, Since γ = 0 then we either have k ≡ 0 or all k(n) are different from 0. In the first case we have a constant (trivially periodic) solution, whereas in the second case we set h(n) = 1 k(n) and obtain The above relation has a solution, where H 2 is an arbitrary constant up to the following restrictions: Before we pass to f observe that in the case γ 2 = 1 we cannot have a periodic solution for h, since h(0) = h(N ) enforces γ = 0, which contradicts our starting point. If γ 2 = 1 the periodicity condition is which is possible only if γ 2N = 1 or H = 0. The solution with H = 0 is nothing else as a constant solution with γ −1 (corresponding to the other choice of the root of the equation (c − γ)γ = 1). We can write explicit form of a non-constant (i.e. with H = 0) solution for f : This form of the solution is very convenient, as it is easy to verify the multiplication property for f :

The real-valued solutions
As we are interested in real metrics G g , we consider real-valued solutions of the above recurrence system. It immediately follows from (5.12) that non-constant real solutions exist only for |H| = 1, i.e. for H = e iφ with some φ. Using the fact that γ satisfies γ 2N = 1, γ 2 = 1 and H 2 = γ 2n+2 , n ∈ Z, we can choose γ = e πi l N and obtain a set of solutions, parametrised by l = 1, ..., N − 1, N + 1, . . . 2N − 1, Some of the solutions are, however, repeated as f 2N −l,φ = f l,−φ . Moreover, for such γ we have c = 2 cos πl N . Note that although we have excluded the case γ 2 = 1, the above formula recovers some of the constant real solutions, which arise for l = 0 (f (n) = 1) and l = N (f (n) = −1), so in fact we can extend the range of l also into l = 0 and l = N . It is also easy to see that in case of the real nonconstant solutions X p cannot be a positive function. Finally, let us observe that in case we do not demand reality of the metric, the formula above is still valid but with φ allowed to be an arbitrary complex number.

The coefficients of the linear connection
In the next step we are going to solve the system of equations following from (5.3) without restricting ourselves to real solutions of X g . Using the first and second equation and (5.4) we obtain a linear dependence between b g and a g −1 : (5.15) Reintroducing 1 + R g −1 a g −1 into the first equation we have: which, after splitting X g + R g b g into (X g − 1) + R g (1 + b g ), is equivalent to Note that 1 + b g cannot vanish at any point since X g cannot vanish at any point, so we can divide both sides by it. Next, substituting we obtain: This has an obvious solution Y g ≡ 0, which gives: and apart from this solution Y g must be invertible at each point. Then, take y g = (Y g ) −1 to obtain: To solve this equation it is sufficient to find just one solution y 0 g of the inhomogeneous equation and a family of solutions of the homogeneous equation: The first problem is solved explicitly by verifying that provided that c = −2. We shall discuss the special case c = −2 later. Next, we solve the homogeneous equation. It is easy to see that all solutions are parametrised by a multiplicative constants κ p , κp, for n ∈ Z N , where κ g are such that y hom g + y 0 g = 0, since we require y g to be invertible. Observe that for the function y g to be periodic we need to have: which, after taking into account that the product of all X g (k) in the non-constant case is γ N gives us: further restricting the possible solutions for X g , which then must be parametrised by an integer l = 0, 1, . . . 2N − 1 such that N + l is always even. From now on we will always assume that N + l is even, and proceed with the further analysis. If we have X g = const then either κ g = 0 or X N g = (−1) N . For real-valued solutions it restricts constant X g to be −1, or, for even N , to be 1. But since here c = −2 the first possibility is not allowed.
Finally we go back to the case c = −2, for which there exists only the constant solution X g = −1. In this case the equation for b g reduces to which, as we already know from the previous subsection, has only one periodic solution b g = 0.
To summarize, we have three possible cases: • X g = −1.
In this case b g = 0.
In this case, combining the results, we have two possibilities where y hom g is expressed in (5.17).
Now, what is left in the last case is the compatibility with (5.4). Indeed, although we had determined possible solutions for b g and b g −1 we must further check whether they are related with each other through (5.4). First observe that if b g is of the first type, then from (5.4) it follows that the solution for b g −1 is, The last expression for b g −1 can be rewritten as which is the solution of the second type with the homogeneous part vanishing. Similarly, inserting the solution for b g of the second type with y hom g = 0 to (5.4), we end up with the solution for b g −1 of the first type.
Our goal is to establish a relation between y hom g −1 and y hom g . We have already discussed cases with vanishing homogeneous parts, and have shown that they are coupled, in the aforementioned sense, to the solutions of the first kind, so from now on we assume that for both g and g −1 we have a solution for b of the second type and with y hom = 0.
Inserting these two solutions into (5.4) we end up with Using (5.17) we can write (5.20) as: which gives, Notice that since X g satisfies (5.6), we have: so the right hand side is independent on n, and the equation imposes a condition on the product of κ g and κ g −1 : To sum up, we have proven the following result: and for B g = 1 + b g we have the following possibilities depending on X g , Case I. If X g = const the only following functions X p are allowed, for l = 1, ..., N − 1, and an arbitrary constant φ such that e 2iφ = e 2lπ N (n+1) . Then with c = 2 cos lπ N there exist three possible solutions, and, provided that Furthermore, the constants κ p and κp are restricted via a constraint: and also requirement that y hom p + y 0 p = 0. Case II. If X g = γ ≡ const: • if γ N = (−1) N and γ = −1 then (5.24) is also a solution.
Notice that for γ = −1 the cases (a) and (b) reduce to the first bullet point.
As the next step let us summarize the restrictions on the possible metrics. As we have computed all possible solutions for we can always choose one of the functions G p , Gp arbitrarily, and then the second one will be determined by the relation above.
Remark 5.1. For the real metric satisfying g = g * , the constant solutions above are restricted to real constant X g , whereas the non-constant solutions are restricted by an additional demand that φ is a real parameter. Only the solutions with X g = const > 0 give the real metric g that equips the module of one-forms with a Hilbert C * -module structure (see Lemma 4.3).
Remark 5.2. If we further assume that the metric is compatible with the differential calculus, ∧g = 0, the solution for X g provides the solution for G p = Gp given by: The only real constant solutions that are compatible with the differential calculus are restricted to X g = 1, and, for even N , also −1, yet only the first one gives a Hilbert C * -module structure. Moreover, no nonconstant solution gives rise to a Hilbert C * -module structure since they are not of constant sign.
We can further assume that in addition to compatibility of the metric with the star structure, the connection itself is star-compatible, i.e. relations in (3.7) are satisfied.
Using the first relation in (5.3) we can express A in terms of B, and then the first relation in (3.7) implies that Observe that since X g satisfies (5.6), the right-hand side of this equation is non-negative. Indeed, using (5.6) we can write 26) and the problem reduces to examine the quadratic equation x 2 − cx + 1 = 0, which has no real roots iff |c| < 2. Hence for those c, the right-hand side is always positive. Interestingly, this is the same range of c for which there exist non-constant solutions for X g . On the other hand, for constant solutions X g combining (5.25) with the Theorem 5.2 we see that in these cases B g has to be equal to 1. Let us further examine which non-constant solutions determined in Theorem 5.2 are allowed when compatibility with the star structure is imposed, so we are concentrate on Case I therein. By a straightforward computation we check that cases (a) and (b) do not fulfil the condition (5.25). So, suppose now we take solutions as in the case (c) with non-zero homogeneous parts y hom g . Using 5.6 again, an the fact that c = −2, the condition (5.25) can be reduced to On the other hand, y hom satisfy (5.20) and R g y hom g = −y hom g X g , so together with the relation above it implies that so we get a restriction for possible star-compatible solutions: Parametrizing the relation for the solution B g : can be rephrased as b 2 − a + r 2 e i(ϕ+ρ) + rb(e iϕ + e iρ ) = 0.
Simple calculations show that b 2 − a = r 2 , so for r = 0 the star-compatibility condition for a connection introduces the following constrains on phases ρ and ϕ: As a result we have the following. Proof. It follows from the computations before that in such a case we have X g = const > 0 and from the above discussion it follows that the star-compatibility of the connection fixes B g to be equal to 1.
We finish with a remark that this corollary is in a complete agreement with the result obtained in [1], where only the case X g = 1 was assumed.

The curvature
In this section we shall compute the curvature of the torsion-free linear connection compatible with the metric g. Though it can be done for arbitrary metrics that satisfy the compatibility connections, we shall restrict ourselves to the case of real metrics that equip the bimodule of one-forms with a Hilbert C * -module structure. This will restrict X p = γ > 0.
Definition 6.1. The Riemannian curvature for a given connection ∇ is a map: defined by the following prescription By a straightforward computation we get the following: Theorem 6.1. The Riemannian curvature for the connection ∇ from Theorem 5.2 is: To define the objects corresponding to Ricci and scalar curvature we need, however, some more structure.

Definition 6.2. Let ι be a bimodule map representing two-forms in
such that the following diagram commutes.
Then, we define: 6) and the Ricci tensor is defined as , where the Sweedler's notation on Ω 1 ⊗ A Ω 1 ⊗ A Ω 1 is used.
Observe that, the above definition uses the metric unlike the usual definition of the Ricci tensor that is metric independent and uses the trace.
Following [5] we can further define the Einstein tensor and the scalar curvature, Definition 6.3.
Obviously, R is an element of the algebra A.
In our case of A = C(Z N ) with N > 4 we observe that the most general form of the lifting map ι is where β ∈ C(Z N ). As an immediate consequence we finally obtain for the Ricci tensor, Since ρ g has a form M g θ g + N g θ g −1 for g = p,p, we get for the scalar curvature and as a result (6.14) We can formulate the main theorem.
On the other hand, for even N in addition to the above ones there are also solutions corresponding to the last point in Case II of Theorem 5.2. In these cases the corresponding Ricci tensor and the scalar curvature are given by In the above formulas either B p = 1 and Bp is given by (5.24) or the other way around. the scalar curvature is R(n) = ± 1 2 (W + (n) − W − (n + 1)), i.e.
, (6.15) with the sign − for the case (a) and + for cases (b) and (c). On the other hand, for the special cases discussed at the end of the previous theorem, the scalar curvature is: In particular for the constant metric G, this curvature vanishes in all these cases.
Remark 6.1. It is interesting to see the continuous limit of the expression (6.15). A simple computation gives that if we denote by g(t) the limit of the G(n) function, for the parametrisation of the curve with t, then the curvature R(t) becomes: .

Examples of metrics and curvatures
It is interesting to see how the scalar curvature depends on the metric. Clearly it vanishes for the constant metric, which can be depicted as an equilateral N -polygon. On the other hand, if we consider a polygon that approximates the ellipse, that is the respective lengths of the sides correspond to the lengths of lines connecting points on the ellipse like depicted on the Fig. 1, we obtain a nontrivial scalar curvature. We can then compute the scalar curvature for the assumed form of the metric, which become as shown on the Fig. 2 and which very closely approximate its continuous limit.
Even more interesting is the inverse problem, of finding the metric such that the scalar curvature is fixed. This shall be treated rather as an exercise in the N → ∞ limit, that is an infinite lattice with the algebra C(Z), as we fix three distances and then compute the rest using the recursive relation arising from the Theorem 6.2. It is clear that we cannot then guarantee periodic solutions and, moreover, the choice of the initial values leads to solutions that differ from the continuous approximations.
We have checked some example cases with the constant scalar curvature. It appears that for the positive scalar curvature (see Fig. 3), and the initial data of equilateral sides we obtain oscillating distances, whereas for the negative (small) curvature (see Fig. 4) the metric tends rapidly to zero.   7 Linear connections and curvature for products of Z N In this section we shall extend the computations of linear connection to the tensor product of two C(Z N ) algebras, which corresponds to the Cartesian product of discrete spaces. Since we consider the minimal differential calculi over both algebras and their natural graded tensor product, most of the results from previous sections can be transferred. In particular, it is easy to see that the only possible metric is the diagonal one, that is for the algebra A 1 ⊗ A 2 , which means that the total metric is the sum of metrics, yet the coefficients can be elements of the full algebra.
For simplicity we shall consider here the product of two algebras, this can be later extended to an arbitrary number of component algebras in the product. Furthermore, we restrict ourselves only to negative metrics, which then allows us to use the results of Theorem 6.2. Let us introduce the notation used in this section. We denote by p and s the generators of the groups Z N and Z M , with their inversesp ands.
Lemma 7.1. The only bimodule metric over C(Z N ) ⊗ C(Z M ) is of the form: where G p , Gp, G s , Gs are functions over Z N × Z M .
Lemma 7.2. The most general linear connection for the minimal differential calculus over C(Z N ) ⊗ C(Z M ) with N, M > 4 is determined by the map σ given by:
Proof. Since the calculus is inner we can apply (3.4). Futhermore, in a completely similar manner as in the Prop. 3.1 we infer that if both N and M are different than 3, α has to be a zero map. As a result, the connection is determined by the bimodule map σ only. Now, form the bimodule structure, as in the proof of Prop. 3.4, we get the exact form of this map, provided that N, M = 4.
Lemma 7.3. The metric compatibility condition, which can be written in general as, ψ a,b g,h θ a ⊗ A θ b leads to the following system of 36 equations which can be divided into six types written explicitly below (where we use the convention that h = g, g −1 ): Some simplification can arise from considering torsion-freeness together with vanishing of the cotorsion, coT ∇ = (d ⊗ id − id ∧ ∇)g, which is implied [29] by torsion-freeness together with metric compatibility. These conditions are much simpler since they are linear and in principle can lead to significant restrictions on possible solutions of the main problem.
In our case the cotorsion-freeness can be written explicitly as a system of 16 equations for functions A, B, C and W : where h = g, g −1 and these indices are taken from {p,p, s,s}.
Observe that first of them can be used to express A in terms of B, and the last one to determine C as a function of W . The second one is a compatibility condition for functions C.
It appears, however, that even using these linear dependencies the resulting set of nonlinear equations is at present beyond the possibilities of exact analytical solutions. Instead we shall concentrate on showing few possible solutions for the metrics and compatible linear connections, in particular we want to answer the question whether for the constant metrics there exist only one compatible linear connection. 7.1 Special cases of linear connection for the torus. We begin with considering a special case with all W being zero, which then enforces all C to be identically 1 and as a consequence the situation almost splits into the two parts related with the two algebras for discrete circles Z N and Z M . Indeed, for W = 0 first three relation from the list for the metric compatibility condition reduce to separate equations for groups Z N and Z M , whose solutions we have already found in the previous section. Furthermore, cotorsion-freeness implies that for such a case we have for all g and h = g, g −1 . Using this result in the fifth condition for metric compatibility we immediately infer that all functions C have to be constantly equal 1. As a result R h G g = G g for all g and h such that h = g, g −1 . We say that in this case the metric is perpendicularly constant. The remaining relations are automatically fulfilled. Furthermore, for W = 0 and C = 1 the connection ∇θ g = a,b Γ g a,b θ a ⊗ A θ b contains only terms Γ g a,b with a, b, g ∈ {p,p} or a, b, g ∈ {s,s} separately. Therefore, the Riemannian curvature splits into the sum of Riemannian curvatures for two discrete circles. Therefore, what we obtain, is the construction of the Riemannian geometry of the discrete torus with the product metric and as a result both Ricci tensor and the scalar curvature have similar behaviour.
7.1.2 The case of the constant metric.
The previous example shows the existence of nontrivial solutions in the case of the product geometry yet does not show that the solutions are unique. Therefore, for the second example we shall ask the question of all linear connections compatible with the constant metric. Suppose now that the metric coefficients satisfy G p = Gp = G s = Gs and are constant. By symmetry arguments we also assume that all A, B, C and W are also constant, which are identical (separately) for all A, B and W s (as there is symmetry in the change of the space), moreover we assume that C ps = C sp = Cps = Csp = C 1 and C ps = Cp s = Cs p = C sp = C 2 . The resulting system of equation is then BA − 1 = 0, and indeed has a unique solution: We infer from that at least in the case of the constant metric (which is the same for each of the components of the torus) there exists a unique metric compatible linear connection with certain symmetries. The more general case, with arbitrary constant coefficients leads to a huge number of nonlinear equations for 20 variables, which is difficult to solve. Therefore, the only method to proceed is step by step.
To see how this study is involved let us consider the most general case, with the assumption that all W s are different from 0. We still assume that G p = Gp = G s = Gs are constant, likewise all C and W and suppose now all W are non-zero. Moreover, we do not impose B to be constant here. The fifth relation in metric compatibility can be now written in the form where the cotorsion-freeness was used in a completely similar manner as we did it in the previous cases. Changing g into g −1 or h into h −1 , it follows that W g = W g −1 and W h = W h −1 . The fourth relation for metric compatibility can be therefore written as so by changing h ↔ h −1 and subtracting resulting equations we get B h = B h −1 and similarly also B g = B g −1 . Therefore the above equation reduces to B g + 1 B h = 1. The third relation for metric compatibility is now of the form, hence by the symmetry of the second term we infer .
Using now B g + 1 B h = 1 it can be reduced to an algebraic equation which has three solutions: Since we still have an analogue of equation (5.16), R g B g = 1 Bg , the first solution is excluded. From the second one we deduce that the function B g can take values only in the set 1 2 , and if B g (n, m) = 1 2 (1 ± i √ 3), then B g (n + 1, m) has to be equal to 1 2 (1 ∓ i √ 3). Obviously such solutions are possible only if N is even. Moreover, from 1 Bg + B h = 1 we can deduce also that B h = B g , hence R h B g = 1 Bg , so we have a similar behaviour also in the second argument. As a result there are two possible solutions: and moreover the existence of such solutions requires both N and M to be even. Furthermore, in such a case we have 2W h W g = −1, which determine the values of W s (and, using cotorsion-freeness, also of Cs). Indeed, using the fifth relation for metric compatibility (which now is of the form W g (2 + W h + W g ) = 0) together with the condition 2W h W g = −1 we get W g = −1 − 3 2 and W h = 3 2 − 1, or with the exchanged role of indices h and g.
Therefore at least one of W s needs to vanish unless both N and M are even when the aforementioned possibility occurs, however, if it is not the case, we shall see that it is not possible that only one of W is zero. Indeed, suppose the contrary, i.e. without loss of generality assume that only W h −1 = 0. First notice that the last relation in cotorsion-freeness implies Applying the above relation (together with R h A h = 1 B h −1 and the analogue of (5.16) which is still valid here) in the fourth condition for metric compatibility we get Since W h −1 = 0 we get B h −1 = 1 or W h = 0. Since we had assumed only one W vanishes, B h −1 = 1.
Applying the same technique to the third condition for metric compatibility as for the fourth one, we infer 1 B g − 1 (B g −1 − 1) + W g (W h + W h −1 ) = 0.
Replacing g with h, and using B h −1 = 1, W h = 0 we end up with W g = −W g −1 . If W g = 0, then also W g −1 . In this case the fifth condition for metric compatibility (after using cotorsion-freeness) reduces to W g −1 2 + W g −1 = 0, so both W g −1 and W g (since W h , W h −1 = 0) are equal to −2. But the only possibility to satisfy W g = −W g −1 is now W g = W g −1 = 0, which is a contradiction. Therefore, the claim is proven. Furthermore, notice that since from W h = W h −1 = 0 we were able to deduce that W g = W g −1 = 0, it follows that if one W vanishes then there exists at least one pair of vanishing W s: W a = W a −1 = 0. Therefore it is not possible that exactly two of W with indices in different algebras vanish simultaneously.
Therefore even in the case of a constant metric, such that the lengths of sides are the same in all direction, the solution is not uniquely determined by the requirement of torsion-freeness and metriccompatibility. In addition to the trivial solution with all W being zero (which reduces to the case discussed in the previous subsection), there are also other possibilities, e.g with W h = W h −1 = 0 and W g = W g −1 = −2. For N and M with even parities, there are even more sophisticated solutions with alternating functions B.

Conclusions and overview
In this paper we posed the question, whether it is possible to classify all linear connections over the minimal differential calculi on the finite cyclic group that are torsion-free and compatible with a given metric. Surprisingly, even though the problem is nonlinear the answer is positive yet only possible for a class of metrics that are either proportional-symmetric (left and right metrics are proportional to each other) or satisfy very special relations that are quantized. However, only the proportional-symmetric solutions are meaningful in the sense of Riemann geometry, as only they can lead to a norm on the space of one-forms. The resulting linear connections yield a nontrivial scalar curvature for the Riemannian geometry of the discretized circle, which has an interesting continuous limit.
The extension of the construction of bimodule connections and compatible metrics to the products of two discretized circles leads to a highly nontrivial set of compatibility conditions and this paper only scratches the surface of the problem. Yet, we were able to show that for the constant metric there exists at least one torsion-free linear connection that is compatible with it. This example shows that torsionfreeness and metric compatibility are not so restrictive conditions as in the classical situation and even in the simplest case we can have a plenty of non-trivial solutions.
There remain two important problems: the uniqueness of the linear connection for a class of nonconstant metrics as well as the relation of the computed scalar curvature to the spectral analysis through the Dirac operator [3,4,21] for discretized models, which we leave for the forthcoming work.