Invariant connections on Euclidean space

We recall and solve the equivalence problem for a at C1 connection ∇ in Euclidean space, with methods from the theory of di erential equations. The problem consists in nding an a ne transformation of Rn taking ∇ to the so called trivial connection. Generalized solutions are found in dimension 1 and some speci c problems are solved in dimension 2, mainly dealing with at connections. A description of invariant connections in the plane is attempted, in view the study of real orbifolds. Complex meromorphic connections are shown in the cone cL(p, q) of a lens-space.


Introduction
We wish to study linear connections ∇ on R n which are less than smooth from the point of view of the differentiable class, i.e. their Christoffel symbols are not C ∞ . We are particularly interested in observing the behaviour of the associated tensors, such as the torsion and curvature, and solving the equivalence problem in the framework of non-smooth connections on smooth manifolds. This is generically as follows: given two manifolds M 1 , M 2 endowed with linear connections ∇ 1 , ∇ 2 , when does there exist a diffeomorphism Φ : M 1 → M 2 such that Φ · ∇ 1 = ∇ 2 . The diffeomorphism is then called an affine transformation.
The equivalence problem is solved in [11] in the category of analytic manifolds with analytic connections, so it seems that the problem should be undertaken with PDE tools. Under mild conditions, we solve it for the case of the trivial connection in R n , leaving aside the demand of analyticity. For one particular example in R 2 we explicitly give the solution.
We also recall invariant connections for some groups of diffeomorphisms, i.e. groups of affine transformations for a same ∇. These are most relevant in the theory of symmetric spaces. Translations plus one isomorphism F invariant ∇ are studied in R 2 , in order to bring curvature and holonomy issues into the theory of orbifolds.
The equivalence problem is an old theme, as we may see e.g. in [9][10][11], yet its importance in geometry remains.
Orbifolds are a generalization of manifolds to include the notion of singularities of the kind of R n /G in 0, where G is a finite subgroup of GL n . Certainly any definition of connection in this new category will agree locally with a G-invariant connection in Euclidean space (cf. [7,8]) . We have shown that R 2 / F , where F is the conjugation map, admits a symplectic connection, torsion free, with non-vanishing curvature. Also we prove all foldings by conjugate-rotations of the plane F (z) = e iθ z admit some specific flat non-trivial connections. We have looked for translation invariant connections since they are easier to find. Though we should leave the translations invariance dependence, to have freedom in coordinates so that the question of which connection-curvature really interprets the orbifold singularity may be well posed.
To finish this article on the quest towards local invariant connections, we treat the case of lens spaces L(p, q) and their cone singularity. Here the case is of meromorphic objects and, indeed, we find a family of such conections, non-flat. We remark this new ∇ is just an unnoticed particular case within the whole subject of [12].

Linear connections
Let M be any paracompact smooth manifold and let X U denote the Lie algebra of smooth vector fields on an open subset U of M .
We recall the notion of a linear connection on a manifold M . It is given by a covariant derivative, i.e. an operator ∇ on the space of pairs of smooth vector fields X, Y defined on M , sending another smooth vector field ∇ X Y on M , and satisfying the following relations: From the first two conditions it follows that ∇ is a local operator (cf. [6]): if two vector fields Y 1 , Y 2 agree on some open subset U , then so do their covariant derivatives. To see this suppose Y = 0 on U , then for each point x ∈ U take a function f ∈ C ∞ M with supp f ⊂ U and f = 1 on a neighborhood of x (these functions exist always). Then f Y = 0 on M , so Contrary to other local operators, as for instance the Lie bracket of vector fields, the covariant derivative of Y ∈ X U induces a well-defined linear map ∇Y : T m M → T m M for any m ∈ U ; for each v ∈ T m M just take a chart (x 1 , . . . , x n ) around m and any smooth functions Then the previous facts and condition (iii) imply that we Y |m -which therefore does not depend nor on the chart, nor on the extension X of v.
The following two tensors are used in the study of linear connections. The torsion and the curvature These are tensors indeed, linear over the C ∞ U ring, as it is easy to prove. One can see the curvature as a measure of how covariant derivatives of Z commute, along the directions X, Y , up to the one along the Lie commutator of X and Y . The connection is called flat if R ∇ = 0. Obviously, T ∇ ∈ Ω 2 (T M ) and R ∇ ∈ Ω 2 (End T M ).
Connections determine geometry of manifolds by their ability to induce parallel displacement. In the tangent bundle they also give the notion of geodesics, i.e. curves γ which satisfy ∇ γ γ = 0 (we may deduce as above that the operator ∇ γ is well defined over a curve γ, i.e., if Y,Ỹ are vector fields on a neighborhood of γ such that Y γ =Ỹ γ , then ∇ γ Y = ∇ γ Ỹ ).
To finish, suppose we have two connections ∇ 1 , ∇ 2 . Then it is trivial to check that their difference is a tensor:

Diffeomorphisms action on connections
We recall here other well known facts about connections.
Let M, N be two manifolds and suppose Φ : M → N is a smooth diffeomorphism. Then Φ induces a linear map X → Φ · X defined by: ∀y ∈ N.
Proof. We just have to evaluate the action of the Lie bracket on smooth Hence for two vector fields on M Notice that, for any h ∈ C ∞ M , we have Φ · (hX) = (h • Φ −1 )Φ · X = (Φ · h)Φ · X, extending notation to functions. Also notice that formula (1.1) can be written as (Φ · X)(Φ · h) = Φ · (X(h)).
Given a diffeomorphism Ψ : An even more surprising property of the 'push-forward' map is that it acts on the space of connections. Given a connection ∇ on M we may define a new connection Φ · ∇ on N by for any Z, W ∈ X N . The only non trivial identity to check is the Leibniz identity: The action on connections under composition of two diffeomorphisms carries canonically, as it should: Let∇ be another connection on N . We recall that a map Φ which satisfies ∇ = Φ · ∇ is called an affine transformation. If it is an affine transformation of M onto itself, with∇ = ∇, then the connection is said to be Φ invariant. All these definitions are in [9] or [11].
As we have been showing, any given tensors transform under the pushforward map in an obvious way.
The following identities are easy to check: Under affine transformations, clearly unparametrized geodesics are taken to geodesics. A map which has such a property is called a projective transformation. This notion has been thoroughly studied in Riemannian geometry. Recently, V. S. Matveev proved the Lichnerowicz-Obata conjecture, stating that a connected group which acts projectively on a closed Riemannian manifold, then acts affinely (cf. the proof and the history of this conjecture in [13,14]). A close question dealing with projective metric structures in real dimension 2, is found in recent [3]. Our last section studies R 2 too.

Connections in R n
We shall now restrict our study to connections in Euclidean space. We change notation a bit and assume F : R n → R n is a diffeomorphism. Also we let (x 1 , . . . , x n ) or (y 1 , . . . , y n ) denote Euclidean coordinates and abreviate the induced vector fields ∂ ∂x i to ∂ i . This is just the vector e i of the canonical basis. Writing F (x) = y then From now on we assume Einstein's summation convention. Let ∇ be any connection. It is determined by the Christoffel symbols: Then the Christoffel symbolsΓ h ij of this new connection satisfy the equation: Proof. We have that On the left hand side we have, letting G = F −1 , Of course, we may write an equation analogous to (1.2) in terms of G = F −1 , since G ·∇ = ∇. Moreover, a given connection on a manifold satisfies such equation, withΓ = Γ, under any coordinate change diffeomorphism.
Given any∇ and ∇, when does there exist a diffeomorphism F which makes the two connections the affine transformation of one another? This is called the equivalence problem. In [11,chapter VI,theorem 7.4] it is proved that a solution to this problem exists locally if the connections have analytic Christoffel symbols and if higher order derivatives of the torsion and the curvature tensors satisfy , for all k = 0, 1, 2, ... and for a linear isomorphism φ : T x 0 M → T y 0 N . Moreover, the problem is solved globally in the restricted context of analytic manifolds M, N . By a local solution it is meant a diffeomorphism F : U → V from a neighborhood U of x 0 onto a neighborhood V of y 0 and such that dF x 0 = φ.
Remark: An interesting consequence of this result is the following. If M is a C ∞ manifold with a C ∞ linear connection such that ∇T ∇ = 0, ∇R ∇ = 0, then M is an analytic manifold and the connection is analytic [11,chapter VI,theorem 7.7]. This shows that all symmetric spaces are analytic manifolds.
Remark: There is another type of transformation of linear connections we want to be aware (this applies generally to connections on vector bundles, cf. [6]). The gauge transformations, which we recall in the case of U open in R n , are defined by a map u : U → GL n and act on connections ∇ = d + Γ almost like an "infinitesimal affine transformation" covering the identity map of the manifold. Namely, they are defined by Before we proceed, we recall here in local coordinates the formula for

Flat connections
It is easy to see the gauge transformation induces a conjugation by u of the curvature tensor, but the same is not true for the torsion. Proof. Let s 0 = (∂ 1 , . . . , ∂ n ). We first show that there is a solution of ∇s = 0, for a smooth frame s : U → (R n ) n , on an open neighborhood U of each point. Writing in matrix notation s = s 0 u and ∇s 0 = s 0 Γ, we have ∇s = (∇s 0 )u + s 0 du = s 0 (Γu + du).
Now Γu + du = 0 is a first order linear differential equation, which has a solution if, and only if, its exterior derivative is zero. Therefore we compute dΓ u − Γ ∧ du + d 2 u = 0 or, equivalently, dΓ + Γ ∧ Γ = 0. But, the reader may care to deduce this is the same as R ∇ = 0. The second part of the result is trivial to check.
In the following, let DF = [ ∂F i ∂x j ]. Here we state an approach to the equivalence problem for the trivial connection.
with k a constant invertible matrix.
Notice that, once we find F , we may incorporate k in u.
2 Existence results for the equivalence problem In R suppose we are given a linear connection ∇ = d + Γ, with Γ a 1-form with values in End R = R. Clearly, a 1-form on the real line corresponds to a function Γ 1 11 such that Γ x (v) = Γ 1 11 (x)v, ∀v ∈ R, and clearly the torsion and the curvature of ∇ both vanish. Nevertheless, we may still try to solve the equivalence problem. According to Proposition 1.2 we look for a diffeomorphism F such that (we let Γ = Γ 1 11 ) Noteworthy, with the most simple non-trivial connection, that is, with Γ a non-zero constant, we obtain the transformation where c 1 , c 2 are constants, which requires further notice on restrictions of the domain.
For the differential equation (2.1) with generic Γ, we may introduce the following weak variational problem.
Remark: The weak formulation (2.2) is obtained by the following computation: When boundary conditions are taken into account, (2.1) becomes a boundary value problem.
Proposition 2.1. Assume that Γ is a continuous real function such that

4)
where V is the set of functions G ∈ H 1 (a, b) such that G(a) = 0.
Proof. Assume that Γ(t)t ≤ α, for all t ∈ R. Otherwise the proof is analogous. Let us concentrate on the existence proof to the mixed boundary value problem (case 3) under the Galerkin method. The cases 1 and 2 are similar and simpler. Let A be the induced operator of the weak variational equality (2.4), i.e., A : V → V defined by Applying the Poincaré inequality, we recall that V is a separable Hilbert . Letting {H k } be a basis of V , we set the finite dimensional space as V N = H 1 , · · · , H N for N ∈ N. Using (2.3) it follows that A is coercive:

Then there exists a Galerkin solution F
for all G ∈ V N and, by density, for all G ∈ V . Taking G = F N in (2.5) we obtain Thus the Galerkin solution satisfies the estimate Thus we can extract a subsequence of F N , still denoted by F N , such that ([a, b]). In order to prove that F is a solution to (2.4) we will pass to the limit in (2.5) for all G ∈ V as N tends to infinity. To pass to the limit the term on the left hand side in (2.5), it is sufficient the weak convergence of F N to F in L 2 (a, b). Notice that this does not allow to pass to the limit the last term on the right hand side in (2.5). So to prove the strong convergence it remains to show that (2.6) First let us identify |F N | 2 as an element of the dual space of C([a, b]). Hence we can extract a subsequence of |F N | 2 , still denoted by |F N | 2 , weak-* convergent to χ in L 1 (a, b). Next passing to the limit (2. Now passing to the limit in (2.5) when G = F N is chosen, we get Finally gathering (2.7) and (2.8) we conclude (2.6).

The dimension n = 2 case
In dimension 2 we will find the integrability condition (1.4), for F belonging to C 2 . As a first case to study, we present the following example.
Example 1. We consider the symmetric and flat connection ∇ = d + Γ given by Γ 1 11 (x, y) = f (x), Γ 2 22 (x, y) = g(y), where f, g are C α , and any other Γ k ij = 0. Then ∇ is flat by trivial reasons. Solving ∇ = F · d with F ∈ C α+2 , implies solving for F 1 the system An analogous system must be satisfied by F 2 . Imposing further ∂ y F 1 = ∂ x F 2 = 0 we see that the problem is equivalent to solving the dimension 1 case. Next we present a non-constant example.

Example 2.
Consider the open set R + × R and a connection given by functions. An easy computation shows that ∇ is flat: Now the group-valued map u may be deduced from Γ = udu −1 = −(du)u −1 , i.e. the equations Henceforth we find that the equation in is the one to be solved, applying Proposition 1.4. Notice F 1 , F 2 = Jac F = 1. This is the case where the map u takes values in SL(2, R).
Find F 1 and F 2 in the forms F 1 (x, y) = e 2(f (x)−g(y)) , F 2 (x, y) = √ 2(f (x)+ g(y)); then we obtain the following equations Then we obtain where c 1 and c 2 are determined according to the domain. Considering c 1 = c 2 = 0, we obtain the function ) , x > 0, y < 0, solving our particular and ilustrative problem.

Remark: In Proposition 3.3, if u is such that
and u 11 and u 22 are functions in F 2 and F 1 , it results This is impossible. Then we conclude that the existence of a solution depends on u.
In conclusion, if we find (F 1 , F 2 ) of class C 2 we have the following restrictions on u: or, equivalently,

In dimension n
Here we state the existence result to Proposition 1.4. In order to adapt the proof of the generalized Frobenius Theorem [15, pp. 167], we rewrite (1.5) as with w denoting the inverse matrix of k. Let us begin by stating the following existence result.
and for any δ Moreover, the solution z verifies Proof. In order to apply the Schauder fixed point theorem [15, pp. 56], let us consider the ball, with radious R > 0, Let us construct the mapping L : ξ → z as follows From (2.9), it follows From (2.10), the derivative of z i with respect to x j verifies Thus, choosing L maps the ball B R into itself. Since L is a continuous mapping, in order to conclude that L is compact it remains to show that L maps bounded sets into relatively compact sets. Indeed, for any M ⊂ C 1 (Q) bounded set and observing that C 1 is compactly imbedded in C, every sequence {z m } ⊂ L(M ) contains a convergent subsequence. Thus the Schauder fixed point theorem guarantees the existence of z ∈ C 1 (Q) such that Lz = z. The derivative (2.12) is a consequence of Lz = z.
Now we are able to adapt the proof of the generalized Frobenius Theorem [15, pp. 167]. Note that the generalized Frobenius Theorem gives two equivalent statements requiring the existence of z ∈ C 2 . Theorem 2.1. Suppose that the assumptions of Proposition 2.2 are fulfilled. If additionally the integrability condition holds ∂u ij ∂ξ p u pq w qm w jl = ∂u ij ∂ξ p u pq w ql w jm (2.13) then there exists F ∈ C 1 satisfying (1.5). Moreover, if F ∈ C 2 then u verifies (2.13).
it satisfies the ordinary differential equation Introducing this relation and successively (2.11) of Proposition 2.2 in (2.15) we obtain Applying the assumption (2.13) it results the linear ODE v ij (t) = ∂u il ∂z p w lq x q v pj . (2.16) Thus the initial condition v ij (0) = ∂ j z i (0, x) = 0 implies that the ODE (2.16) has the unique solution v ≡ 0. Setting F (x) = z(1, x) and using (2.14) we get which concludes the proof of Theorem 2.1. Finally, for F ∈ C 2 , the condition (2.13) is due to the Schwartz property of functions of class C 2 .
If we change u to the matrix valued map u k, then we realize the integrability condition (2.13) is in fact the one on the torsion stated in (1.4).

Invariant linear connections
Given a linear connection ∇ on a manifold M , one may define the subgroup Diff (M, ∇) of affine transformations of ∇. It is still a problem to find its dimension, as well as that of the orbit of ∇ under Diff (M ) in the space of linear connections.
One may also try to determine the linear connections on a manifold M which are invariant under a given set of diffeomorphisms. If we have a Lie group G, then it is easy to produce G-left invariant connections as bilinear maps g × g → g, where g is the Lie algebra of left invariant vector fields (cf. example 2, section 1.2).
Translation invariant connections in R n are those for which Γ k ij are all constants. This is trivial to deduce from (1.2) applied to any map A homothety invariant connection is one for which as we may see taking F (x) = λx in the usual equation (1.5). If we want ∇ invariant for all λ, then ∇ is possibly very curved at the origin and certainly flat at infinity.

Over-determined systems of translation invariant connections in R 2
Now we are going to find linear connections in R 2 which are invariant for all translations plus one more single isomorphism F (x 1 , x 2 ) = (ax 1 + bx 2 , cx 1 + dx 2 ). In view of the case of orbifolds, we are going to assume det F = ±1 (we want the group generated by F to be finite). One can reduce the system restricting to some particular subspace of linear connections. For instance, torsion free: then the system (3.2) reduces to 6 equations in 6 variables, because Γ k 12 = Γ k 21 , k = 1, 2. Indeed, (1.2) is symmetric in i, j if ∇ = d + Γ is torsion free.

Metric connections.
A second case is that of metric connections with torsion (without torsion there is only the trivial, Levi-Civita connection): Thus there are only two unknowns and the system (3.2) is given by In order to have rk S < 2 we must have, e.g., Then we find non-trivial F given by F a,± (x, y) = ±(x, −y), or F b,± (x, y) = ±(y, x).
Proposition 3.1. F a,± and F b,± are the only non-trivial isomorphisms F of the plane for which there exist non-trivial metric, translation and F invariant connections.
The proof follows from the system above and the curvature computations are trivial. Notice we may state corresponding results for the minus cases.
Symplectic connections. Another interesting type of conections is that of symplectic torsion free connections: Γ k ij is totally symmetric when contracted with the 2-form ω = dx ∧ dy (see e.g. [2]), arising from the parallelism of ω, ∇ω = 0. This is the same as Γ 1 i1 = −Γ 2 i2 or, equivalently, Γ i ∈ sl(2, R). In sum, The rank of the essentially 6x4 matrix is less than 4 in situations our 'computer' does not obtain a pleasant result. But the case a = −d = 1, c = b = 0 is a solution. Then the connections are given by (3.3) and Γ 1 12 = Γ 2 11 = 0. According to (1.3) we find Complex connections. We also have the case of complex or gl(1, C)connections: This gives an over-determined system as above, still unsolved according to its rank. If we moreover demand Γ torsion free, then the system reduces to 2 unknowns: The condition for rk < 2 remains to be deduced, but if we require F ∈ GL(1, C), that is F (x, y) = (ax + by, −bx + ay), then F = Id is the only isomorphism which admits that kind of invariant connections. If we look for F of the previous kind, that is F (x, y) = (ax + cy, cx − ay) then the equations for rk < 2 resume to the vanishing of Equivalently, a 2 + c 2 = 1. Since we were hoping for det F = ±1 the result is automatic; thus we may write a = cos θ, c = sin θ, to find the condition (cos 2θ − cos θ)Γ 1 11 + (sin 2θ + sin θ)Γ 1 12 = 0.
In sum we have proved the following. The curvature is trivial since Γ is constant and a type (1, 0) form; since there are no type (2, 0) forms on the complex line, R ∇ = dΓ + Γ ∧ Γ = 0.
In truth, all translation invariant complex connections in C are flat, cf. formulae (1.3,3.4).

Invariant connections on orbifolds cL(p, q)
Let ∇ be a holomorphic connection in C n with coordinates (z 1 , . . . , z n ), i.e. its Christoffel symbols for ∇ ∂z i ∂ z j are holomorphic functions. Then the equations of an affine holomorphic transformation F are again determined by system (1.2) but with the x j replaced by holomorphic coordinates z j = x j + iy j . Indeed, since F · ∂ z j = 0, we must have where as usual ∂ z j = 1 2 (∂ x j − i∂ y j ), ∂ z j = ∂ z j . We recall the lens space L(p, q) = S 3 /Z p , the orbit space for the action of F (z 1 , z 2 ) = (az 1 , dz 2 ) on the 3-sphere, with a, d ∈ C such that a p = 1, d = a q . In the study of the cone with a singularity cL(p, q) = {λz : z ∈ L(p, q), λ ∈ R + } = C 2 / F , there are invariant connections with meromorphic coefficients, as we shall see in the following example.
Example. cL(p, q) admits a meromorphic connection with Christoffel symbols For the proof, notice that, although ∇ is not translation invariant, we may still formally use system (3.2) viewing the Γ's composed with F on the left hand side. Then essentially two types of equation appear: aΓ 1 11 • F = Γ 1 11 , a 2 Γ 2 11 • F = dΓ 2 11 , and these equations have obvious solutions.
The search of meromorphic connections on orbifolds was studied in [12]; anyway our example seems to be original. The use of connections in this context has appeared in [7,8] We remark that the classification of orbifold singularities with complex structure is still an open problem and there are various approaches to it either through the Riemannian or the complex perspective -cf. [1,4,5] and the references therein to see the wealth of examples and geometries one might continue searching for.