Affine analogues of the Sasaki-Shchepetilov connection

Abstract For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.


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Maria Robaszewska in such a way, that the integrability condition dΩ − Ω ∧ Ω = 0 is equivalent to the structural equations dω 1 = ω 2 1 ∧ ω 2 , dω 2 = −ω 2 1 ∧ ω 1 and dω 2 1 = ω 1 ∧ ω 2 of a pseudospherical surface (K = −1). This sl(2, R)-valued 1-form Ω itself can be interpreted as the connection form of a connection on some principal SL(2, R)bundle. The condition dΩ − Ω ∧ Ω = 0 means that the curvature of this connection vanishes. In this respect the connection Ω differs from the Levi-Civita connection of the considered pseudospherical metric. On the other hand, Ω appeared to be somehow related to the Levi-Civita connection, because the Levi-Civita connection form 0 −ω 2 1 ω 2 1 0 "is contained" in Ω. As might be expected, the question of finding the geometric interpretation of Ω occurred.
In the paper [10] A.V. Shchepetilov explained the geometric meaning of the Sasaki connection. Using an equivalent representation of Ω, so(2, 1)-valued, he constructed a flat connection ∇ on the vector bundle T M ⊕ E, where T M is the tangent bundle and E = R × M is a trivial one-dimensional vector bundle (our notation is slightly different from that in [10]) Here g is a metric on M , ∇ is its Levi-Civita connection, f ∈ C ∞ (M ) is a section of E and X, Y are vector fields on M . Shchepetilov considered also manifolds with metric of constant positive curvature K = +1. The corresponding flat connection ∇ on T M ⊕ E is The aim of this paper is to construct a similar flat connection ∇ for a twodimensional manifold with non-metrizable locally symmetric connection ∇ and with ∇-parallel volume element. Our main motivation for research is as follows. Firstly, manifold with locally symmetric linear connection can be thought of as a generalization of a constant sectional curvature Riemannian manifold. Secondly, sometimes more important than (M, g) or (M, ∇) alone is an embedding of M into R 3 . For example, every isometric embedding of a pseudospherical surface (M, g) into R 3 corresponds to some particular solution of the sine-Gordon equation. Therefore restriction to those non-flat locally symmetric connections which are induced on hypersurfaces in R n+1 is legitimated. If such hypersurface f is degenerate and its type number r is greater than 1, then around each generic point of M there exists a local cylinder decomposition which contains as a part a non-degenerate hypersurface in R r+1 with some locally symmetric connection (see [4]). On the other hand, if f is non-degenerate and n > 2, then ∇ is the Blaschke connection, ∇h = 0, S = ρ id, ρ = const, ρ = 0 and f (M ) is an open part of a quadric with center [4]. Similarly as in the second proof of Berwald theorem in [3] one can then define a pseudo-scalar product G in R n+1 such that , G(f * X, ξ) = 0 and G(ξ, ξ) = ρ, where ξ is the affine normal. It is easy to check that relative to this pseudo-scalar product f is a hypersurface of constant sectional curvature ρ. If f is non-degenerate, n = 2 and the induced locally symmetric connection satisfies the condition dim im R = 2, then there also exists a pseudo-scalar product on R n+1 = R 3 relative to which f has constant Gaussian curvature and ξ is perpendicular to f [6].

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On the contrary, if f : M → R n+1 is of type number 1 or if f : M → R 3 is nondegenerate and dim im R = 1, then the connection as a connection of 1-codimensional nullity (dim ker R = n − 1) is not metrizable [7], therefore we have reason for generalizing Shchepetilov's construction. The present paper deals with the case n = 2.

Preliminaries
Let M be a connected two-dimensional real manifold and let ∇ be a locally symmetric connection on M , satisfying the condition dim im R = 1, where for and R is the curvature tensor of ∇. Such connections were studied by B. Opozda in [5]. Opozda proved that for every p ∈ M there is a coordinate system (u, v) around p such that where ε ∈ {1, −1}. A local coordinate system in which a locally symmetric connection ∇ is expressed by (3) will be called a canonical coordinate system for ∇ [5]. Here we assume that a ∇-parallel volume element vol exists on the whole M .
It follows, that for every p ∈ M we can find around p a local basis (X 1 , X 2 ) of T M , satisfying the conditions: Ric(X 2 , X 2 ) = ε and vol(X 1 , X 2 ) = 1.
For example, on the domain of canonical coordinates (u, v) as in (3) we may take X 1 = 1 c ∂ u and X 2 = ∂ v , where c is the non-zero constant such that vol = c du ∧ dv. Let ω 1 , ω 2 be the dual basis for (X 1 , X 2 ). The local connection form is (ω i j ) = 0 ω 1 2 0 0 and the structural equations are dω 1 = −ω 1 2 ∧ ω 2 , dω 2 = 0 and The following proposition is easy to check.

Proposition 2.1
Let M be a two-dimensional manifold with locally symmetric connection ∇ satisfying condition dim im R = 1. Let ω 1 , ω 2 and ω i j be the dual basis and the local connection forms for some local basis of T M satisfying the condition (4). Then each of the following four 1-forms Ω i

Maria
Robaszewska In the construction of P and Ω in [8] and in the present paper we consider the left action of H on Q: a * q : ∈ H, and some left action of G on P . Another possible way is to consider traditionally a right action, but we have then −Ω instead of Ω.

The connections on the vector bundle T M ⊕ E
We will use the definition of the covariant derivative of a section of an associated bundle which comes from [1], and is described for example in [2]. Since we consider here the left action of G on P and the right action of G on R N , z * c := c −1 z, some details may be different from that of [1] and [2].
Let T M be the tangent bundle of M and let E be the trivial bundle, Since G acts transitively on fibres of P , there exists b ∈ G Affine analogues of the Sasaki-Shchepetilov connection To each local section η of an associated vector bundle P × G R N corresponds some mapping η : P | U → R N -called the Crittenden mapping -which satisfies the condition η(bp) = η(p) * b −1 . Since we have actually defined the right action of G on R N using the left action, x * c := c −1 x, we can write this condition simply as η(bp) = b η(p). By definition of the Crittenden mapping, [(p, η(p))] = η(π(p)). Conversely, to each mapping η : Let X be a vector field on M . For every connection form Ω i from Proposition 2.1 we will find the covariant derivative ∇ X η of a local section η of T M ⊕E. and with the (1, 1) tensor field L such that vol(LX, Y ) = Ric(X, Y ) for every X, Y .
Proof. By definition of the covariant derivative, the Crittenden mapping corresponding to ∇ X η is equal to X H ( η), where X H is the horizontal lift of X to P | U . We use a local section Let Ω be the connection form on P . The local connection form is τ * Ω = Ω σ . We have , where the right-invariant vector field B = −Ω σ (X x ), which we easily obtain from the condition Ω(X H τ (x) ) = 0: The first part of X H τ (x) ( η) is equal to , X x (Ψ)). [42]

Maria Robaszewska
The second part is Here (b t ) is 1-parameter subgroup of G generated by B. It follows that For Ω σ = Ω 1 we obtain We have also because V 1 is a local section of ker Ric. We obtain finally If we take Ω σ = Ω 2 , then we obtain from (5)
Note that for every Z we have For the second connection we finally obtain the global formula For Ω σ = Ω 3 we have For Ω σ = Ω 4 we obtain Proof. We will compute
But Ric is symmetric, ∇R = 0 implies ∇ Ric = 0, and for each two-dimensional manifold For the connection (8) we obtain LX)) .

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For the connection (9) we obtain Note that im L ⊂ ker Ric. For (10) we have

Some remarks about interpretation of ∇
As is shown in [10], in the metric case using (at least local) embedding of (M, g) with K = ±1 into Euclidean or pseudoeuclidean space E we may identify ∇ with the restriction of the flat connection on T E = E × E to E × M and identify the trivial one-dimensional summand E with the normal bundle of the surface.
We consider now the case of non-metrizable locally symmetric connection on M , dim M = 2. Let f : M → R 3 be an immersion and let ∇ be the connection induced on M by f and the transversal vector field ξ. If we identify the bundle f * (T M )⊕R ξ with T M ⊕E, then to the vector field f * (Y )+Ψξ corresponds the section Y ⊕ Ψ of T M ⊕ E. The Gauss and Weingarten formulae yield that to where h is the affine fundamental form, S is the shape operator and τ is the transversal connection form (see [3] for the definitions). We look for f and ξ such that D = ∇. Comparing the right-hand side of (14) with that of (6) and (8) for the section 0 ⊕ 1 gives τ = 0, which means that we may restrict ourselves to equiaffine transversal vector fields. Furthermore, since h is always symmetric and vol is anti-symmetric, we see that there are no f and ξ which allow to identify in the above described way the connection (8) with the standard connection D on the bundle R 3 × M .
As concerns (6), it should be h = Ric, which implies that we should consider some realization of ∇ on a degenerate surface f with the type number tf equal to 1. Such realizations were described by B. Opozda in [7]. Using a general description given in Proposition 6.2 of [7] and claiming that ξ = −f , we easily obtain the following particular local realizations of ∇ and Here u, v is some fixed local canonical coordinate system for ∇. The volume element vol = du ∧ dv is the element induced by (f, ξ) from R 3 . For a centro-affine immersion (f, ξ = −f ) and n = 2 we have SX = X and . It follows that using the immersion (15) or (16) we may identify (6) with the standard connection D.
Similarly as it was for (6), the above immersion f is degenerate. By definition (see [3]), an immersion f : M 2 → R 4 is non-degenerate if the symmetric bilinear function G σ is non-degenerate. For a local frame field σ = (X 1 , X 2 ) the function G σ is defined by the formula (cf [3]) It is impossible to obtain in a similar way the connection (10), because vol is anti-symmetric.

Some further remarks
In general, to each immersion (f, ξ) and to each local basis σ = (X 1 , X 2 ) of T M corresponds some GL(3, R)-valued 1-form Ω σ Here ω i j are local connection forms of the induced connection and S = S 1 (·)X 1 + S 2 (·)X 2 is the shape operator. The condition dΩ σ − Ω σ ∧ Ω σ = 0 is equivalent to the fundamental Gauss, Codazzi and Ricci equations. The formula (5) gives on T M ⊕ E a flat connection D described by formula (14).
The considered in the present paper 1-forms Ω i were constructed as satisfying additional condition Ω i = Aω 1 + Bω 2 + Cω i j with constant A, B and C. For given Ω σ such constant A, B and C may not exist, in such a case the connection D is always different from ∇. For example, (M, ∇) can be affinely immersed also as a non-degenerate surface in R 3 . Such immersions and transversal fields are described in [5]. If we use one of them, then we obtain D different from (6) and (8).
For each given connection ∇ on M , for each (1, 1) tensor field A and (0, 2) tensor field α we can define some connection ∇ A,α on T M ⊕ E by the formula We may look for such connections ∇ for which there exist A and α such that ∇ A,α is flat.
It is easy to compute

Summary
For a locally symmetric connection ∇ with one-dimensional im R we have constructed two flat connections on the vector bundle T M ⊕ (R × M ) and two flat connections on T M ⊕ (R 2 × M ). From each pair only one connection may be identified with the standard connection in R N , N = 3 or N = 4, after suitable local embedding of (M, ∇) into R N . Those embeddings are degenerate.