O ct 2 01 9 Geometric Regularity Results on B kα , β-Manifolds , I : Affine Connections

In this paper we consider the existence problem of affine connections on C-manifolds M whose coefficients are as regular as one needs. We show that if M admits a suitable subatlas, meaning a B α,β-structure for a certain presheaf of Fréchet spaces B and for certain functions α and β, then the existence of such regular connections can be established. It is also proved that if the B α,β-structure is actually nice (in the sense of [1]), then a multiplicity result can also be obtained by means of Thom’s transversality arguments.


Introduction
As it is very well known, any paracompact C k -manifold (M, A), with k ≥ 2, admits many affine connections ∇ whose coefficients are of class C k−2 [2].Recall that this very standard result is obtained as follows.Let Conn k−2 (T M ) be the set of affine connections of class C k−2 in M .One first shows that Conn k−2 (T R n ) in nonempty for every k.One then show that each C k -map f : N → M induces a pullback function f * : Conn k−2 (T M ) → Conn k−2 (f * T N ), from which one conclude that open sets of R n admit affine connections of class C k−2 , for every k.In particular sets C k -diffeomorphic to open sets of R n admits C k−2 -connections.Then, for a given open covering U by coordinate systems ϕ i : U i → R n of M one takes a subordinate partition of unity ψ = (ψ i ) i and one defines ∇ = i ψ i • ∇ i , where ∇ i ∈ Conn k−2 (T U i ), so that Conn k−2 (T M ) is nonempty.Since the difference of C k−2 -connections is C k (M )-linear, one conclude that Conn k−2 (T M ) is an affine space of Ω 1 (M ; End(T M )).
In many situations, e.g, in gauge theory and when we need certain uniform bounds on the curvature, it is desirable to work with connections ∇ whose coefficients are not only C k−2 , but actually L p -integrable or even uniformly bounded [3,4].In other situations, e.g, in the study of holomorphic geometry, we consider holomorphic or hermitian connections over complex manifolds, whose coefficients are holomorphic [5,6].If M has a G-structure, we can also consider G-principal connections ω on the frame bundle F M , which induce an affine connection ∇ ω in M , whose coefficients ω c b (∂ a ) = Γ c ab are such that for every a, Γ a ∈ C k−2 (U ) ⊗ g, where (Γ a ) cb = Γ c ab [7].Interesting examples of these are the symplectic connections, which are specially important in formal deformation quantization of symplectic manifolds [8,9].As a final example, a torsion-free affine connection defines a L 3 -space structure in a smooth manifold M (in the sense of [10]) iff its coefficients satisfy a certain system of partial differential equations [11].
Thus, if ρ represents some "regularity condition", e.g, one of the previous ones, let Conn ρ (T M ) ⊂ Conn k−2 (T M ) be the subset of those affine connections whose coefficients satisfy ρ.It is then natural to ask when Conn ρ (T M ) is nonempty or when it remains an affine space.More precisely, it is natural to ask when the construction above can be applied ipsis litteris in order to prove that Conn ρ (T M ) is an affine space.This is a nontrivial question, since the existence of such a regular connection may imply geometric and topological obstructions on M [4,6].In this article we attack this problem in the context of B k α,β -structures introduced in [1].More precisely, for us the "regularity condition ρ" is represented by sets Γ ⊂ Z ≥0 and S ⊂ Γ, by integer functions α 0 : S → Γ and β 0 : S → [2, k], and by a family B = (B i ) i∈Γ of presheaves of nuclear Fréchet spaces B i : Open(M ) op → NFre on M .Saying that an affine connection ∇ on (M, A) satisfies the "regularity condition ρ" described by these data means that there exists a subatlas B ⊂ A and an intersection structure presheaf (ISP) X 0 over M such that for every ϕ : U → R n in B ρ the induced coefficients (Γ ϕ ) c ab are such that ∂ µ (Γ ϕ ) c ab ∈ B α 0 (i) (U ) ∩ X 0 (U ) C k−β 0 (i) (U ) for every i ∈ S and every µ such that |µ| = β 0 (i).The space of these regular connections is denoted by All above examples of connections satisfying regularity conditions can be put in this language.
We prove that, given this "regularity condition" ρ = (B, S, α 0 , β 0 , X 0 ), if the structural presheaf B is suitable and if (M, A) admits a (B k α,β , X)-structure compatible with α 0 and β 0 , then there exists an ISP X 0 , compatible with X, such that for every (θ, ϑ) close enough to (α 0 , β 0 ) the space Conn k θ,ϑ (M ; X 0 ) is nonempty.In more details, we prove the following result: Theorem A. Let (M, A, B k α,β (X)) be a (B k α,β , X)-manifold, with k ≥ 2, and suppose that the structural presheaf B is such that B k α,β (X) admits a nice distributive (α ′ 0 , β ′ 0 ; j)-connection of degree r ≥ 2, say for a proper full ISP X ′ .Then for every ordinary pair (θ, ϑ) and every z ∈ Γ ≥0;β ′ 0 there exists a (B k θ,ϑ , X ′ |Γ ≥0;β ′ 0 −z )-connection in M .The result above is about existence of regular connections.The second natural question is about multiplicity, i.e, what is the size of their subspace.About this problem we prove the expected fact that the difference ) is an affine space over the R-vector space Ω 1 α 0 ,β 0 (M ; X 0 ) ⊂ Ω 1 (M ) of these 1-forms.This is not enough for ensuring the existence of many linearly independent connections, since a priori we have no information on the the space Ω 1 α 0 ,β 0 (M ; X 0 ).Nonetheless, we also prove a less obvious fact: in the smooth context, i.e, for k = ∞, if the structural presheaf B satisfies an additional closure condition, then there exists a map which has no local fixed points.More precisely, we say that two affine connections ∇ and ∇ are locally different and we write ∇ ≁ ∇ if for every ϕ : U → R n in B and every a, b, c = 1, ..., n there exists U abc ⊂ U such that Γ c ab (p) = Γ c ab (p) for each p ∈ U abc .We say that F has no local fixed points if F (∇) ≁ ∇ for every ∇ ∈ Conn ∞ θ,ϑ (M ; X 0 ).The existence of such F is consequence of the following theorem.
Theorem B. In addition to the hypotheses of Theorem A, suppose that M is smooth and has countably many connected-components, that B admits a nice distributive (α ′ 0 , β ′ 0 ; j)-connection of degree r ≥ 2, say in a proper ISP X ′ , and that B is (θ, ϑ|Γ ≥0;β ′ 0 −z )-strong in X ′ for some ordinary (θ, ϑ) and some z ∈ Γ ≥0;β ′ 0 .Then, for every The paper is organized as follows.In Section 2 we quickly present the needed background for our main results, which are basically concepts introduced in [1].These concepts can be understood in categorical terms by means of functors and commutative diagrams for natural transformations.Here, in order to make a more direct relation with the results of the next sections, these concepts will be revisited in the objectwise approach, via functions and equations.In Section 3 the notion of (B k α 0 ,β 0 , X 0 |S)-connection is formally introduced and Theorem A is proved.In Section 4 the additional conditions appearing in Theorem B are discussed and a proof of Theorem B is given.
A distributive Γ-space consists of a Γ-space B endowed with two maps ǫ, δ : Γ ≥0 × Γ ≥0 → Γ ≥0 and continuous linear maps , where ⊗ denotes the projective tensor product, which are both left-compatible and right-compatible, in the sense that for every x ∈ B i , y ∈ B j and z ∈ B k , and every i, j, k ∈ Γ ≥0 the following equations hold (notice that the first two makes sense only because of the last two): We also require that + ii coincides with the sum in B i .The pairs ( * , ǫ) and (+, δ) define a multiplicative structure and an additive structure in B, respectively.We have a category NFre Γ, * ,+ of distributive Γ-space whose morphisms between (B, +, * , ǫ, δ) and (B ′ , + ′ , * ′ , ǫ ′ , δ ′ ) are pairs (f, µ) which are simultaneously a morphism of (Γ, ǫ)-spaces (B, ǫ) → (B ′ , ǫ ′ ) and (B, δ) → (B ′ , δ ′ ), in such a way that A full Γ-ambient is a triple (X, γ, γ Σ ), where X is a category with pullbacks and γ and γ Σ are faithful functors γ : NFre Γ → X and γ Σ : NFre Γ×Γ → X.The second functor extends to (B, ǫ)-spaces by defining γ Σ (B, ǫ) = γ Σ (B ǫ ).Let X, X ǫ ∈ X and given spans of monomorphisms γ(B) ֒→ X ←֓ γ(B ′ ) and ǫ be the corresponding pullbacks.If B and B ′ are actually distributive Γ-spaces, we have additional spans for their multiplicative and additive structures, as presented below for the multiplicative case.The full data consisting of the full Γ-ambient and the spans above is called a full intersection structure between B and B ′ and denoted by X.The object X is the base object of X and the tuple (X, X ǫ , X * , X + ) is the full base object of X.
We say that a full Γ-ambient is proper if for every set of spans as above there are a Γ-space B ∩ X B ′ and (Γ × Γ)-spaces pb( * , * ′ ; X * ) and pb(+, The triple consisting of those representing objects in NFre Γ × NFre Γ×Γ × NFre Γ×Γ is the intersection space between B and B ′ in X.We say that B and B ′ have nontrivial intersection in X if both B ∩ X B ′ , pb( * , * ′ ; X * ) and pb(+, + ′ ; X + ) have positive real dimension.As established in [1], this is the case if X is a standard vectorial intersection structure and γ(B) and γ(B ′ ) have nonempty intersection when regarded as sets (see there for more details).
All the discussion above can be internalized in the presheaf category of presheaves in R n .For instance, we define a presheaf of (Γ, ǫ)-spaces in R n as a functor B : Open op (R n ) → NFre Γ,Σ and a presheaf of distributive Γ-spaces as a functor B : Open and (K l ) some nice sequence of compact sets.The distributive structure is given by pointwise addition/multiplication, with ǫ(i, j) = min(p(i), p(j)) = δ(i, j).Of special interest are the cases m = n, m = 1 and p the intersection is closed by pointwise multiplication by a certain space of smooth functions.If it is possible to choose A with nontrivial intersersection with the sub-presheaf we say that we have a presheaf of (B, k, α, β)-functions in (X, m) and write Throughou this paper we will work with presheaves of (B, k, α, β|S)-functions in (X, m) for m being some multiple of n, i.e, m = r • n for some r > 0. Thus, when there is no risk of confusion we will omit m, leaving it implicit in each context.
A C k -manifold is a paracompact Hausdorff topological space M endowed with a maximal atlas A, say with charts ϕ i : As proved in [1], under certain hypothesis on B, α and β, these additional structures exist in any for every ϕ ∈ A and φ ∈ A ′ .If S = Γ ≥0 we talk about (B k α,β , X)-structures, (B k α,β , X)-manifolds and (B k α,β , X)-morphisms.Remark 2.2.Throughou this paper we will always assume that sets Γ parametrizing the (Γ, ǫ)spaces have a large enough degree, as required for making sense of the definitions and results of the next sections.Sometimes an estimative for the degree will be explicitly given; in other cases the degree will be implicit.This hypothesis on the degree, however, is not much restrictive.Indeed, if B is any Γ-space with degree deg(Γ) and X ⊂ Z ≥0 is any subset of integers we can define another Γ Thus, given a (Γ, ǫ)-space we can always extend it by adding trivial (Γ, ǫ)-spaces with arbitrarily large degree.

Existence
• From now on we will assume that the (B k αβ , X)-structures ) be the category whose objects are coordinate systems (U, ϕ) ∈ B k α,β (X) and there exists a unique morphism (U, ϕ) ) be the category whose objects are continuous functions f : U → R, where (U, ϕ) ∈ D(B k α,β (X)).Similarly, there exists a morphism g : In the following we will work with a 3-parameter family Ω c ab of them, with a, b, c = 1, ..., n.We will write (Ω ϕ ) c ab to denote Ω c ab (U, ϕ).
Theorem 3.1.Let X be a proper full ISP and let (M, A, B k α,β (X)) be a nice (B k α,β , X)-manifold, with k ≥ 2. Given functions α 0 and β 0 as above, there exists a connection ∇ in M whose coefficients in each (ϕ, U ) ∈ B k α,β belongs to where and (ǫ, δ) is the distributive structure of B.
Proof.Recall that any family of X) or not), so that the existence of ∇ is locally established.In order to globalize it, let (ϕ s , U s ) s be an open covering of M by coordinate systems (not yet in B k α,β (X)) and let ∇ s be a connection in U s with coefficients (f s ) c ab .Let (ψ s ) be a partition of unity subordinate to (ϕ s , U s ) and recall that ∇ = s ψ s • ∇ s is a globally defined connection in M .We assert that if ϕ s are in B k α,β (X), then ∇ is a well-defined connection whose cofficients belong to (1) for certain U ′ .Notice that the coefficients (Γ s ) c ab of ∇ in a fixed ϕ s are obtained in the following way: let N (s) be the finite set of every s ′ such that For each s ′ ∈ N (s) we can do a change of coordinates and rewrite (f s ′ ) c ab in the coordinates ϕ s as given by the functions where by abuse of notation By hypothesis (f s ′ ) l mn is a (B, k, α 0 , β 0 )-function in X.Furthermore, since ϕ s ∈ B k α,β (X) we have where U s ′ ss ′ = ϕ s ′ (U ss ′ ).This is well-defined since β(2) ≤ k.Due to the compatibility between the additive/multiplicative structures of B α and C k−β , the first and the second terms of the right-hand sice of (3) belong to respectively, so that the left-hand side of (3) belongs to where -presheaf in X, the sum (4) belongs to the same space.Repeating the process for all coverings in B k α,β (X) and noticing that α ′ and β ′ do not depend on s, s ′ , we get the desired result.
We would like to extend the connection ∇ in the last theorem in order to get a genuine (B k α ′ ,β ′ , X|S)-connection, where α ′ and β ′ are functions such that α ′ (0) = α ′ 0 and β ′ (0) = β ′ 0 .We will prove that this can be done under certain proper hypotheses on the underlying nice C k,β n,αpresheaf at the cost of making a change in the ambient presheaf X.
Given B and B ′ presheaves of Γ-spaces and given functions θ, ϑ, θ ′ , ϑ ′ : Γ ≥0 → Γ ≥0 , let X and Y be proper ISP which are both between B θ and B ′ θ ′ , and between B ′ ϑ and B ′ ϑ ′ .We say that X and Y are compatible if they are defined over the same Γ-ambient.A (θ, ϑ; θ ′ , ϑ ′ )-connection between B and We are interested in certain "universal connections", as follows.Let (X, γ, γ Σ ) be a full Γ-ambient and let ∩ X,γ,γ Σ (B, B ′ ) be the presheaf category of presheaves of base spaces of proper ISP between B and B ′ which are compatible and defined over (X, γ, γ Σ ).Furthermore, let ∩ X,γ,γ such that s(D(X)) = X and such that the diagram below commutes for every f = (θ, θ ′ , ϑ, ϑ ′ ) ∈ O × Q.Thus, the dotted arrow exists (compare with the definition of connection between ISP given in [1]).If this universal arrow is created by the functor γ (e.g, if γ create pullbacks), then there exists ξ : connection for which these universal arrows are created.
Proof.First of all, notice that Γ ≥0 (β ′ 0 ; j) = Γ ≥0;β ′ 0 for every j.Furthermore, let W ⊂ Γ ≥0;β ′ 0 × Γ ≥0 be the subset of each (z, l) such that z + l ∈ Γ ≥0;k , so that W [z] is the set of every 0 Finally, apply Lemma 3.1.Suppose now that B and B ′ are actually presheaves of distributive Γ-spaces and X, Y are proper full ISP.In this case, define a distributive (θ, ϑ; θ ′ , ϑ ′ )-connection as one which preserves the multiplicative and additive structures, in the sense that there are objectwise monomorphisms making commutative the diagram below, and also objectwise monomorphisms A distributive (α 0 , β 0 ; j)-connection in B is a (α 0 , β 0 ; j)-connection in B which is distributive when regarded as a (D O , D Q )-connection.We say that a distributive (α 0 , β 0 ; j)-connection is nice if in the diagram below the dotted arrows, arising as natural fatorizations of the upper and lower paths in the diagram, are equal for every θ ∈ O and ϑ ∈ Q, where u arises from universality of pullbacks (the non-identified arrows are inclusions).In other words, if the (α 0 , β 0 ; j)-connection preserves bump-functions.Let (M, A, B k α,β (X)) be a (B k α,β , X)-manifold.A (α 0 , β 0 ; j)-connection of degree r in the (B k α,β , X)-structure B k α,β (X) is a (α 0 , β 0 ; j)-connection in the preasheaf B which preserves the transition functions ϕ ji in B k α,β (X) and also their derivatives ∂ µ ϕ ji ,with |µ| ≤ r, in the same sense that it preserves bump functions.A nice (α 0 , β 0 ; j)-connection in B k α,β (X) is one whose underlying (α 0 , β 0 ; j)-connection in B is nice.
By the hypothesis we have a (α ′ 0 , β ′ 0 ; j)-connection ξ, so that from Corollary 3.1 for any ordinary pair (θ, ϑ), the image (Γ ξ ϕ ) c ab of the coefficients (4) by ξ θϑ belong to We assert that (Γ ξ ϕ ) c ab also define a connection.This follows the fact that ξ is nice and has degree r ≥ 2. Indeed, we have , where in the first step we used (3), in the second we used compatibility between the distributive structures of B and C k− and also the distributive properties of ξ, and in the third step we used that ξ has degree r ≥ 2. Thus, by the same kind of arguments, , which clearly define an affine connection ξ θϑ (∇) in M .Since they belong to (6), it follows that ξ θϑ (∇) is actually a (B k θ,ϑ , X ′ |Γ ≥0;β ′ 0 −z )-connection.
As commented in Section 1, this is not enough to conclude the existence of many linearly independent (B k α 0 ,β 0 , X 0 |S)-connections.Indeed, we need to analyze the dimension of the space of (B k α 0 ,β 0 , X 0 |S)-forms of degree one.Instead of doing this, with eyes in future applications, we will follow a more direct approach.
Notice that under the assumption of a distributive nice (α ′ 0 , β ′ 0 ; j)-connection for B, the proof of Theorem 3.1 suggests that a nice (B k α,β , X)-manifold has not only one, but actually many of those B k θϑ -connections, for every ordinary (θ, ϑ).Indeed, for each choice of (B, k, α 0 , β 0 )-functions f c ac covering M by charts in B k α,β (X), the expression (4) defines a affine connection in M , which is really a (B k θϑ , X ′ )-connection, due to Proposition 3.1.But different families of functions could a priori define the same (B k θϑ , X ′ )-connection.In other words, Theorem 3.1 and Proposition 3.1 give us the functions in the diagram below, where the first set is that of all 3-parameter families f c ab of presheaves of (B, k, α 0 , β 0 )-functions in X and the second one is the set of affine connections in M whose coefficients belong to (1).It happens that the composition map usually does not have a trivial kernel (each choice of a (α ′ 0 , β ′ 0 ; j)-connection induces one of these composition maps).
Psh k α 0 ,β 0 ;3 (R n ; X) The next result reveals that under certain additional assumptions every (B k θϑ , X ′ )-connection ∇ can be deformed into another (B k θϑ , X ′ )-connection ∇ which differs from ∇ in at least one nonempty open set.In order to be more precise, notice that diagram (7) can be completed to the following noncommutative diagram, where the first vertical arrow is the inclusion and the other two take a connection ∇ and give the 3-parameter family presheaf of local expressions.Furthermore, Psh 3 (R n ) is just the set of 3-parameter family of presheaves of continuous maps.
We say that Ω = (Ω c ab ) and Λ = (Λ c ab ) in Psh 3 (R n ) are locally different (and we write Ω ≁ Λ) if for every a, b, c = 1, ..., n and every U ⊂ R n there exists a nonempty open set U a,b,c ⊂ U such that Ω c ab (p) = Λ c ab (p) for every p ∈ U a,b,c .We say that two (B k θϑ , X ′ |S)-connections ∇ and ∇ are locally different if their underlying 3-parameter family presheaves are locally different, i.e, if (∇) ≁ (∇).A map F assigning to each (B k θϑ , X ′ |S)-connection ∇ another (B k θϑ , X ′ |S)-connection F (∇) is has no local fixed points if for every ∇ we have F (∇) ≁ ∇.In the following we will show that under certain addition assumptions there exists one map which has no local fixed points.This will be done by making use of Thom's transversality, so that we will need to assume k = ∞.We will also need a closure condition.Indeed, given functions θ : S → Γ ≥0 and ϑ : S → [0, k], with S ⊂ Γ ≥0 , we say that a B k α,β -presheaf B is (θ, ϑ|S)-strong in a proper ISP X ′ between B θ and C k−ϑ if for every U and every i ∈ S, the image of the projection map Theorem 4.1.In addition to the hypotheses of Theorem 3.1, assume that M is smooth and has countably many connected-components, that B admits a nice distributive (α ′ 0 , β ′ 0 ; j)-connection of degree r ≥ 2, say in a proper ISP X ′ , and that B is (θ, ϑ|Γ ≥0;β ′ 0 −z )-strong in X ′ for some ordinary (θ, ϑ) and some z ∈ Γ ≥0;β ′ 0 .In this case, for any 3-parameter family Ω c ab of presheaves of are nonzero in some nonempty U ⊂ U , where Γ c ab is the presheaf of coefficients of ∇.
Proof.First of all we notice that if f, g : V → R are two transversal real smooth functions defined on an open set V ⊂ R n , then the set V = of points p ∈ V such that f (p) = g(p) cannot be dense in V .Indeed, V = is given by f ∩ p g = f −1 (p) ∩ g −1 (p).Since f ⋔ g, this a (n − 2)-submanifold of V and therefore of R n , but n − (n − 2) = 2 ≥ 1, so that f ∩ p g cannot be dense in V and thus in R n .On the other hand, transversality is a generic property, so that if f is not transversal to g, we can modify it a bit in order to get f such that f ⋔ g.Now, since we are assuming k = ∞ and since B admits a connection in which B is (θ, ϑ|Γ ≥0;β ′ 0 −z )-strong in X ′ , each B θ(i) (V ) ∩ X ′ (V ) C ∞ (V ) can be regarded as a closed set in C ∞ (V ), so that any dense set X in C ∞ (V ) is also dense in B θ(i) (V ) ∩ X ′ (V ) C ∞ (V ).Thus, any f in that intersection can be regarded as a smooth real function and small modifications of it (regarded as a smooth function) remains in B θ(i) (V ) ∩ X ′ (V ) C ∞ (V ).Now, given a countable covering of M by charts (U s , ϕ s ) ∈ B ∞ α,β (X), which exists since M is paracompact and has countably many connected-components, let r 0 = min Γ ≥0;β ′ 0 −z and consider (Ω ϕs ) c ab ∈ B θ(r 0 ) (U N (s) ) ∩ X ′ (N (s) ) C ∞ (U N (s) ), regard each of them as a smooth function and take the sum g c = ab (Ω ϕs ) c ab .Let (f s ) l mn be (α 0 , β 0 )-functions in X defining a (B ∞ θϑ , X ′ |Γ ≥0;β ′ 0 −z )connection, whose existence is ensured by Corollary 3.1.Consider the corresponding functions (Γ ϕs ) c ab ∈ B θ(r 0 ) (U N (s) ) ∩ X ′ (N (s) ) C ∞ (U N (s) ), regard them as smooth function and also take the sum f c ϕs = ab (Γ ϕs ) c ab .By the above we can assume f c ϕs ⋔ g c ϕs , otherwise we can modify (f s ) l mn a bit in order to get this.Therefore, the set U c s,= in which f c ϕs (p) = g c ϕs (p) is not dense, so that there exists p ∈ ϕ s (U ) and small neighborhood B s,p ∋ p such that B s,p ∩ U = = ∅.Let U c the union of all those B s,p .Finally, take U as the intersection of U c for all c.That this intersection is nonempty follows from the fact that C ∞ (V ) is a Baire space for every V and that the covering (U s , ϕ s ) s is enumerable.Taking U a ′ ,b ′ = U a ′ ∩ U b ′ and noticing that this intersection is nonempty, the proof is done.
where O, Q ⊂ Mor(Γ ≥0 ; Γ ≥0 ) are set of functions.Given maps D O , D Q , defined in O × O and in Q × Q ′ , respectively, and assigning to each pair θ, θ ′