Cartan--Whitney Presentation, Non-smooth Analysis and Smoothability of Manifolds: On a theorem of Kondo--Tanaka
aa r X i v : . [ m a t h . DG ] A p r CARTAN–WHITNEY PRESENTATION, NON-SMOOTH ANALYSIS ANDSMOOTHABILITY OF MANIFOLDS: ON A THEOREM OFKONDO–TANAKA
SIRAN LI
Abstract.
Using tools and results from geometric measure theory, we give a simple new proofof the main result in [17] (Theorem 1.3 in K. Kondo and M. Tanaka, Approximation of LipschitzMaps via Immersions and Differentiable Exotic Sphere Theorems,
Nonlinear Anal. (2017),219–249), as well as the converse statement. It explores the connections between the theoryof non-smooth analysis à la
F. H. Clarke and the existence of special systems of Whitney flat -forms with Sobolev regularity on certain families of homology manifolds. Introduction
The smoothability of topological manifolds has long been a question at the heart of dif-ferential and geometric topology:
Given a manifold with structures of weak regularity, e.g., atopological, homology, or Lipschitz manifold, does it admit a smooth structure?
Foundational works on smoothability of manifolds by Whitehead [31] and Cairns [2] (alsosee Pugh [22] for an alternative, modern proof), Stallings [26], Shikata [24, 25], Moise [20]and Kirby–Siebenmann [16], among many others, provide deep insights into the structures ofmanifolds. As popularised by Gromov in [11], they address the simple yet fundamental question:“what is a manifold”. The geometrical and topological developments in this line culminate inthe discovery of exotic ( i.e. , homeomorphic but not diffeomorphic) structures; see Milnor [19],Freedman [9], Donaldson–Sullivan [7] and Gromov [10].On the other hand, using the techniques from geometric measure theory, an analytic ap-proach has been developed to tackle the smoothability problem. Sullivan [27, 28, 29] initiatedthe programme of detecting the smoothability of a Lipschitz manifold using the notion of a “mea-surable cotangent bundle”. Its sections ϑ are identified with flat forms , which were introducedby Whitney in his theory of geometric integration theory [30]. Roughly speaking, ϑ is a localcoframe with weak regularity and an essentially nondegenerate volume density, and the integra-tion of ϑ along segments gives rise to a branched covering map F ϑ . Heinonen–Rickman [14] andHeinonen–Sullivan [15] proved that the local smoothability of a Lipschitz manifold is equivalentto that the local degree of F ϑ = 1 ; furthermore, Heinonen–Keith [13] established its equivalencewith the Sobolev regularity condition ϑ ∈ W , .In a recent paper [17], a brand-new perspective has been adopted by Kondo–Tanaka toapproach the smoothability problem. It connects F. H. Clarke’s theory of non-smooth analysis Date : April 2, 2019.2010
Mathematics Subject Classification.
Primary: 49Q15, 58A05, 57R10, 57R12, 57R55; Secondary: 49J52,90C56.
Key words and phrases.
Smoothability of Manifolds; Non-smooth Analysis; Singular Points; Lipschitz Maps;Cartan–Whitney Presentations; Flat Forms. [4, 5]), originally developed for applications in optimisation and control theory, to the approxi-mation of Lipschitz maps by diffeomorphisms. The Main Theorem . in [17] is as follows: Theorem 1.1.
Let M be an n -dimensional compact Riemannian manifold, and let N be an ν -dimensional Riemannian manifold, where ≤ n ≤ ν . Then, a Lipschitz map F : M → N isapproximable by smooth immersions if sing Cl F = ∅ . sing Cl F denotes the singular set of F in the sense of Clarke [4, 5]; see Definition 2.2. Therigorous definitions for relevant geometric-analytic notions shall be given in § . The proof in [17]may be viewed as an intricate generalisation of the classical arguments by Grove–Shiohama [12].In this note we present a simple, new proof of Theorem 1.2, which also establishes itsconverse at the same strike. Our proof is based on the geometric measure theoretic studies onthe smoothability problem (see [27, 28, 29, 14, 15, 13]). In particular, we make crucial use of theresults due to Heinonen–Keith [13]. We hope it may provide an avenue for further explorationson the linkages between Clarke’s non-smooth analysis [4, 5] and geometric measure theory.In summary, we shall prove: Theorem 1.2.
In the setting of Theorem 1.1, the Lipschitz map F : M → N is approximableby smooth immersions if and only if sing Cl F = ∅ . Alternative analytic approaches, besides the geometric measure theoretic and non-smoothanalytic ones, have also been developed to study the smoothability problem; see Ball–Zarnescu[1] and the references cited therein. This paper and the subsequent developments also address theapplications of manifold smoothability theorems to the modelling and analysis of liquid crystals.2.
Background
In this section we briefly discuss some preliminary materials on non-smooth analysis andgeometric measure theory. For comprehensive treatments, we refer to [4, 5, 17] on the formertopic and to [8, 30, 6] on the latter.
Smoothability.
A topological manifold M is said to possess a ℘ -structure ( ℘ ∈ { Lipschitz, C k,α , smooth= C ∞ , analytic... } ) if and only if there is an atlas { ( U α , φ α ) : α ∈ I} of M suchthat all the transition maps φ α ◦ φ − β : R n ⊃ φ β ( U α ∩ U β ) −→ φ α ( U α ∩ U β ) ⊂ R n have ℘ -regularity. By definition, a topological manifold has a C -structure. A ℘ -manifold M issaid to be smoothable if there is a sub-atlas with respect to which M admits a C ∞ -structure. Cartan–Whitney presentation.
Let
O ⊂ R ν be an open subset. A k -form ω is said to be a (Whitney) flat k -form if and only if ω has measurable coefficients and ω, dω ∈ L ∞ ( O ) . Here dω is understood in the weak ( i.e. , distributional) sense.In what follows the definition of Cartan–Whitney presentations will be given. Let us firstrecall a prototypical case: Consider an n -dimensional topological manifold M , and let U be anopen neighbourhood of some point p ∈ M with a well-defined orientation. Then there exists a ocal coframe { θ , . . . , θ n } ∈ Γ( T ∗ U ) such that the differential n -form θ ∧ . . . ∧ θ n agrees withthe orientation and is nondegenerate: for a constant c > , Z U θ ∧ . . . ∧ θ n d V g M ≥ c Vol g M ( U ) > . (2.1)Here and throughout, for a fibre bundle E over M , Γ( E ) denotes the space of its sections. Also, d V g M is the volume measure induced by g M .The notion of Cartan–Whitney presentations serves as a generalisation of the n -form θ ∧ . . . ∧ θ n in the above. Let X be a metric space that is a “nice” n -dimensional subset of R ν , ν ≥ n .Let U be an open neighbourhood of a fixed point p in X . A (local) Cartan–Whitney presentationnear p consists of an n -tuple ρ = ( ρ , . . . , ρ n ) of flat -forms defined in an R ν -neighbourhood O of p , such that O ∩ X ⊂ U and ⋆ ( ρ ∧ . . . ∧ ρ n ) ≥ c ′ > a.e. on O ∩ X (2.2)for some constant c ′ > .What does it mean by “nice” for X ? On one hand, in the above definition ρ is defined on O ⊂ R ν , so we have to ensure that its local restrictions to X make sense. On the other hand, U ⊂ X needs to have a good sense of orientation, so that the Hodge-star in (2.2) is well-defined.Indeed, by the work [23] of Semmes, the following conditions ensure that X is nice enough tomake sense of the above definition of Cartan–Whitney presentation: X is a locally Ahlfors n -regular,locally linearly contractible homology n -manifold, n ≥ . (2.3)Here, X is locally Ahlfors n -regular if and only if X has Hausdorff dimension n , and for everycompact K ⋐ X there exist numbers r K > , C K ≥ such that C − K r n ≤ H n ( B ( x, r )) ≤ C K r n for each metric ball B ( x, r ) ⊂ X with x ∈ K and r < r K . X is locally linearly contractibleif and only if for every compact K ⋐ X there exist numbers r ′ K > , C ′ K ≥ such that everymetric ball B ( x, r ) ⊂ X with x ∈ K and r < r ′ K contracts to a point inside B ( x, C ′ K r ) . Finally, X is homology n -manifold if and only if it is separable, metrisable, locally compact, locallycontractible, and that for each x ∈ X the following identity on homology groups holds: H • ( X, X ∼ { x } ; Z ) ∼ = H • ( R n , R n \ { } ; Z ) . (2.4)The quadruple ( C K , r K ; C ′ K , r ′ K ) is called the local data of X on K . One may refer to §§ –3 in Heinonen–Keith [13] for detailed discussions. The punchline is: a local Cartan–Whitneypresentation ρ can be defined on X ⊂ R ν with weak regularity as in (2.3). Sobolev space.
Next let us define the Sobolev space W , on X satisfying (2.3): it is the normcompletion of Lipschitz functions φ : X → R with respect to k φ k , := k φ k L ( X ) + k ap Dφ k L ( X ) , where the approximate differential ap Dφ is a.e. defined on X as in . . of Federer [8]. Measureble cotangent structure; the theorem of Heinonen–Keith on smoothability.
Let X be as in (2.3). A result due to Cheeger [3] implies that X is n -rectifiable. For any flat -form ω defined on a open subset O ⊂ R ν such that O ∩ X =: U is non-empty, one can define he restriction ω U as a map from U to T ∗ x U . The space T ∗ x U is viewed as a measurable sectionof T ∗ U ⊂ T ∗ R ν , i.e. the measurable cotangent bundle ; see § . in [13] and p.303, Theorem 9Ain [30].Furthermore, in the pioneering works [27, 28, 29] Sullivan introduced the notion of a mea-surable cotangent structure . It consists of a pair ( E, ι ) , where E is an oriented rank- n Lipschitzvector bundle over X , and ι is a module map over Lip( X, R ) from Lipschitz sections of E to flat -forms on X , such that the following holds: If σ , . . . , σ n : X → E are Lipschitz sections suchthat σ ∧ . . . ∧ σ n determines the chosen orientation on E , then for every α (index of an oriented,trivialised atlas { U α } ) the flat n -form ι ( σ ) | α ∧ . . . ∧ ι ( σ n ) | α = τ α dx ∧ . . . ∧ dx n satisfies ess inf K τ α > in each compact subset K ⋐ U α . In practice, one considers E = T ∗ U α as in the above paragraph.The main result in [13] can be summarised as follows: Theorem 2.1 (Theorems 1.1 and 1.2, [13]) . (1) If X ⊂ R ν as in (2.3) admits a Cartan–Whitney presentation in W , ( X ) , then it is locally bi-Lipschitz parametrised by R n .(2) An oriented Lipschitz manifold is smoothable if and only if it admits a measurable cotan-gent structure with local frames in W , ( X ) . Non-smooth analysis.
Let ( X, d X ) and ( Y, d Y ) be metric spaces. For a function φ : X → Y ,one can define its Lipschitz norm as usual with respect to the metrics d X , d Y . As an example,let ( M, g M ) and ( N, g N ) be n - and ν -dimensional Riemannian manifolds, and denote by dist M and dist N the Riemannian distance functions on M , N induced by the Riemannian metrics,respectively. The Lipschitz norm of φ : M → N is k φ k Lip(
M,N ) := sup x = y, x,y ∈ M dist N Ä φ ( x ) , φ ( y ) ä dist M ( x, y ) . By Rademacher’s theorem, at d V g M - a.e. point x ∈ M a Lipschitz function φ : M → N haswell-defined differential d x φ : T x M → T φ ( x ) N .For φ : ( X, d X ) → ( Y, d Y ) as above, its generalised differential is the set-valued function: ∂φ ( x ) := conv (cid:16)n lim i →∞ d x i φ : d x i φ exists and dist M ( x i , x ) → o(cid:17) . Here conv denotes the convex hull. Note that for any x ∈ X , each element of m ∈ ∂φ ( x ) can beidentified with a matrix; thus we may introduce the following Definition 2.2.
The singular set of φ à la Clarke ( [4, 5] ) is sing Cl φ := n x ∈ X : there exists m ∈ ∂φ ( x ) that is not of the maximal rank o . A function φ : X → Y is said to be approximable by smooth immersions if for any ǫ > there exists a smooth immersion ι ǫ : X → Y such that dist N ( φ ( x ) , ι ǫ ( x )) ≤ ǫ for any x ∈ X , andthat k ι ǫ k Lip(
X,Y ) ≤ (1 + ǫ ) k φ k Lip(
X,Y ) . 3. Proof
This section is devoted to the proof of Theorem 1.2. We shall establish the more general heorem 3.1. Let X ⊂ R ν be a locally Ahlfors n -regular, locally linearly contractible homological n -manifold; ν ≥ n ≥ . Assume the existence of a Cartan–Whitney presentation in W , ( X ) .Let F : X → R ν ′ be a Lipschitz map, ν ′ ≥ n . Then the image F ( X ) is a smoothable topological n -manifold if and only if sing Cl F = ∅ .Proof for Theorem 3.1 ⇒ Theorem 1.2.
Let M be an n -dimensional compact Riemannian man-ifold. It admits a C k -isometric embedding into R ν , for k ≥ and ν large enough, by Nash’stheorem [21]. Cover M by finitely many coordinate charts and fix one such chart U . Denote by { ∂ / ∂x , . . . , ∂ / ∂x n } ⊂ Γ( T U ) a local orthonormal frame on U , and let { dx , . . . , dx n } ⊂ Γ( T ∗ U ) bethe corresponding coframe. Clearly { dx , . . . , dx n } are flat -forms with sup ≤ i ≤ n k dx i k L ∞ ( U ) ≤ C < ∞ and d ( dx i ) = 0 . Moreover, since dx ∧ . . . ∧ dx n is a volume n -form, hence Eq. (2.2) isverified. Thus, { dx , . . . , dx n } constitute a Cartan–Whitney presentation in C ∞ ( U ) ⊂ W , ( M ) .Furthermore, since F : M → N is Lipschitz and F ( U ) is precompact in the manifold N ,by shrinking U if necessary, we can take F ( U ) lying in one single geodesic normal ball B on N .As the exponential map on B is a C ∞ -diffeomorphism, by composing with it we may assumethat F maps into the Euclidean space R ν ′ .Finally, by the local nature of the statement of Theorem 1.2, it remains to show that underthe assumption that F ( U ) satisfies (2.3), F | U is approximable by smooth immersions if and onlyif F ( U ) is smoothable. For the forward implication, we may utilise verbatim the arguments onp.32 in [13]; in particular, the proof of Eqs. (8.9) and (8.10) and an application of the results in[18, 27]. For the converse, one may pass to a sub-atlas of the C ∞ -structure and take ι ǫ ≡ F .Thus Theorem 1.2 follows. (cid:3) We are now ready to show Theorem 3.1. Heuristically, the key idea is that sing Cl F = ∅ prevents F from pinching necks. Proof of Theorem 3.1.
Fix an open neighbourhood U ⊂ X , on which there is a given Cartan–Whitney presentation ρ = ( ρ , . . . , ρ n ) ∈ W , ( U ) . Let F : U → R ν ′ be a Lipschitz map. Again,by the local nature of the statement, we may assume that F ( U ) is orientable and show that F ( U ) is smoothable if and only if sing Cl ( F | U ) = ∅ . In the sequel we view F as defined on U .To this end, consider the pushforward n -tuple of flat -forms: F ρ = Ä F ρ , . . . , F ρ n ä . In view of Theorem 2.1 (2), it then suffices to prove the equivalence between sing Cl F = ∅ and thefollowing two conditions altogether: F ρ defines a measurable cotangent structure à la Sullivan[27, 28, 29] on F ( U ) , and that F ρ ∈ W , ( R ν ′ ) . (3.1)The Sobolev regularity condition (3.1) is automatic, as W , -tensors are preserved underpushforward via Lipschitz functions. In the sequel, we show that sing Cl F = ∅ if and only if F ρ yields a measurable cotangent structure, momentarily assuming that F ( U ) satisfies (2.3).Let us first suppose sing Cl F = ∅ and deduce that F ρ induces a measurable cotangentstructure. By definition, for each x ∈ U , every element of the generalised differential ∂F ( x ) isof maximal rank. Denote by E the set of points on U where dF do not exist; H n ( E ) = 0 byRademacher’s theorem. In addition, clearly a necessary condition for sing Cl F = ∅ is that the ifferential dF : T U → T R ν ′ (defined in the distributional sense; see § ) is invertible at H n - a.e. point on U .Our goal is to show that ess inf F ( U ) ⋆ F Ä ρ ∧ . . . ∧ ρ n ä ≥ c > (3.2)under the assumption: ess inf U ⋆ Ä ρ ∧ . . . ∧ ρ n ä ≥ c > . (3.3)Suppose (3.2) were false. Then, for any ǫ > there would be a set Σ ⊂ U with H n (Σ) > and ⋆F ( ρ ∧ . . . ∧ ρ n ) < ǫ on Σ . Without loss of generality we may take Σ to be the metricball B ( p, r ) ⊂ U with an H n -null set Γ removed, such that Γ ⊃ B ( p, r ) ∩ E . After passing tosubsequences if needed, one can find a convergent sequence of points { q i } ⊂ Σ ∼ Γ such that q j → q ∈ B ( p, r ) and that det Ä dF ( q j ) ä < ǫ / c , (3.4)where c is as in (3.3). Indeed, observe the identity ⋆F ( dx ∧ . . . ∧ dx n ) = det dF ¶ ⋆ Ä ρ ∧ . . . ∧ ρ n ä© wherever dF is invertible; the determinant is well-defined as F is Lipschitz. Thus, in view ofEq. (3.3), we have ⋆F ( dx ∧ . . . ∧ dx n ) ≥ c det dF outside an H n -null set on U . Thus Eq.(3.4) follows. However, in the limits of ǫ ց and j = j ( ǫ ) ր ∞ , Eqs. (3.3), (3.4) and the rank-nullity theorem imply (via a diagonalisationargument) that dF cannot be of maximal rank at the limiting point q . Thus q ∈ sing Cl F , whichyields a contradiction.Conversely, assume that F is not approximable by smooth immersions; we shall find apoint q ′ in sing Cl F . By definition, it suffices to find a sequence { q ′ j } ⊂ U ∼ E such that q ′ j → q ′ and that lim j ր∞ dF ( q ′ j ) has rank less than n . Indeed, again due to Theorem 2.1, thenon-approxmability of F leads to ess inf F ( U ) ⋆ F Ä ρ ∧ . . . ∧ ρ n ä = 0 . (3.5)In view of the Lipschitzness of F and the precompactness of U , the essential infimum in Eq. (3.5)is attained at a point. That is, for some q ′ ∈ U there holds ⋆F ( ρ ∧ . . . ∧ ρ n ) Ä F ( q ′ ) ä = 0 .Moreover, for any ǫ > , there exists a sequence { q ′ j } ⊂ U ∼ E depending possibly on ǫ , suchthat q ′ j → q ′ (hence F ( q ′ j ) → F ( q ′ ) ) and that ⋆F ( ρ ∧ . . . ∧ ρ n ) Ä F ( q ′ j ) ä < ǫ. Since ( ρ , . . . , ρ n ) is a Cartan–Whitney presentation on U , by Eq. (3.3) we may assume that allthe q ′ j chosen above satisfy ⋆ Ä ρ ∧ . . . ∧ ρ n ä ( q ′ j ) ≥ c > . By an analogous estimate as for Eq. (3.8), we can again bound det dF ( q ′ j ) ≤ ǫ / c . In particular, the determinant of dF are well-defined at each point q ′ j . By sending ǫ ց , j = j ( ǫ ) ր ∞ and using a standard diagonalisation argument, we find that any pointwise ubsequential limit m of dF ( q ′ j ) verifies rank m ≤ n − . So, as { q ′ j } converges to q ′ , we immediately obtain that q ′ ∈ sing Cl F .We are now left to prove that F ( U ) satisfies the structural assumptions in (2.3). For thispurpose, we shall make crucial use of Eq. (3.2) established above. Indeed, by the Lipschitznessof F let us rewrite Eqs. (3.3) and (3.2) as c ≤ ess inf U ⋆ Ä ρ ∧ . . . ∧ ρ n ä ≤ ess sup U ⋆ Ä ρ ∧ . . . ∧ ρ n ä ≤ C , (3.6) c ≤ ess inf U ⋆ F Ä ρ ∧ . . . ∧ ρ n ä ≤ ess sup U ⋆ F Ä ρ ∧ . . . ∧ ρ n ä ≤ C . (3.7)where C depends on the flat norm of ρ , and C additionally on the Lipschitz norm of F .Indeed, Eqs. (3.6)(3.7) imply that < λ := c C ≤ (cid:12)(cid:12)(cid:12) det( dF ) (cid:12)(cid:12)(cid:12) ≤ C c =: Λ < ∞ H n − a.e. on U. (3.8)As a result, given any metric ball “ B ( x, r ) ⊂ F ( U ) , we can find radii < r − < r + such that thefollowing inclusions hold outside at most a H n -null set : F Ä B ( x, r − ) ä ⊂ “ B ( x, r ) ⊂ F Ä B ( x, r + ) ä . (3.9)Here and hereafter, we always use “ B ( • , • ) to denote the metric balls in F ( U ) ; the notation B ( • , • ) is reserved for the metric balls in U .To proceed, notice that one may take r ± to be equal to r modulo a multiplicative factordepending only on Λ and λ . Thus, by the area formula ([8], 3.2.20), there exists a constant < c < ∞ depending only on Λ , λ , the Lipschitz norm of F and the local data of X such that c r n ≤ H n ( B ( x, r )) ≤ c r n . This gives the local n -Ahlfors regularity of F ( U ) . The local linearcontractibility of F ( U ) follows similarly from (3.9).Finally, Let B ⊂ U be the set on which Eq. (3.8) fails. Then Eq. (2.4) clearly holds on U ∼ B , since F is a homeomorphism thereon and hence leaves H • ( X, X ∼ {∗} ; Z ) invariant. For ∗ ∈ B , by excision we have H • (cid:16) F ( U ) ∪ {∗} ∼ B , F ( U ) ∼ B ; Z (cid:17) ∼ = H • (cid:16) F ( U ) , F ( U ) ∼ {∗} ; Z (cid:17) . Utilising the facts that F is Lipschitz (hence continuous) on U , that F is a bi-Lipschitz homeo-morphism onto F ( U ) ∼ B by Eq. (3.8), and that U is a Lipschitz n -manifold modulo reparametri-sations (thanks to Theorem 2.1 (1)), we deduce that F ( U ) ∪ {∗} ∼ B deformation retracts onto F ( U ) ∼ B . This allows us to compute the relative homology from the reduced homology ‹ H • : H • (cid:16) F ( U ) ∪ {∗} ∼ B , F ( U ) ∼ B ; Z (cid:17) ∼ = ‹ H • F ( U ) ∪ {∗} ∼ B F ( U ) ∼ B ; Z ! ∼ = H • Ä R n ; R n ∼ { } ; Z ä . Hence F ( U ) is a homology n -manifold.The proof is now complete. (cid:3) Acknowledgement . 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Ann. of Math. (1961), 154–211. Siran Li: Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston,Texas, 77251-1892, USA • Department of Mathematics, McGill University, Burnside Hall, 805Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada.