Weak stationarity of a matrix valued differential form at superdensity points of its vanishing set

A property of weak stationarity of a matrix valued differential form at superdensity points of its vanishing set is proved. This result is then applied in the context of the Maurer–Cartan equation.


Introduction
The main result of this work (cf.Theorem 3.1) establishes a property of weak stationarity of a matrix valued continuous differential form at the superdensity points of its vanishing set.To make this statement more understandable, we now recall very briefly some definitions and properties (referring the reader to Section 2, for a more complete presentation).Let us consider an M -dimensional C k manifold M and recall that a matrix valued C p differential h-form on M is a square matrix whose entries are C p differential h-forms on M. The classical formalism for differential forms, i.e., wedge product, exterior differentiation, integration and pullback, extends naturally to matrix valued differential forms (cf.Section 2.2).In this extended formalism it is easy to introduce a notion of distributional exterior derivative, which will be denoted by δ (cf.Definition 3.1).We also recall that, if E is a subset of M, then P ∈ M is said to be an m-density point of E relative to M if there is a C 1 chart (W, Φ) such that P ∈ W and where L M and B r (Φ(P )) are, respectively, the Lebesgue measure on R M and the ball of radius r centered at Φ(P ).We observe that this definition does not depend on the choice of the coordinate chart (cf.Section 2.4).
We are now able to state more precisely than before the result in Theorem 3.1: Let M be an M -dimensional C 2 manifold and let γ be a matrix valued C 0 differential form on M which has the distributional exterior derivative δγ of class C 0 .Then we have (δγ) Q = 0, whenever Q is an (M + 1)-density point of {P ∈ M | γ P = 0}.
In Section 4, by a simple application of Theorem 3.1, we provide a new proof of the following property in the context of Frobenius theorem about distributions (cf.[ We have the following well-known theorem, due to Cartan (cf [9, Theorem 1.6.10]):Let M be a smooth manifold and let φ be a g-valued smooth differential 1-form on M verifying the Maurer-Cartan equation Then for all P ∈ M there exist a neighborhood U of P and a smooth map Relatively to this context, we will provide a structure result for the sets under the assumption that φ does not verify the Maurer-Cartan equation (1.1).In particular, let M be an M -dimensional C 2 manifold and let φ be a R L×L -valued C 1 differential 1-form on M such that (dφ) Q = −(φ ∧ φ) Q for all Q ∈ M. Obviously this condition prevents the possibility of φ being locally a C 1 pullback of Γ G (cf. Remark 5.1).More interesting information on the content of {f * Γ G = φ| U } is given in Corollary 5.2, namely:

Basic notation and notions
2.1.Basic notation.The coordinates of R M are denoted by (x 1 , . . ., x M ) so that dx 1 , . . ., dx M is the standard basis of the dual space of R M .For simplicity, we set Let L M and H s denote, respectively, the Lebesgue measure and the s-dimensional Hausdorff measure on R M .The open ball of radius r centered at x ∈ R M will be denoted by B r (x).Let R L×L be the vector space of all L × L real matrices and Gl(L, R) be the Lie group of nondegenerate matrices in R L×L .The Lie algebra of Gl(L, R) will be denoted by gl(L, R).Since R L×L R L 2 we can denote the natural coordinates on Gl(L, R) by the matrix notation (z ij ).
2.2.Manifolds, differential forms.In relation to this topic, we will adopt the notations commonly used in the main bibliographic references (see, e.g., [10,12]).We report here, quickly, just a few of them.
is any local representation of ω, then f α is of class C k (resp.C k c , i.e., C k with compact support).For any given P ∈ M, we will use the standard notation ω P instead of ω(P ).As we did for real-valued maps, let us set {ω = 0} := {P ∈ M | ω P = 0} for simplicity.The set of all Let M be a C k imbedded submanifold of a C k manifold N and let ι : , the restriction of ω to M) will be denoted by ω| M .
We also need matrix-valued differential forms, i.e., matrices whose entries are differential forms.
For the sake of convenience, we will sometimes (e.g. in Section 5 below) refer to the members of Mat L C p F h (M) by simply calling them C p differential h-forms as well.The subset of Mat L C p F h (M) whose members have all the entries in , then we define Observe that d is linear and d The exterior product of of two matrix-valued differential forms A trivial computation shows that differentiating the exterior product of matrix-valued differential forms yields the usual formula (provided k ≥ 1): Let us recall that a C 1 Riemannian manifold (M, g) with the associated Riemannian distance function is a metric space whose topology coincides to the original manifold topology, cf.[10,Theorem 13.29].Hence one can define the corresponding s-dimensional Hausdorff measure H s g , cf. [8, Section 2.10.2],[13,Chapter 12].The open metric ball of radius r centered at P ∈ M will be denoted by B g (P, r).

2.3.
Hausdorff measure on manifolds.For the convenience of the reader, we recall the following well-known properties of the Hausdorff measure H s g on a C 1 Riemannian manifold (N , g): for all Borel sets B ⊂ N , where V g denotes the standard volume form of (N , g), cf.[ (1) For every (2) For every C 1 Riemannian metric g on N , one has H s g (E) = 0.
(3) There exists a C 1 Riemannian metric g on N such that H s g (E) = 0. (1) There is a C 1 chart (W, Φ) of N such that P ∈ W and (2) For every C 1 Riemannian metric g on N , one has (3) There exists a C 1 Riemannian metric g on N such that Definition 2.1.If any or, equivalently, all of the conditions of Proposition 2.2 are satisfied, then we say that P is an m-density point of E (relative to N ).The set of all m-density points of E is denoted by E (m) , cf. [5].
Remark 2.1.Let N and E be as in Proposition 2.2.The following facts occur: . In particular, one has E (m) ⊂ E (N )  for all m ∈ [N, +∞).• Let {E j } j∈J be any family of subsets of N and m ∈ [N, +∞).
-One has -If J is countable infinite, then (2.1) can fail to be true, e.g., N = R 2 and Remark 2.2.For convenience of the reader, we recall some known results in the special case when N = R N (which actually could be easily generalized):

The main result
Throughout this section M and k will denote, respectively, a M -dimensional manifold and the regularity class of M. We will assume k ≥ 1, if not otherwise stated.
From Remark 3.1 we get immediately the following proposition.
Then µ is uniquely determined.Definition 3.1.Let the assumptions of Proposition 3.1 be verified.Then we say that λ has the distributional exterior derivative (DED) in Mat L C 0 F h+1 (M).The latter is defined as δλ := (−1) h+1 µ, so that , with respect to the C l topology.Hence in Definition 3.1 we can equivalently assume that (3.1) holds for all ϕ ∈ Mat L C l c F M −h−1 (M).
The following propositions state some expected properties.We observe that the first three are trivial.
Then, for all a, b ∈ R, the matrix-valued differential form aλ . Then, by Definition 3.1 and Remark 3.2 (with l = k − 1), we obtain Proof.
Then for all x ∈ supp(ϕ) there exists an open set V (x) ⊂ M and a countable family {f If we extend every f arbitrarily to all of M and define Hence, letting j → +∞, we obtain The conclusion follows from the arbitrariness of ϕ.
Let us now state and prove the main result.
Proof.First of all, set for simplicity B r := B r (0) ⊂ R M and let ρ ∈ (0, 1).Then consider g ∈ C 2 c (B 1 ) such that 0 ≤ g ≤ 1, g| Bρ ≡ 1 and For r > 0, define g r ∈ C 2 c (B r ) as and observe that (for all x ∈ B r and i = 1, . . ., M ) by Definition 2.1.Now set for simplicity U := Φ(U) and let θ ∈ Mat L C 2 F M −1−h (U ) be chosen arbitrarily.Obviously there must be (F hence, for all i, j, we have (provided r is small enough)

Recalling (3.3), we obtain
On the other hand, the triangle inequality yields It follows that Then, by first letting r → 0+ (and recalling (3.4)) and then letting ρ → 1−, we obtain F (ij) θ (0) = 0 (for all i, j).The conclusion follows from the identity (3.5) and the arbitrariness of θ.
The following simple corollary of Theorem 3.1 will be useful below.
Observe that γ is the Maurer-Cartan form of Gl(L, R).Recall that Γ G is left-invariant, takes values in g and satisfies the Maurer-Cartan equation, that is and assume that the following property holds: For all P ∈ M there exist a neighborhood U of P and a C 1 map f : U → G such that f * Γ G = φ| U .Then, first of all, φ takes values in g.Moreover, by Proposition 3.3, Proposition 3.6 and (5.1), one has Relative to the opposite implication, it is well known that a g-valued smooth differential 1-form satisfying the Maurer-Cartan equation is always, at least locally, a smooth pullback of the Maurer-Cartan form.In fact the following theorem holds, cf [9, Theorem 1.6.10].
Theorem 5.1 (Cartan).Let M be a smooth manifold and let φ be a g-valued smooth differential 1-form on M satisfying the identity dφ = −φ ∧ φ.Then for all P ∈ M there exist a neighborhood U of P and a smooth map f : U → G such that f * Γ G = φ| U .Moreover, if f 1 , f 2 : U → G are any two smooth maps with this property, then there exists a ∈ G such that f 2 (Q) = af 1 (Q) for all Q ∈ U.

5 , Theorem 1 . 3 ]
and [6, Corollary 5.1]): Let D be a non-involutive C 1 distribution of rank M on a C 2 manifold N .Then, for every M -dimensional C 1 open submanifold M of N , the tangency set of M with respect to D has no (M + 1)-density points relative to M. Section 5 presents an application of Theorem 3.1 in the context of Maurer-Cartan equation.To explain what we are talking about, let us first consider a matrix Lie subgroup G of Gl(L, R) with Lie algebra g and denote its Maurer-Cartan form by Γ G .Recall that Γ G is a left-invariant g-valued smooth differential 1-form on G and be approximated in measure by uniformly N -dense closed subsets of U .More precisely: For all C < L N (U ) there exists a closed set F ⊂ U such that L N (F ) > C and F (m) = ∅ for all m > N (obviously one hasF (N ) ⊂ F and L N (F \ F (N ) ) = 0), cf.[4, Proposition 5.4].• Let N ≥ 2 and E ⊂ R N be a set of finite perimeter, so that H N −1 (∂ * E) <+∞ (where ∂ * E is the reduced boundary of E, cf.[11, Theorem 15.9]).Then L N (E \ E (m 0 ) ) = 0, withm 0 := N + 1 + 1 N − 1 , cf.Theorem 1 in [7, Section 6.1.1](compare also [2, Lemma 4.1]).Moreover, the number m 0 is the maximum order of density common to all sets of finite perimeter.More precisely, the following property holds (cf.[3, Proposition 4.1]): For all m > m 0 there exists a compact set F m of finite perimeter in R N such that L N (F m ) > 0 and F (m) m = ∅.

Theorem 4 . 1 ( 5 .
Theorem 3.2 of[6]).Let ω ∈ C 1 F h (N ) and f : U ⊂ R M → N (where U is open) be a C 1 map.ThenU ∩ {dλ = f * ω} (M +1) ⊂ {f * dω = 0} for every λ ∈ C 1 F h−1 (U ).Proof.Observe that µ := (dλ) ∈ Mat 1 C 0 F h (U ) has the DED in Mat 1 C 0 F h+1 (U ) and δµ = 0, by Remark 3.3.Hence and by Corollary 3.1 (with L = 1) we get (f * dω)Q = 0 for all Q ∈ A (M +1) f,(ω),µ = U ∩ {dλ = f * ω} (M +1) .Remark 4.1.If O is a family of linearly independent C 1 differential 1-forms defining a distribution D of rank M on N (cf.[10,Chapter 19]), then, for all y ∈ N , the M -plane D y is the only M -dimensional integral element of O at y, i.e., V M (O) y = {D y }.Hence: • The set I(f, O) coincides with the tangency set of f (U ) with respect to D; • The condition (4.1) is verified if and only if D is non-involutive at each point of N , cf. [10, Proposition 19.8].Thus the structure identity (4.2) proves that if D is non-involutive at each point of N then the following property holds: For every M -dimensional C 1 open submanifold M of N , the tangency set of M with respect to D has no (M + 1)-density points relative to M, cf.[5, Theorem 1.3] and [6, Corollary 5.1].Applications II, The context of Maurer-Cartan form Let us consider any matrix Lie subgroup G of Gl(L, R) with Lie algebra g ⊂ gl(L, R) and let ι : G → Gl(L, R) be the inclusion map.Then let γ ∈ Mat L C ∞ F 1 (Gl(L, R)) be defined at z = (z ij ) ∈ Gl(L, R) as γ z := (z ij ) −1 (dz ij ) and define the Maurer-Cartan form of G as Γ

Remark 5 .
1 shows that, if M is a C 2 manifold and φ ∈ Mat L C 1 F 1 (M), the occurrence of condition (dφ) Q = −(φ ∧ φ) Q , for all Q ∈ M (5.2)prevents the possibility of φ being locally a C 1 pullback of the Maurer-Cartan form Γ G .Thus, whatever the choice of C 1 map f : U ⊂ M → G, the set {f * Γ G = φ| U } cannot have interior points.In Corollary 5.2 below we provide a structure result for this set, under assumption (5.2), by using superdensity.Now we provide an application of Corollary 3.1, which is the natural counterpart in this context of Theorem 4.1 in Section 4. Theorem 5.2.Let M be a M -dimensional C 2 manifold and let φ ∈ Mat L C 0 F 1 (M) have the DED in Mat L C 0 F 2 (M).Moreover, let U ⊂ M be open and consider a C 1 map f: U → G. Then (δφ) Q = −(φ ∧ φ) Q for all Q ∈ U ∩ {f * Γ G = φ| U } (M +1) .Proof.Let Q ∈ U ∩ {f * Γ G = φ| U } (M +1) and observe that (f * Γ G ) Q = φ Q , (5.3) by continuity.We observe also that, by Proposition 3.2, φ| U has the DED in Mat L C 0 F 2 (U) and δ(φ| U ) = (δφ)| U .If we now apply Corollary 3.1 with M := U, N := G, ω := Γ G , µ := φ| U , then we get [13,ection 3.2.46],[13,Proposition12.6].•If M is a C 1 imbedded submanifold of N and g M denotes the induced metric, then one has H s g If g denotes the standard Euclidean metric on R N , then one obviously has H s g = H s .In particular, H N g is the N -dimensional Lebesgue measure.