DIFFERENTIAL FORMS ON CORNERS

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1 DIFFERENTIAL FORMS ON CORNERS SERAP GÜRER AND PATRICK IGLESIAS-ZEMMOUR ref. We show that, for the subset diffeology, differential forms defined on halfspaces or corners of Euclidean spaces, are the restrictions of a differential forms defined on an open neighborhood of the corner in the ambient Euclidean space. Usually, smooth maps from corners K n = {(x 1,..., x n ) x i 0} into the real line R are just defined as restrictions of smooth maps, defined on some open neighborhood of the corner [Cer61] [Dou62] etc. This heuristic becomes a theorem in diffeology where K n is equipped with the subset diffeology. Indeed every map from K n to R such that composed with a smooth parametrisation 1 P: U R n, taking its values in K n, is smooth, is the restriction of a smooth maps defined on some open neighborhood of the corner [PIZ13, 4.16]. It is always a progress when a convention, based on mathematicians intuition, becomes a theorem in a well defined axiomatic. Here the axiomatic is the theory of Diffeology. Noticing that C (K n, R) is just the space of differential 0-forms Ω 0 (K n ), it is legitimate to ask about the behavior of differential k-forms on K n, that is, Ω k (K n ) a it is defined in [PIZ13, 6.28]. In this paper we prove the following theorem (art. 4): Theorem. Every differential form on the corner K n is the restriction of a smooth form on an open neighborhood of K n in R n. Precisely, the pullback : j : Ω k (R n ) Ω k (K n ) is surjective, where j denotes the inclusion from K n into R n. Date: July 25, This research is partially supported by Tübitak, Career Grant No. 115F410. The authors thank the Institut d Études Politiques d Aix en Provence for its hospitality in 2016 and A parametrisation is just a map defined on an open subset of an Euclidean space. 1

2 2 SERAP GÜRER AND PATRICK IGLESIAS-ZEMMOUR Let us just remind that a differential k-form on a diffeological space X is a mapping α that associates with each plot P in X, a smooth k- form α(p) on dom(p), such that the smooth compatibility condition α(f P) = F (α(p)) is satisfied, where F is any smooth parametrisation in dom(p). Smooth Structure on Corners 1. Corners as Diffeologies. We denote by K n the corner K n = {(x i ) n i=1 R n x i 0, i = 1,..., n}. And we equip it with the subset diffeology. A plot in K n is just a regular smooth parametrization in R n but taking its values in K n. (A) The corner K n is the diffeological n-power of the half-line K = [0, [ R, equipped with the subset diffeology. (B) The corner K n is embedded in R n, and closed. That is, the D- topology of the induction K n R n coincides with the induced topology 2 of R n, see [PIZ13, 2.13]. (C) Let X 0 = {0} X 1 X n = K n be the natural filtration of K n, where the levels X j are defined by X j = {(x i ) n i=1 K n there exist i 1 < < i n j such that x il = 0}. Then, the stratum S j = X j X j 1 is the subset of points in R n that have j, and only j, coordinates strictly positive. The strata S j are equipped with the subset diffeology 3. { } S j = (x i ) n i=1 R n There exist i 1 < < i j such that x il > 0,. and x m = 0 for all m / {i 1,..., i j.} Then, S j is D-open in X j, j 1. As a subset of X j, S j is the (diffeological) sum of ( n j ) connected components indexed by a string of j ones and n j zeros. Proof. For the first item, it s immediately by definition. Considering the second item: for any subset U K n open for the induced topology, there exists (by definition) an open subset O R n such that U = O K n. Then, for all plots P in K n, P 1 (U) = P 1 (O) is open, because plots are continuous. On the other hand, let U K n be D-open. Then, sq 1 (U) R n is open, where sq : R n K n is the map sq (x i ) n i=1 = (x2 i )n i=1. And 2 The standard topology of R n is the D-topology of its standard smooth structure. 3 Recall that, by transitivity of subset diffeology, to be a subspace of Sl or K n or of R n is identical.

3 DIFFERENTIAL FORMS ON CORNERS 3 sq 1 (U) K n is open for the induced topology of R n. Now, the map sq restricted to K n is an homeomorphism. Hence, since U = sq (sq 1 (U) K n ), U is open for the induced topology of R n. Therefore the D-topology of the induction coincides with the induced topology, as we claimed. For the third item: let x X j, then the number ν of coordinates of x that are 0 is at least n j, i.e. ν n 1. Next, if x X j and x / X j 1, then ν n j and ν < n j + 1, thus, ν = n j. Therefore, X j X j 1 is the subset of points in R n that have exactly n j coordinates equal to 0 and the other j strictly positive: Consider now a point x = (x 1,..., x n ) S j S j 1. Since the j nonzero coordinates of x are strictly positive, there exists ε > 0 such that x i ε > 0, for all non-zero coordinate of x. The open n-parallelepiped C x =]x 1 ε, x 1 + ε[ ]x n ε, x n + ε[ R n contains x, and C x S j S j S j 1. Thus, S j S j 1 = C x S j. x S j S j 1 Now, let P : U S j be a plot for the subset diffeology. Then, P 1 (S j S j 1 ) = x Sj S j 1 P 1 (C x S j ), but P 1 (C x S j ) = P 1 (C x ) since val(p) S j. Next, since P is smooth as a map into R n and C x is open, P 1 (C x ) is open and then P 1 (S j S j 1 ) is open. Therefore, S j S j 1 is D-open in S j. 2. Smooth Maps on Corners. It has been proved that a map f : K n R, is smooth in the sense of diffeology, if and only if it is the restriction of a smooth map F defined on some open neighborhood O of K n into R [PIZ13, 4.16]. That is, f C (K n, R) if and only if, f = F K n and F C (O, R). 3. The Square Function Lemma. Let sq : R n K n be the smooth parametrisation: sq (x 1,..., x n ) = (x 2 1,..., x 2 n). Then sq : Ω k (K n ) Ω k (R n ) is injective. That is, for all α Ω k (K n ), if sq (α) = 0, then α = 0. Proof. Note that each component of S j S j 1 is isomorphic to R j. Hence, if sq (α) = 0, since sq sq 1 (S j S j 1 ) is a 2-folds covering over S j S j 1, α S j S j 1 = 0. that is, for all plot Q in S j S j 1, α(q) = 0. Let then, for some j 1, P j : U j S j be a plot. In view of what precedes, the subset O j = P 1 j (S j S j 1 ) is open, and α(p j O j ) = α(p j ) O j = 0. By continuity, α(p j ) O j = 0, where O j is the closure of O j. Let then U j 1 = U j O j and P j 1 = P j U j 1. Then, U j 1 is open and P j 1 : U j 1 S j 1 is a plot. This construction gives a descending recursion, starting with

4 4 SERAP GÜRER AND PATRICK IGLESIAS-ZEMMOUR any plot P: U K n, by initializing P n = P, U n = U and S n = K n. One has P j = P U j, U j 1 U j, the recursion ends with a plot P 0 with values in S 0 = {0}, and α(p 0 ) = 0 since P 0 is constant. Therefore α = Differential Forms On Corners. The previous article (art. 2) deals with smooth real functions on corners, that is, Ω 0 (K n ). It is a particular case of the more general theorem: Theorem. Any differential k-form on the corner K n, equipped with the subset diffeology of R n, is the restriction of a smooth differential k-form defined on some open neighborhood of the corner. Precisely, the pullback : j : Ω k (R n ) Ω k (K n ) is surjective, where j denotes the inclusion from K n to R n. Proof. Let ω Ω k (K n ) and K n = {(x i ) n i=1 x i > 0, i = 1,..., n}. One has ω K n = a i1...i k (x 1,..., x n ) dx i1 dx ik, with i j = 1,..., n and a i1...i k C ( K n, R). Recall that sq : (x i ) n i=1 (xi 2)n i=1, then sq (ω) = A i1...i k (x 1,..., x n ) dx i1 dx ik, where A i1...i k C (R n, R). Let ε j : (x 1,..., x j,..., x n ) (x 1,..., x j,..., x n ), then sq ε j = sq and (sq ε j ) (ω) = ε j (sq (ω)), that is, sq (ω) = ε j (sq (ω)). Hence, ε j (sq (ω)) = A i1...i k (x 1,..., x j,..., x n ) dx i1 dx ik Then, Hence, Thus, A i l j i l j i 1 < j <i k A i1...j...i k (x 1,..., x j,..., x n ) dx i1... dx j dx ik. (x 1,..., x j,..., x n ) = A i1...i k (x 1,..., x j,..., x n ), A i1...j...i k (x 1,..., x j,..., x n ) = A i1...j...i k (x 1,..., x j,..., x n ). A i1...j...i k (x 1,..., x j = 0,..., x n ) = 0. A i1...j...i k (x 1,..., x j,..., x n ) = 2x j A i1...j...i k (x 1,..., x j,..., x n ), with A i1...j...i k C (R n, R). Therefore, there are real smooth functions  i1...i k defined on R n such that A i1...i k (x 1,..., x n ) = 2 k x i1... x ik  i1...i k (x 1,..., x n ).

5 Now, implies Hence, = DIFFERENTIAL FORMS ON CORNERS 5 sq (ω K n ) = sq (ω) {x i 0} 2 k x i1... x ik a i1...i k (x1, 2..., xn) 2 dx i1 dx ik 2 k x i1... x ik  i1...i k (x 1,..., x n ) dx i1 dx ik.  i1...i k (x 1,..., x n ) = a i1...i k (x 2 1,..., x 2 n) for x i 0, i = 1,..., n. Thus (x 1,..., x n )  i1...i k (x 1,..., x n ), which belongs to C (R n, R), is even in each variable. Therefore, according to Schwartz Theorem [Sch75] 4, there exist a i1...i k C (R n, R), such that One deduces:  i1...i k (x 1,..., x n ) = a i1...i k (x 2 1,..., x 2 n). a i1...i k (x 1,..., x n ) = a i1...i k (x 1,..., x n ), for all (x 1,..., x n ) K n. Then, defining the k-form ω on R n by ω = a i1...i k (x 1,..., x n ) dx i1 dx ik, one has already ω K n = ω K n. Let us show that ω K n = ω. That is, let us check that for all plot P: U R n, P (ω) = ω(p). Actually, one has Indeed: sq (ω) = sq (ω K n ). sq (ω) = A i1...i k (x 1,..., x n ) dx i1 dx ik = 2 k x i1... x ik  i1...i k (x 1,..., x n ) dx i1 dx ik = 2 k x i1... x ik a i1...i k (x1, 2..., xn) 2 dx i1 dx ik. And, on the other hand: sq (ω K n ) = 2 k x i1... x ik a i1...i k (x1, 2..., xn) 2 dx i1 dx ik. 4 Which is a generalisation of a famous Whitney Theoreme [Whi43]

6 6 SERAP GÜRER AND PATRICK IGLESIAS-ZEMMOUR Thus, sq (ω ω K n ) = 0. Therefore, according to the previous lemma (art. 3), ω ω K n = 0. And indeed, ω is the restriction of the smooth k-form ω on K n. 5. An Exemple Of Application. Among the possible applications of the theorems above (art. 3) and (art. 4), there is already one worthy of mention. It is about the description of closed 2-form, invariant with respect to the action of a Lie group. As it has been showed in particular in the classification of SO(3)-symplectic manifolds [Igl84, Igl91], any closed 2-form form ω on a manifold M, invariant by a compact group 5 G, is characterized by its moment map μ: M G (we assume the action Hamiltonian), and for each moment map, a closed 2-form ε Z 2 (M/G). Let us be precise: the space of closed 2-forms Z 2 (M) is a vector space, the space of G-equivariant maps from M to G is also a vector space. Then, the map associating its moment map 6 μ with each invariant closed 2-form ω is linear. What we claim is that the kernel of this map is exactly Z 2 (M/G), where M/G is equipped with the quotient diffeology. Now, if an equivariant map is easy to conceive, it is more problematic for a differential form on the space of orbits, which is generally not a manifold. This is where the above theorem can help, because it happens that M/G is not far to be a manifold with boundary or corners, as show the following example. Let us consider the simple case of M = R 2n equipped with the standard symplectic form ω = n i=1 dq i dp i. It is invariant by the group SO(2, R) n acting naturally, each factor on its respective copie of R 2. The (diffeological) quotient space Q n = R 2n / SO(2, R) n is the n-th power of Q = R 2 / SO(2, R). Let X = (X i ) n i=1 with X i = (q i, p i ). There is a natural smooth bijection j 2n : Q n K n, given by j 2n : class(x) ( X i 2 ) n i=1. It turns out that this smooth bijection 7 induces, by pullback, an injection j 2n from Ωk (K n ) into Ω k (Q n ). Thus, thanks to (art. 4), for each 2-form ε on the quotient Q n there exists a 2-form ε on R n, such that ε = j 2n (ε). And the 2-form ω is characterized by μ and ε K n, with ε Ω k (R n ). Proof. Let us prove that j 2n is injective. Let x = (x i) n i=1 Rn and ι n : x (x i, 0) n i=1 from Rn into R 2n. Let j n : R n /{±1} n K n be defined by j n : class(x) sq (x) = (x 2 i )n i=1. Then, j n = j 2n ι, where ι is the 5 There could a diffeological generalisation possible here to non compact group. 6 The manifold M is supposed to be connected. To have a unicity of the moment maps we decide to fix their value to 0 at some base point m 0 M, for example. 7 Which is not a diffeomorphism [PIZ07].

7 DIFFERENTIAL FORMS ON CORNERS 7 projection of ι, from R n /{±1} n to Q n. R n ι R 2n class class R n /{±1} n ι Q n = R 2n / SO(2, R) n K n = [0, [ n But sq = j n class and we know that sq = class jn : Ω k (K n ) Ω k (R n ) is injective (art. 3), thus jn : Ω k (K n ) Ω k (R n /{±1}) is injective. On the other hand, j n = j 2n ι, then jn = ι j2n. Since j n is injective, j2n is necessarilly injective too. References [Cer61] Jean Cerf. Topologie de certains espaces de plongements. Bulletin de la SMF, 89: , [Dou62] Adrien Douady. Variétés à bord anguleux et voisinages tubulaires. In Séminaire Henri Cartan, number 14, pages [Igl84] Patrick Iglesias. Classification des SO(3)-variétés symplectiques de dimension 4. Preprint CPT-84/PE.1673, Centre de Physique Théorique, Marseille Luminy, France, [Igl91] Les SO(3)-variétés symplectiques et leur classification en dimension 4. Bull. Soc. Math. France, 119(4): , First publication [Igl84]. [PIZ07] Patrick Iglesias-Zemmour. Dimension in diffeology. Indagationes Mathematicae, 18(4), [PIZ13] Diffeology. Mathematical Surveys and Monographs. The American Mathematical Society, vol. 185, RI USA, [Sch75] Gerald W. Schwarz. Smooth functions invariant under the action of a compact lie group. Topology, 14:63 68, [Whi43] Hassler Whitney. Differentiable even functions. Duke Math. Journal, pp , vol. 10, Patrick Iglesias-Zemmour, Institut de Mathématique de Marseille, CNRS France & Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel. Serap Gürer, Galatasaray University, Ortaköy, Çraan Cd. No:36, Beikta/stanbul, Turkey. j 2n