Integration of non linear conservation laws?

Transcription

1 Integration of non linear conservation laws? Frédéric Hélein, Institut Mathématique de Jussieu, Paris 7 Advances in Surface Theory, Leicester, June 13, 2013

2 Harmonic maps Let (M, g) be an oriented Riemannian manifold of dimension m. Let (N, h) be a Riemannian manifold of dimension n. A map u : M N is harmonic if g u i + Γ i jk [h](u) αu j β u k g αβ = 0. Alternatively: d u ( du) = 0, where is the Hodge operator: T M Λ m 1 T M;

3 Harmonic maps Let (M, g) be an oriented Riemannian manifold of dimension m. Let (N, h) be a Riemannian manifold of dimension n. A map u : M N is harmonic if g u i + Γ i jk [h](u) αu j β u k g αβ = 0. Alternatively: d u ( du) = 0, where is the Hodge operator: T M Λ m 1 T M; du is an (m 1)-form with values in u T N, where u T N := the vector bundle over M whose fiber at x is T u(x) N.

4 Harmonic maps Let (M, g) be an oriented Riemannian manifold of dimension m. Let (N, h) be a Riemannian manifold of dimension n. A map u : M N is harmonic if g u i + Γ i jk [h](u) αu j β u k g αβ = 0. Alternatively: d u ( du) = 0, where is the Hodge operator: T M Λ m 1 T M; du is an (m 1)-form with values in u T N, where u T N := the vector bundle over M whose fiber at x is T u(x) N. u is the pull-back image by u of the Levi-Civita connection on T N ;

5 Harmonic maps Let (M, g) be an oriented Riemannian manifold of dimension m. Let (N, h) be a Riemannian manifold of dimension n. A map u : M N is harmonic if g u i + Γ i jk [h](u) αu j β u k g αβ = 0. Alternatively: d u ( du) = 0, where is the Hodge operator: T M Λ m 1 T M; du is an (m 1)-form with values in u T N, where u T N := the vector bundle over M whose fiber at x is T u(x) N. u is the pull-back image by u of the Levi-Civita connection on T N ; d u is the covariant differential with respect to u.

6 Integration of the harmonic maps equation Question: can we integrate equation d u ( du) = 0,? Example (use of Noether theorem): assume N = S n, the round sphere in R n+1. Then the harmonic map equation reads: g u + u 2 u = 0. But it can also be written (denoting u du = (u i du j u j du i ) ij ): d( u du) = 0 = u du = db, where b : M is a (m 2)-form with values in so(n + 1). What can we do in the general case?

7 Energy-momentum tensor Let (M, g) be a m-dimensional (pseudo-)riemannian manifold and T = T µ ν µ dx ν be a tensor. Assume that: Can we integrate this equation? Difficult question. µ T µ ν = 0, This problem is of the same type as the preceding one (harmonic maps). Consider τ := T µ ν ( µ vol g ) dx ν, an (m 1)-form with values in T M (i.e. a section of (Λ m 1 T M) T M). Then condition µ T µ ν = 0 is equivalent to: d τ = 0.

8 A general framework Let M be an m-dimensional manifold; This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0.

9 A general framework Let M be an m-dimensional manifold; let V be a rank n vector bundle over M; This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0.

10 A general framework Let M be an m-dimensional manifold; let V be a rank n vector bundle over M; let g be a metric on V ; This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0.

11 A general framework Let M be an m-dimensional manifold; let V be a rank n vector bundle over M; let g be a metric on V ; let be a connection on V which respects the metric g; This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0.

12 A general framework Let M be an m-dimensional manifold; let V be a rank n vector bundle over M; let g be a metric on V ; let be a connection on V which respects the metric g; let ϕ be a p-form on M with values in V. This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0.

13 A general framework Let M be an m-dimensional manifold; let V be a rank n vector bundle over M; let g be a metric on V ; let be a connection on V which respects the metric g; let ϕ be a p-form on M with values in V. This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0. a geometric framework where this make sense;

14 A general framework Let M be an m-dimensional manifold; let V be a rank n vector bundle over M; let g be a metric on V ; let be a connection on V which respects the metric g; let ϕ be a p-form on M with values in V. This gives us a data (M, V, g,, ϕ). We say that (M, V, g,, ϕ) is integrable if Problem: integrate ϕ, i.e. find d ϕ = 0. a geometric framework where this make sense; a way to integrate in this setting.

15 Embedding of a covariant conservation law Consider the following problem: Let (M, V, g,, ϕ) be integrable. In particular d ϕ = 0. Find N N and a fiber bundle embedding s.t. T : V M R N (x, v) (x, T x (v)) x M, T x is an isometric embedding of (V x, g x ) in (R N,, ); Here T ϕ is the p-form on M with values in R N s.t. x M, (T ϕ) x is obtained by the composition (T x M) p V x R N (v 1,, v p ) ϕ x (v 1,, v p ) T x [ϕ x (v 1,, v p )]

16 Embedding of a covariant conservation law Consider the following problem: Let (M, V, g,, ϕ) be integrable. In particular d ϕ = 0. Find N N and a fiber bundle embedding s.t. T : V M R N (x, v) (x, T x (v)) x M, T x is an isometric embedding of (V x, g x ) in (R N,, ); T ϕ is closed. Here T ϕ is the p-form on M with values in R N s.t. x M, (T ϕ) x is obtained by the composition (T x M) p V x R N (v 1,, v p ) ϕ x (v 1,, v p ) T x [ϕ x (v 1,, v p )]

17 Formulation using moving frame Let e 1,, e n be (local) sections of V s.t. x M, (e 1 (x),, e n (x)) is an orthonormal basis of V x (a moving frame). Then is encoded in the connection 1-forms ω a b s.t. e a = e b ω a b ;

18 Formulation using moving frame Let e 1,, e n be (local) sections of V s.t. x M, (e 1 (x),, e n (x)) is an orthonormal basis of V x (a moving frame). Then is encoded in the connection 1-forms ω a b s.t. e a = e b ω a b ; ϕ is encoded by p-forms ϕ a s.t. ϕ = e a ϕ a ;

19 Formulation using moving frame Let e 1,, e n be (local) sections of V s.t. x M, (e 1 (x),, e n (x)) is an orthonormal basis of V x (a moving frame). Then is encoded in the connection 1-forms ω a b s.t. e a = e b ω a b ; ϕ is encoded by p-forms ϕ a s.t. ϕ = e a ϕ a ; equation d ϕ = 0 reads: dϕ a + ω a b ϕb = 0, 1 a, b n.

20 Formulation using moving frame Let e 1,, e n be (local) sections of V s.t. x M, (e 1 (x),, e n (x)) is an orthonormal basis of V x (a moving frame). Then is encoded in the connection 1-forms ω a b s.t. e a = e b ω a b ; ϕ is encoded by p-forms ϕ a s.t. ϕ = e a ϕ a ; equation d ϕ = 0 reads: dϕ a + ω a b ϕb = 0, 1 a, b n. If T : V M R N and E a := T x (e a ), T ϕ = T (e a ϕ a ) = E a ϕ a ;

21 Formulation using moving frame Let e 1,, e n be (local) sections of V s.t. x M, (e 1 (x),, e n (x)) is an orthonormal basis of V x (a moving frame). Then is encoded in the connection 1-forms ω a b s.t. e a = e b ω a b ; ϕ is encoded by p-forms ϕ a s.t. ϕ = e a ϕ a ; equation d ϕ = 0 reads: dϕ a + ω a b ϕb = 0, 1 a, b n. If T : V M R N and E a := T x (e a ), T ϕ = T (e a ϕ a ) = E a ϕ a ; and d(t ϕ) = 0 reads: dϕ a + ω a b ϕb = 0 and ω i b ϕb = 0, i > n (here (E 1,, E n, E n+1,, E N ) is an orthonormal basis of R N and ω α β := de β, E α.

22 Example: harmonic maps into the sphere revisited Consider again u : M S n and du: an (m 1)-form with values in u TS n M R n+1. Then setting T : u TS n M so(n + 1) (x, v) (x, u(x) v) we have: u harmonic d(tu) = 0.

23 Source of inspiration: the isometric embedding problem Let (M, g) be a Riemannian manifold. Find N N and an isometric embedding of (M, g) in the Euclidean vector space (R N,, ), i.e. a map u : M R N s.t. g µν = µ u, ν u. Observation: let ϕ be the 1-form on M with values on T M defined by: x M, ϕ x := identity map: T x M T x M. ϕ is the soldering form. Then d ϕ = 0 because the Levi-Civita connection is torsion free.

24 The isometric embedding problem (alternative formulation) Hence (M, T M, g, ϕ) is integrable! Let u : M R N be an isometric embedding (Cartan Kähler, Nash,...), then du x = du x ϕ x, so if we let then: T := (Id M, du) : T M M R N, du = T ϕ (= d(t ϕ) = 0) and T is a solution of our problem.

25 A local result Theorem [Nabil KAHOUADJI, arxiv: ] Assume p = m 1, let (M, V, g,, ϕ) be integrable and real analytic. Assume that ϕ has a constant rank. Then for each x M, there exists a neighbourhood O of x and a real analytic map of vector bundles T : V O O R n+(m 1)(n 1) s.t. T is isometric and d(t ϕ) = 0. The proof uses the Cartan Kähler theory. Example: if m = n = 4 and p = 3 and ϕ = τ = energy-momentum tensor, T τ is a closed 3-form with values in R 13. Hence T τ = du, where U is a 2-form. The case 1 < p < m 1 is open.

26 Some remarks This does not answer the question of integrating ϕ in a concrete way: if ϕ = τ = energy-momentum, then the numerical values of U (s.t. du = T τ) has no physical meaning;

27 Some remarks This does not answer the question of integrating ϕ in a concrete way: if ϕ = τ = energy-momentum, then the numerical values of U (s.t. du = T τ) has no physical meaning; (However this may be useful in the context of harmonic maps;)

28 Some remarks This does not answer the question of integrating ϕ in a concrete way: if ϕ = τ = energy-momentum, then the numerical values of U (s.t. du = T τ) has no physical meaning; (However this may be useful in the context of harmonic maps;) Main lesson: we should consider the problem the other way around, i.e. contemplate the geometry behind the data (M, V, g,, ϕ).

29 Postulates (or credo) There is a geometry associated to each integrable data (M, V, g,, ϕ);

30 Postulates (or credo) There is a geometry associated to each integrable data (M, V, g,, ϕ); For an arbitrary p, in this geometry, the elementary objects are (p 1)-dimensional.

31 Postulates (or credo) There is a geometry associated to each integrable data (M, V, g,, ϕ); For an arbitrary p, in this geometry, the elementary objects are (p 1)-dimensional. We call an integrable data (M, V, g,, ϕ) a (p 1)-puzzle;

32 Postulates (or credo) There is a geometry associated to each integrable data (M, V, g,, ϕ); For an arbitrary p, in this geometry, the elementary objects are (p 1)-dimensional. We call an integrable data (M, V, g,, ϕ) a (p 1)-puzzle; The condition d ϕ = 0 is an analogue of a torsion free condition;.

33 Functorial remarks If (N, V, g,, ϕ) is a (p 1)-puzzle and if u : M N is a smooth map, we define the pull-back image of (N, V, g,, ϕ) by u to be the (p 1)-puzzle: u (N, V, g,, ϕ) := (M, u V, u g, u, ϕ du). Then if (N, V, g,, ϕ) is integrable, u (N, V, g,, ϕ) is integrable.

34 The case p = 1 We assume that the rank of ϕ is constant, equal to k. if m = k = n, then ϕ x is a vector space isomorphim, which induces an isomorphism T M V. Through this isomorphism g gives a Riemannian metric on M and the Levi-Civita connection;

35 The case p = 1 We assume that the rank of ϕ is constant, equal to k. if m = k = n, then ϕ x is a vector space isomorphim, which induces an isomorphism T M V. Through this isomorphism g gives a Riemannian metric on M and the Levi-Civita connection; if m = k < n, then ϕ x is injective, its image H x := ϕ x (T x M) is a subpsace of V x, which induces a vector subbundle H of V. Then we can identify (M, V, g,, ϕ) with a Riemannian manifold together with a (formal) second fundamental form in an n-dimensional Riemannian manifold (modelled on the total space of the normal bundle H in V );

36 The case p = 1 We assume that the rank of ϕ is constant, equal to k. if m = k = n, then ϕ x is a vector space isomorphim, which induces an isomorphism T M V. Through this isomorphism g gives a Riemannian metric on M and the Levi-Civita connection; if m = k < n, then ϕ x is injective, its image H x := ϕ x (T x M) is a subpsace of V x, which induces a vector subbundle H of V. Then we can identify (M, V, g,, ϕ) with a Riemannian manifold together with a (formal) second fundamental form in an n-dimensional Riemannian manifold (modelled on the total space of the normal bundle H in V ); if m > k = n, then ϕ x is onto but has a nontrivial kernel the most interesting case...

37 The case p = 1 and m > k = n Consider K x := Kerϕ x, it gives a rank m n distribution.

38 The case p = 1 and m > k = n Consider K x := Kerϕ x, it gives a rank m n distribution. This distribution is integrable because of: dϕ a = ω a b ϕb and by Frobenius Theorem. Hence there exists (locally) a foliation of M by (m n)-dimensional submanifolds Σ on which ϕ vanishes.

39 The case p = 1 and m > k = n Consider K x := Kerϕ x, it gives a rank m n distribution. This distribution is integrable because of: dϕ a = ω a b ϕb and by Frobenius Theorem. Hence there exists (locally) a foliation of M by (m n)-dimensional submanifolds Σ on which ϕ vanishes. The restriction of the connection on each leaves Σ of the foliation is flat!

40 The case p = 1 and m > k = n Consider K x := Kerϕ x, it gives a rank m n distribution. This distribution is integrable because of: dϕ a = ω a b ϕb and by Frobenius Theorem. Hence there exists (locally) a foliation of M by (m n)-dimensional submanifolds Σ on which ϕ vanishes. The restriction of the connection on each leaves Σ of the foliation is flat! Actually we can construct locally a metric g on M/Σ (the quotient by the foliation) s.t. (M, V, g,, ϕ) is the pull-back image by the projection M M/Σ of (M/Σ, T (M/Σ), g,, solder).

41 Idea of the proof Let X and Y be two vector fields on M s.t. ϕ(x ) = 0 and [X, Y ] = 0. Then dϕ a (X, Y ) = X ϕ a (Y ) Y ϕ a (X ) ϕ a ([X, Y ]) = X ϕ a (Y ) 0 0, and ω a b ϕb (X, Y ) = ω a b (X )ϕb (Y ) ω a b (Y )ϕb (X ) = ω a b (X )ϕb (Y ) 0. Thus 0 = (dϕ a + ω a b ϕb )(X, Y ) = X ϕ a (Y ) + ω a b (X )ϕb (Y ) = ( X ϕ(y )) a. Hence the restriction of ϕ(y ) to a leaf is paralell. Lastly we can construct a family (Y 1,, Y n ) of such vector fields, s.t. (ϕ(y 1 ),, ϕ(y n )) is a paralell frame on Σ.

42 Isometric embedding of 0-puzzles Using the preceding we can embedd isometrically locally any 0-puzzle (M, V, g,, ϕ) if ϕ has a constant k. We consider the foliation by leaves Σ which are tangent to the distribution kerϕ and construct a Riemannian manifold (N, h) of dimension n s.t. (M, V, g,, ϕ) is the pull-back of (N, T N, h, [h], solder) by the isometric maps: (M, g) onto (M/Σ, g) injective (N, h) Lastly one can embedd isometrically (N, h) (R N,, ). Then (M, V, g,, ϕ) is the pull-back of (R N, T R N,,, d, Id) by the composition of these maps. The differential of this map gives an isometric embedding of (M, V, g,, ϕ).

43 For p > 1? Largely open Try to understand known examples and results: the isometric embedding result of N. Kahouadji for p = m 1;

44 For p > 1? Largely open Try to understand known examples and results: the isometric embedding result of N. Kahouadji for p = m 1; the soldering 1-form on a Riemannian manifold is the differential of the identity map from the fluid M to the rigid (M, g);

45 For p > 1? Largely open Try to understand known examples and results: the isometric embedding result of N. Kahouadji for p = m 1; the soldering 1-form on a Riemannian manifold is the differential of the identity map from the fluid M to the rigid (M, g); the curvature 2-form Ω satisfies the Bianchi identity (dω + [ω Ω] = 0) and is exact as being dω [ω ω];

46 For p > 1? Largely open Try to understand known examples and results: the isometric embedding result of N. Kahouadji for p = m 1; the soldering 1-form on a Riemannian manifold is the differential of the identity map from the fluid M to the rigid (M, g); the curvature 2-form Ω satisfies the Bianchi identity (dω + [ω Ω] = 0) and is exact as being dω [ω ω]; on a 4-dimensional manifold, if τ is the stress-energy tensor, solving the Einstein equation R µν s 2 g µν = T µν amounts to find a soldering 1-form α s.t. 1 2 ɛ λµνσω λµ α ν = τ σ. The left hand side looks roughly like the exterior product of two exact forms.

47 Further questions integrate d ( F ) = 0 (Yang Mills);

48 Further questions integrate d ( F ) = 0 (Yang Mills); would it mayful useful for string theory, branes theory?

49 References F. Hélein, Applications harmoniques, lois de conservation et repères mobiles, Diderot eds., 1996; Harmonic maps, conservation laws and moving frames, Cambridge, N. Kahouadji, Construction of local conservation laws by generalized isometric embeddings of vector bundles, arxiv: F. Hélein, Manifolds obtained by soldering together points, lines, etc., in Geometry, topology, quantum field theory and cosmology, C. Barbachoux, J. Kouneiher, F. Hélein, eds, collection Travaux en Cours (Physique-Mathématiques), Hermann 2009, p