# CARTAN S GENERALIZATION OF LIE S THIRD THEOREM

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1 CARTAN S GENERALIZATION OF LIE S THIRD THEOREM ROBERT L. BRYANT MATHEMATICAL SCIENCES RESEARCH INSTITUTE JUNE 13, 2011 CRM, MONTREAL

2 In many ways, this talk (and much of the work it reports on) owes its existence to an article I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. Part I (the transitive groups), JournalD AnalyseMath. 15 (1965), Simple examples (now called pseudogroups) include: (1) Diffeomorphisms (2) Volume-preserving diffeomorphisms (3) Symplectomorphisms and Contactomorphisms (4) Biholomorphisms, etc. Part II never appeared, and so there was no mention of gauge groups (one of Cartan s examples of an intransitive infinite simple group), and some of the more delicate issues that Cartan faced (with only partial success) were avoided.

3 In many ways, this talk (and much of the work it reports on) owes its existence to an article I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. Part I (the transitive groups), JournalD AnalyseMath. 15 (1965), Simple examples (now called pseudogroups) include: (1) Diffeomorphisms (2) Volume-preserving diffeomorphisms (3) Symplectomorphisms and Contactomorphisms (4) Biholomorphisms, etc. Cartan s work on the intransitive case has been (partially) taken up by many different people, and it has given rise to the theory of what are now called Lie algebroids and Lie groupoids.

4 Lie s Third Theorem: If L is a finite-dimensional, real Lie algebra, then there exists a Lie algebra homomorphism λ : L Vect(L) satisfying λ(x)(0) = x for all x L. Dual Formulation: Let δ : L Λ 2 (L )bealinearmap. Ifitsextension δ :Λ (L ) Λ (L )asagradedderivationofdegree1satisfiesδ 2 =0, then there is a DGA homomorphism φ : ( Λ (L ),δ ) ( Ω (L), d ) satisfying φ(α)(0) = α. Basis Formulation: If Cjk i = Ci kj (1 i, j, k n) areconstants,then there exist linearly independent 1-forms ω i (1 i n) onr n satisfying the structure equations dω i = 1 2 Ci jk ω j ω k if and only if these formulae imply d ( dω i) =0as a formal consequence.

5 Ageometricproblem:Classify those Riemannian surfaces (M 2,g)whose Gauss curvature K satisfies the second order system for some functions a and b. Hess g (K) =a(k)g + b(k)dk 2 Analysis Writing g = ω ω 2 2,thestructureequationsyield dω 1 = ω 12 ω 2 dω 2 = ω 12 ω 1 dω 12 = Kω 1 ω 2 dk = K 1 ω 1 + K 2 ω 2 and the condition to be studied is encoded as ( ) ( ) ( dk1 K2 a(k)+b(k) 2 K1 b(k) K = ω dk 2 K K b(k) K 1 K 2 a(k)+b(k) K 1 Applying d 2 =0tothesetwoequationsyields ( a (K) a(k)b(k)+k ) K i =0 fori =1, 2. )( ω1 Thus, unless a (K) =a(k)b(k) K, suchmetricshavek constant. ). ω 2

6 Conversely, suppose that a (K) = a(k)b(k) K. Does there exist a solution (N 3,ω)tothefollowingsystem? dω 1 = ω 12 ω 2 dω 2 = ω 12 ω 1 ω 1 ω 2 ω 12 0, ω = ω 1 ω 2 dω 12 = Kω 1 ω 2 ω 12 dk K 1 K 2 0 ω 1 dk 1 = 2 a(k)+b(k) K 1 b(k) K 1 K 2 K 2 ω 2. 2 dk 2 b(k) K 1 K 2 a(k)+b(k) K 1 ω 12 Note: d 2 =0is formallysatisfied forthesestructureequations. K 1 Answer: AtheoremofÉ. Cartan [1904] implies that a solution (N 3,ω) does indeed exist and is determined uniquely (locally near p, up to diffeomorphism) by the value of (K, K 1,K 2 )atp.

7 Cartan s result: Suppose that Cjk i = Ci kj and F i α (with 1 i, j, k n and 1 α s) arereal-analyticfunctionsonr s such that the equations dω i = 1 2 Ci jk (a) ωj ω k and da α = Fi α (a) ωi formally satisfy d 2 =0. Then,foreveryb 0 R s,thereexistsanopen neighborhood U of 0 R n,linearlyindependent1-formsη i on U, anda function b : U R s satisfying dη i = 1 2 Ci jk(b) η j η k, db α = F α i (b) η i, and b(0) = b 0. Up to local diffeomorphism of (R n, 0), the pair (η, b) isgerm-unique. Remark 1: Cartan assumed that F =(Fi α)hasconstantrank,butit turns out that, for a solution (η, b) withu connected, F (b) = ( Fi α(b)) always has constant rank anyway. Remark 2: Cartan worked in the real-analytic category and used the Cartan-Kähler theorem in his proof, but the above result is now known to be true in the smooth category. (Cf. P. Dazord)

8 The Hessian equation example: dω 1 = ω 12 ω 2 dω 2 = ω 12 ω 1 ω 1 ω 2 ω 12 0, ω = ω 1 ω 2 dω 12 = Kω 1 ω 2 ω 12 dk K 1 K 2 0 ω 1 dk 1 = 2 a(k)+b(k) K 1 b(k) K 1 K 2 K 2 ω 2. 2 dk 2 b(k) K 1 K 2 a(k)+b(k) K 1 ω 12 K 1 d 2 =0isformallysatisfiedwhena (K) =a(k)b(k) K. Remark: The F -matrix either has rank 0 (when K 1 = K 2 = a(k) =0) or 2 (all other cases). The rank 0 cases have K constant. The rank 2 cases have a 1-dimensional symmetry group and each represents a surface of revolution.

9 Modern formulation of Cartan s theory: A Lie algebroid is a vector bundle a : Y A over a manifold A endowed with a Lie algebra structure {, } :Γ(Y ) Γ(Y ) Γ(Y ) and a bundle map α : Y TA that induces a Lie algebra homomorphism on sections and satisfies the Leibnitz compatibility condition {U, fv } = α(u)(f) V + f {U, V } for f C (A) andu, V Γ(Y ). In our case, take a basis U i of Y = R s R n with a : Y R s the projection and define {U j, U k } = C i jk(a) U i and α(u i )=F α i (a) a α. The formal condition d 2 = 0 ensures that α induces a Lie algebra homomorphism and satisfies the above compatibility condition. A solution isab : B n A covered by a bundle map η : TB Y of rank n that induces a Lie algebra homomorphism on sections.

10 Some geometric applications: 1. Classification of Bochner-Kähler metrics. (B, 2001) 2. Classification of Levi-flat minimal hypersurfaces in C 2.(B,2002) 3. Classification of isometrically deformable surfaces preserving the mean curvature. (originally done by Bonnet in the 1880s) 4. Surfaces with prescribed curvature operator. (B, 2001) 5. Classification of projectively flat Finsler surfaces of constant flag curvature (B, 1997). 6. Classification of austere 3-folds. (B, 1991) 7. Classification of the exotic symplectic holonomies. (originally done by Schwachhofer, et al.) Many more examples drawn from classical differential geometry.

11 AgeneralizationofCartan stheorem.consider equations dη i = 1 2 Ci jk(h) η j η k dh a = ( F a i (h)+a a iα(h)p α) η i. C i jk, F a i,anda a iα (where 1 i, j, k n, 1 a s, and1 α r) are specified functions on a domain X R s. Assume: (1) The functions C, F,andA are real analytic. (2) The tableau A(h) = ( A a iα (h)) is rank r and involutive, withcartan characters s 1 s 2 s q >s q+1 =0forallh R s. (3) d 2 =0reducestoequationsoftheform 0=A a iα(h) ( dp α + B α j (h, p) η j) η i for some functions Bj α.(torsion absorbable hypothesis) Then: Modulo diffeomorphism, the general real-analytic solution depends on s q functions of q variables. Moreover, one can specify h and p arbitrarily at a point. Remark: The proof is a straightforward modification of Cartan s proof in the case r =0(i.e.,whenthereareno freederivatives p α ).

12 Example: Riemannian 3-manifolds with const. Ricci eigenvalues. For simplicity, assume the eigenvalues are all distinct, with c 1 >c 2 >c 3. Then there exist ω i,φ i so that g = ω ω ω 3 2 and dω 1 = φ 2 ω 3 φ 3 ω 2 dω 2 = φ 3 ω 1 φ 1 ω 3 dω 3 = φ 1 ω 2 φ 2 ω 1 dφ 1 = φ 2 φ 3 + c 1 ω 2 ω 3 dφ 2 = φ 3 φ 1 + c 2 ω 3 ω 1 dφ 3 = φ 1 φ 2 + c 3 ω 1 ω 2 2 nd Bianchi implies that there are functions a i and b i so that φ 1 = a 1 ω 1 +(c 1 c 3 ) b 3 ω 2 +(c 1 c 2 ) b 2 ω 3 φ 2 = a 2 ω 2 +(c 2 c 1 ) b 1 ω 3 +(c 2 c 3 ) b 3 ω 1 φ 3 = a 3 ω 3 +(c 3 c 2 ) b 2 ω 1 +(c 3 c 1 ) b 1 ω 2. d(dω i )=0,thenyields9equationsforda i,db i. These can be written in the form da i = ( A ij (a, b)+p ij ) ωj db i = ( B ij (a, b)+q ij ) ωj, where q ii =0andq ij = p jk /(c i c j )when(i, j, k) aredistinct.

13 dω 1 = φ 2 ω 3 φ 3 ω 2 dω 2 = φ 3 ω 1 φ 1 ω 3 dω 3 = φ 1 ω 2 φ 2 ω 1 φ 1 = a 1 ω 1 +(c 1 c 3 ) b 3 ω 2 +(c 1 c 2 ) b 2 ω 3 φ 2 = a 2 ω 2 +(c 2 c 1 ) b 1 ω 3 +(c 2 c 3 ) b 3 ω 1 φ 3 = a 3 ω 3 +(c 3 c 2 ) b 2 ω 1 +(c 3 c 1 ) b 1 ω 2. da i = ( A ij (a, b)+p ij ) ωj db i = ( B ij (a, b)+q ij ) ωj, where q ii =0andq ij = p jk /(c i c j )when(i, j, k) aredistinct. This defines an involutive tableau (in the p-variables) of rank r =9and with characters s 1 =6,s 2 =3,ands 3 =0. Cartan scriteriaaresatisfied, so the desired metrics depend on three functions of two variables. Asimilaranalysisapplieswhenc 1 = c 2 c 3,showingthatsuchmetrics depend on one function of two variables. When c 1 = c 2 = c 3,theCartananalysisgivestheexpectedresultthat the solutions depend on a single constant.

14 General Holonomy. A torsion-free H-structure on M n (where H GL(m) anddim(m) =n) satisfiesthefirststructureequation dω = φ ω where φ takes values in h GL(m) andthesecondstructureequation dφ = φ φ + R(ω ω) where R takes values in K 0 (h), the space of curvature tensors. Bianchi becomes dr = φ R + R (ω) where R takes values in K 1 (h) K 0 (h) m. Second Theorem: For all groups H satisfying Berger s criteria except the exotic symplectic list, K 1 (h) isaninvolutivetableauandtheaboveequations satisfy Cartan s criteria. Ex: For G 2 GL(7, R), the tableau K 1 (g 2 )hass 6 = 6 >s 7 =0,sothe general metric with G 2 -holonomy depends on 6 functions of 6 variables.

15 Other examples. 1. (Cartan, 1926) The metrics in dimension 4 with holonomy SU(2) depend locally on 2 functions of 3 variables. 2. (Cartan-Einstein, ) Analysis of Einstein s proposed unified field theory via connections with torsion. 3. (Cartan, 1943) The Einstein-Weyl structures in dimension 3 depend on 4 functions of 2 variables. 4. (B, 1987) The Ricci-solitons in dimension 3 depend on 2 functions of 2 variables. (More generally, Ric(g) =a(f) g +b(f)df 2 +c(f) Hess g (f).) 5. (B, 2008) The solitons for the G 2 -flow in dimension 7 depend on 16 functions of 6 variables.

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