Convex real projective structures and Weil's local rigidity Theorem
aa r X i v : . [ m a t h . G T ] J un CONVEX REAL PROJECTIVE STRUCTURES AND WEIL’S LOCALRIGIDITY THEOREM
INKANG KIM AND GENKAI ZHANGA
BSTRACT . For an n -dimensional real hyperbolic manifold M , we calculate the Zariskitangent space of a character variety χ ( π ( M ) , SL ( n + 1 , R )) , n > at Fuchisan loci toshow that the tangent space consists of cubic forms. Furthermore we prove the Weil’slocal rigidity theorem for uniforml hyperbolic lattices using real projective structures.
1. I
NTRODUCTION
A flat projective structure on an n -dimensional manifold M is a ( RP n , P SL ( n +1 , R )) -structure, i.e., there exists a maximal atlas on M whose transition maps are restrictionsto open sets in RP n of elements in P SL ( n + 1 , R ) . Then there exist a natural holonomymap ρ : π ( M ) → P SL ( n + 1 , R ) and a developing map f : ˜ M → RP n such that ∀ x ∈ ˜ M , ∀ γ ∈ π ( M ) , f ( γx ) = ρ ( γ ) f ( x ) . In this paper, since we will consider projective structures deformed from hyperbolicstructures, all holonomy representations will lift to SL ( n + 1 , R ) . An RP n -structure isconvex if the developing map is a homeomorphism onto a convex domain in RP n . Itis properly convex if the domain is included in a compact convex set of an affine chart, strictly convex if the convex set is strictly convex. Surprisingly, while many people areworking on global structures of the Hitchin component, it seems that the local structurehas been neglected. This is the starting point of this article. We shall first compute thecohomology H ( π ( M ) , ρ, s l ( n + 1 , R )) of a Fuchsian point ρ which corresponds to ahyperbolic structure π ( M ) → SO ( n, ⊂ SL ( n + 1 , R ) . See Section 2.1 for details.The cohomology is described in terms of quadratic and cubic forms. We shall use theresult of Labourie [9] where he proved that a convex projective flat structure on M definesa Riemannian metric and a cubic form on M . Theorem 1.1.
Let ρ : π ( M ) → SO ( n, ⊂ SL ( n + 1 , R ) , n > , be a repre-sentation defining a real hyperbolic structure on a closed n-manifold M . Let α ∈ H ( π ( M ) , ρ, s l ( n + 1 , R )) . Then α is represented by a cubic form. Keywordsandphrases. Zariski tangent space, real projective structure, Weil’s local rigidity. Research partially supported by STINT-NRF grant (2011-0031291). Research by G. Zhang is supportedpartially by the Swedish Science Council (VR). I. Kim gratefully acknowledges the partial support of grant(NRF-2014R1A2A2A01005574) and a warm support of Chalmers University of Technology during hisstay.
For n = 2 , an element in H ( π ( M ) , ρ, s l (3 , R )) is represented by a sum of a quadraticform and a cubic form when ρ defines a convex real projective structure. This is due to[10, 9]. In this case both the global and local structures have been studied intensively.Recently [8] we have been able to construct mapping class group invariant K¨ahler metricon the Hitchin component of SL (3 , R ) , this is also part of our motivation of the presentpaper. We also mention that Labourie [9] has computed the cohomology H ( π ( M ) , R ) where π ( M ) acts on R through ρ and the defining representation of SL (3 , R ) .As a corollary of our technique, we show the Weil’s local rigidity theorem for uniformreal hyperbolic lattices for dimension > . Theorem 1.2.
Let M = Γ \ SO ( n, /SO ( n ) be a compact hyperbolic manifold. If n > then H (Γ , so ( n, . Acknowledgement
We are grateful for the anonymous referee for the careful readingof an earlier version of this paper and many useful comments.2. T
ANGENT SPACE AT F UCHSIAN LOCUS OF CONVEX PROJECTIVE STRUCTURES ONA MANIFOLD M Projective structure.
The notion of projective structures can be formulated in termsof a flat connection as follows, see [9] for details. Consider a trivial bundle E = M × R n +1 where M is an n -dimensional manifold. Let ω be a volume form on R n +1 and let ∇ be aflat connection on E preserving ω . Let ρ be the holonomy representation of ∇ . A section u of E is identified with a ρ -equivariant map from ˜ M to R n +1 . A section u is said to be ∇ -immersed if the n -form Ω u defined by Ω u ( X , · · · , X n ) = ω ( ∇ X u, · · · , ∇ X n u, u ) is non-degenerate. Then u is ∇ -immersed if it is a non-vanishing section and if theassociated ρ -equivariant map [ u ] from ˜ M to RP n is an immersion.Hence it follows that such a pair ( ∇ , u ) gives rise to a flat projective structure. Labouriereformulated this as a pair of torsion free connection ∇ T on M with a symmetric 2-tensor h on M as follows. One can associate a connection ∇ on E = T M ⊕ L , where L is atrivial bundle M × R , defined by(2.1) ∇ X (cid:18) Yλ (cid:19) = (cid:18) ∇ TX Xh ( X, · ) L X (cid:19) (cid:18) Yλ (cid:19) = (cid:18) ∇ TX Y + λXh ( X, Y ) + L X ( λ ) (cid:19) . Here L X ( λ ) = Xλ denotes the differentiation. Labourie [9] showed that if ∇ is flatand ∇ T preserves the volume form defined by h , then ∇ gives rise to a flat projectivestructure. He further showed that h is positive definite if the structure is properly convex.We will use this final form of projective structure in this paper to carry out the explicitcalculations. EIL’S LOCAL RIGIDITY THEOREM 3
Tangent space of convex projective structures.
Let M be an n -dimensional man-ifold and Γ = π ( M ) its fundamental group. Let ρ be a representation of Γ into SL ( n +1 , R ) defining a convex projective structure on M . There is a flat connection ∇ on theassociated R n +1 -bundle E preserving a volume form as in the previous section.The flat connection ∇ on E defines also a connection on the dual bundle E ∗ , g l ( E ) = E ⊗ E ∗ , the bundle of endomorphisms of E , and further on s l ( E ) , the trace free endomor-phisms, since ∇ preserve the volume form on E , by the Leibniz rule and the commutativerelation with the contractions.Write temporarily F for any of these flat bundles with fiber space F . We fix conventionthat if F = E ∗ or s l ( E ) ⊂ g l ( E ) = E ⊗ E ∗ , we write then a F -valued one-form α as α : ( X, y ) → α ( X ) y with the first argument being tangent vector and the secondargument being element of F . For conceptual clarity we recall that given ∇ E on E theconnection ∇ E ∗ on E ∗ -valued sections is defined by the equation X ( α ( y )) = ( ∇ E ∗ X α )( y ) + α ( ∇ EX y ); whereas the connection on sections of s l ( E ) is defined by(2.2) ∇ EX ( α ( y )) = ( ∇ slX α )( y ) + α ( ∇ EX y ) . We shall abbreviate them all as ∇ X .The flat connection ∇ on E as well as its induced connection ∇ T induces exteriordifferentiation d ∇ and d ∇ T on -forms, locally defined as d ∇ ( X ω i dx i ) = X ∇ ω i dx i , where ω i is a local section. For notational convenience we shall write all of them just as d ∇ ; no confusion would arise as it will be clear from the context which sections they areacting on. We will freely write a (0 , -tensor as α ( X ) Y = α ( X, Y ) . We shall need thefollowing formula for the exterior differentiation on a End ( E ) -valued one-form: ( d ∇ α )( X, Z ) y = ( ∇ X α )( Z ) y − ( ∇ Z α )( X ) y = ∇ X ( α ( Z ) y ) − α ( ∇ X Z ) y − α ( Z )( ∇ X y ) − ( ∇ Z ( α ( X ) y ) − α ( ∇ Z X ) y − α ( X )( ∇ Z y ))= ∇ X ( α ( Z ) y ) − α ( Z )( ∇ X y ) − ( ∇ Z ( α ( X ) y ) − α ( X )( ∇ Z y )) − α ([ X, Z ]) y (2.3)since ∇ is flat, in particular ∇ T is torsion free.We shall describe the cohomology in terms of some symmetry conditions of certaintensors. Let g be a Riemannian metric on M with the Levi-Civita connection ∇ g , thecorresponding exterior differentiation being denoted by d g . We consider following threeconditions for (0 , -tensors α , in which case α is a quadratic form: (q1): α is symmetric, (q2): trace-free with respect to g , (q3): α is d ∇ g closed, d ∇ g α = 0 , INKANG KIM AND GENKAI ZHANG and the following four conditions for a
End ( T M ) -valued one-form, in which case α is a cubic form: (c1): α ( X ) Y is symmetric in X, Y , α ( X ) Y = α ( Y ) X , (c2): α ( X ) is symmetric with respect to g , α ( X ) ∗ = α ( X ) , (c3): α is d ∇ g closed, d ∇ g α = 0 , equivalently the cubic form g ( α ( X ) Y, Z ) is closed, (c4): α ( X ) is trace-free.The following theorem is proved in [9, Theorems 3.2.1 & Proposition 4.2.3]. Theorem 2.1.
Suppose M admits a properly convex projective structure. Then there issplitting of the bundle (2.4) E = T M ⊕ L where L is a trivial bundle M × R and a Riemannian metric g on M such that the flatconnection ∇ on E is given by (2.5) ∇ X (cid:18) Yλ (cid:19) = (cid:18) ∇ TX Xg ( X, · ) L X (cid:19) (cid:18) Yλ (cid:19) = (cid:18) ∇ TX Y + λXg ( X, Y ) + L X ( λ ) (cid:19) . Here L X ( λ ) = Xλ is the differentiation, ∇ T is a torsion-free connection on T M pre-serving the volume form of the Riemannian metric g and ∇ g is the Levi-civita connectionof g , such that the tensor c , defined by g (( ∇ TX − ∇ gX ) Y, Z ) = c ( X, Y, Z ) , satisfies thecondition (c1-c4). In other words such a flat connection ∇ determines a metric g on T M and thus on thebundle
T M ⊕ L , g ⊕ | · | : ( X, λ ) g ( X, X ) + λ such that the connection ∇ takes the form ∇ X = (cid:20) ∇ gX L X (cid:21) + (cid:20) Q ( X ) XX ♯g (cid:21) where ∇ TX − ∇ gX = Q ( X ) , hence the first part is a skew-symmetric (i.e. orthogonal)connection and the second part is a symmetric form.We shall now compute the tangent space of the deformation space at Fuchsian locusinduced from the natural inclusion SO ( n, ⊂ SL ( n + 1 , R ) . A hyperbolic n -manifoldcan be viewed as a real projective manifold and it can be deformed inside the convex realprojective structures for any dimension n ≥ . The component containing the hyperbolicstructures constitutes the strictly convex real projective structures [2, 3, 6]. Indeed, anyconvex real projective structures whose holonomy group is not contained in SO ( n, hasa Zariski dense holonomy group in SL ( n + 1 , R ) , see [1].Hence from now on, we shall assume Q = 0 and then g is a hyperbolic metric on M with constant curvature − . All the covariant differentiations below will be the oneinduced by ∇ g . EIL’S LOCAL RIGIDITY THEOREM 5
Theorem 2.2.
Let ρ : π ( M ) → SO ( n, ⊂ SL ( n + 1 , R ) , n > be a representationdefining a hyperbolic structure on the compact n-manifold M . Let g be the hyperbolicmetric determined by ∇ . Then there exists an injective map from H ( π ( M ) , ρ, s l ( n +1 , R )) into the space of cubic forms satisfying (c1-c4). The proof will be divided into several steps.Let α be a one-form representing an element of H ( π ( M ) , ρ, s l ( n + 1 , R )) , realizedas an element of Ω ( M, s l ( E )) . Write it as X α ( X ) = (cid:20) A ( X ) B ( X ) C ( X ) D ( X ) (cid:21) ∈ s l ( E ) , under the splitting (2.4), where A ∈ Ω ( M, End(
T M )) , B ∈ Ω ( M, T M ) , C ∈ C ∞ ( M, T ∗ M ⊗ T ∗ M ) and X → D ( X ) = − tr A ( X ) is a one-form.We first observe for the covariant differentiation of a section of s l ( E ) u = (cid:18) a bc e (cid:19) is the one-form(2.6) X → ∇ X u = (cid:18) ( ∇ gX a ) · + c ( · ) X − g ( X, · ) b ∇ gX b + eX − a ( X ) g ( X, a · ) + ( ∇ gX c )( · ) − e ( X, · ) ( Xe ) + g ( X, b ) − c ( X ) (cid:19) , acting on a section y = ( Y, λ ) , where the dots denote the variable Y . These are exactforms. Lemma 2.3.
Up to exact forms we can assume B = 0 , D = 0 . Proof.
We prove first that we can choose u so that α + ∇ u has its entries B ( X ) = bX, D = 0 where b is a scalar function. Indeed we choose u = (cid:18) a c e (cid:19) ,a = B , c the one-form c = de + D . The form α + ∇ u then has the desired form: X α ( X ) + ∇ X u = (cid:20) A ( X ) + ( ∇ gX B ) · + c ( · ) X B ( X ) + eX − B ( X ) C ( X ) + g ( X, a · ) + ( ∇ gX c ) · − g ( X, · ) e D ( X ) + L X ( e ) − c ( X ) (cid:21) = (cid:20) A ′ ( X ) eXC ′ ( X ) 0 (cid:21) , as claimed. We write the new form α + ∇ u as α with its entries B = b Id, D = 0 .Next we take v = (cid:18) a Id c − na (cid:19) , a := 1 n + 1 b, c := − nda = − nn + 1 db, the same calculation above shows that α + ∇ v has its B = 0 , D = 0 . (cid:3) INKANG KIM AND GENKAI ZHANG
We shall need the precise formula of the diagonal part in the lower triangular form,keeping tract of the computations we find α + ∇ ( u + v ) has the form(2.7) (cid:18) A ( X ) + ∇ gX B + ( Xb ) I + f X ∗ (cid:19) , b = − n + 1 tr B, f = db Lemma 2.4.
Let α be lower triangular with B = 0 , D = 0 . The covariant derivatives d g A and d g C are related to A and C by (2.8) d g A )( X, Z )( Y ) + ( C ( Z ) Y ) X − ( C ( X ) Y ) Z and (2.9) g ( X, A ( Z ) Y ) − g ( Z, A ( X ) Y ) + ( d g C )( X, Z )( Y ) , and there hold the symmetric relations: (2.10) A ( X ) Z = A ( Z ) X , (2.11) C ( X ) Z = C ( Z ) X .
Proof.
We write an arbitrary section y of E as y = ( Y, λ ) = ( y T , y n ) , the tangen-tial and respectively normal component. To write down the condition on the closed-ness d ∇ α ( X, Z ) = 0 in terms of the components
A, C , we recall (2.3). The condition d ∇ α ( X, Z ) = 0 is then d ∇ α ( X, Z ) y = v ( X, Z ; y ) − v ( Z, X ; y ) − α ([ X, Z ]) y = v ( X, Z ; y ) − v ( Z, X ; y ) − (cid:18) A ([ X, Z ]) YC ([ X, Z ]) Y (cid:19) . (2.12)where v ( X, Z ; y ) := ∇ X ( α ( Z ) y ) − α ( Z )( ∇ X y ) = (cid:18) ∇ gX ( A ( Z ) Y ) + ( C ( Z ) Y ) X − A ( Z )( ∇ gX Y + λX ) g ( X, A ( Z ) Y ) + X ( C ( Z ) Y ) − C ( Z )( ∇ gX Y + λX ) (cid:19) . Here we have used the fact that B = 0 , D = 0 in the computations.Recall that d g A = d ∇ g A is defined by ( d g A )( X, Z )( Y ) = (2.13) ∇ gX ( A ( Z ) Y ) − A ( Z )( ∇ gX Y ) − ( ∇ gZ ( A ( X ) Y ) − A ( X )( ∇ gZ Y )) − A ([ X, Z ]) Y which is a well-defined End ( T M ) -valued 2-form, and ( d g C )( X, Z )( Y )= L X ( C ( Z ) Y ) − C ( Z )( ∇ gX Y ) − ( L Z ( C ( X ) Y ) − C ( X )( ∇ gZ Y )) − C ([ X, Z ]) Y (2.14)is the Riemannian exterior differential of the form C , and is an element of Ω ⊗ Ω . Thefirst two equations (2.8) -(2.9) are obtained from (2.12) by putting y = ( Y, . EIL’S LOCAL RIGIDITY THEOREM 7
Correspondingly we have, taking y = (0 , ,(2.15) A ( X ) Z − A ( Z ) X = 0 and(2.16) C ( X ) Z − C ( Z ) X = 0 , resulting the symmetric relations (2.10)- (2.11). (cid:3) Let α be the bilinear form α ( X, W ) = C ( X, W ) := C ( X ) W and α the End ( T M ) -valued one-form α ( Y ) X := 12 ( A ( Y ) X + A ∗ g ( Y ) X ) . Then α ( X, W ) is symmetric in X and W , hence it satisfies (q1). α ( Y ) is symmetricwith respect to g and trace free, since α ( Y ) ∈ sl ( E ) , α ( Y ) = tr A ( Y ) + D ( Y ) =tr A ( Y ) , hence it satisfies the conditions (c1) and (c4). Lemma 2.5.
Let α be of the above form with B = 0 , D = 0 . Then we have C = 0 and α = 0 and α satisfies the conditions (c1)-(c4) for n > .Proof. The equation (2.9) combined with (2.10) implies that g ( X, A ( Y ) Z ) − g ( Z, A ( Y ) X ) + ( d g C )( X, Z ) Y = 0 . In other words(2.17) ( A ∗ g ( Y ) − A ( Y )) X = − (( d g C )( X, · )( Y )) ♯g where the lowering of the index in the right hand side is with respect to the secondvariable. Since g is parallell with respect to ∇ g then (( d g C )( X, · )( Y )) ♯g is an exact End ( T M ) -valued one form. This is not obvious and requires proof. Indeed, let C ♭ be the End ( T M ) -valued 0-form, C ♭ ( X ) = X i C ( X, Z i ) Z i where { Z i } is a local orthonormal frame of T M . We claim that(2.18) (( d g C )( X, · )( Y )) ♯g = ( d gY C ♭ )( X ) . By definition we have the identity section Id = P i Z i ⊗ Z ♯i and, ∇ gY Id = X i ( ∇ gY Z i ⊗ Z ♯i + Z i ⊗ ∇ gY ( Z ♯i ) = X i ∇ gY Z i ⊗ Z ♯i + X i Z i ⊗ ( ∇ gY Z i ) ♯ , namely, for any Z (2.19) X i g ( Z i , Z ) ∇ gY Z i + X i g ( ∇ gY Z i , Z ) Z i . INKANG KIM AND GENKAI ZHANG
Here we have used the fact that ∇ gY commutes with ♯ , ∇ gY ( Z ♯i ) = ( ∇ gY Z i ) ♯ . By definition,LHS of (2.18) is LHS = X i (( d g C )( X, Z i )( Y )) Z i = X i (( ∇ gY C )( X, Z i )) Z i = X i Y ( C ( X, Z i )) Z i − X i C ( ∇ gY X, Z i ) Z i − X i C ( X, ∇ gY Z i ) Z i . Here C ∈ C ∞ ( M, T ∗ M ⊗ T ∗ M ) is a zero form, hence ( d g C )( X, Z i )( Y ) = ( ∇ gY C )( X, Z i ) .On the other hand, RHS = ( ∇ gY C ♭ )( X ) = ∇ gY ( C ♭ ( X )) − C ♭ ( ∇ gY X )= ∇ gY X i C ( X, Z i ) Z i ! − X i C ( ∇ gY X, Z i ) Z i = X i Y ( C ( X, Z i )) Z i + X i C ( X, Z i ) ∇ gY Z i − X i C ( ∇ gY X, Z i ) Z i . To treat the second term we compute its inner product with any Z ; it is X i C ( X, Z i ) g ( ∇ gY Z i , Z ) = C ( X, X i g ( ∇ gY Z i , Z ) Z i )= − C ( X, X i g ( Z i , Z ) ∇ gY Z i )= − g ( X i C ( X, ∇ gY Z i ) Z i , Z ) where the second equality is by (2.19). Hence X i ( C ( X, Z i )) ∇ gY Z i = − X i C ( X, ∇ gY Z i ) Z i , proving RHS = LHS and hence confirming (2.18).The form α can now be written as α ( Y ) = 12 ( A ∗ g ( Y ) + A ( Y )) = A ( Y ) + 12 ( A ∗ g ( Y ) − A ( Y )) with the second term ( A ∗ g ( Y ) − A ( Y )) being exact, which implies that d g α = d g A .The equation (2.8) can now be written as(2.20) d g α )( X, Z )( Y ) + ( C ( Z ) Y ) X − ( C ( X ) Y ) Z where α ( X ) is trace-free and symmetric with respect to g . This in turn implies that themap Y → C ( Z, Y ) X − C ( X, Y ) Z is symmetric, g ( C ( Z, Y ) X − C ( X, Y ) Z, W )= g ( Y, C ( Z, W ) X − C ( X, W ) Z ) . (2.21)Let { Z i } be an orthonormal basis, Y = Z = Z i , and summing over i , we get(2.22) (tr g C ) g ( X, W ) + ( n − C ( X, W ) = 0 . EIL’S LOCAL RIGIDITY THEOREM 9
Taking again the trace we find(2.23) tr g C = 0 . Hence α satisfies (q2).Substituting this into the previous formula we get ( n − C ( X, W )) = 0 , and consequently(2.24) C ( X, W ) = 0 if n > .For n > it follows from (2.24) and (2.17) that(2.25) A ∗ g ( Y ) = A ( Y ) i.e, A ( Y ) is symmetric with respect to g . Consequently α ( Y ) = A ( Y ) = A ∗ g ( Y ) . Hence α ( Y ) X = A ( Y ) X = A ( X ) Y = α ( X )( Y ) by Equation (2.10). This proves that α satisfies (c1)-(c2), (c4) for n > . The equation (2.20) combined with C = 0 impliesfurther d g α = 0 . Hence α satisfies all the conditions to be a cubic form. (cid:3) We prove now that the map from α to the cubic form in Lemma 2.5 is injective. Lemma 2.6.
Let n > and suppose α ∈ H ( π ( M ) , ρ, sl ( n + 1 , R )) . Then the map α α from H ( π ( M ) , ρ, sl ( n + 1 , R )) to the cubic form α is injective.Proof. In Lemma 2.5, we showed that if α is represented by a sl ( E ) -valued one formwith B = D = 0 , then C = 0 and the associated cubic form is α ( Y ) = A ( Y ) . Hence ifthe associated cubic form vanishes, A = 0 . This implies that α is represented by an exact1-form by Equation (2.7), hence it is a zero element in the cohomology. (cid:3) This finishes the proof of Theorem 2.2.Now we prove the Weil’s local rigidity theorem, H (Γ , so ( n, , n > , [12,Chapter VII], [13] as an application of our technique. For n = 2 , it is well-known that thecohomology H (Γ , so (2 , is determined by quadratic forms satisfying (q1-q3), namelyreal part of holomorphic quadratic forms; see e.g. [4]. Theorem 2.7.
Let M = Γ \ SO ( n, /SO ( n ) be a compact hyperbolic manifold. (1) If n > then H (Γ , so ( n, . (2) If n = 2 then H (Γ , so ( n, is given by the space of quadratic forms satisfying(q1-q3). Proof.
Let α represent an element in H (Γ , so ( n, viewed as an element in Ω ( M, so ( n, .The elements in so ( n, are of the form (cid:20) a bc (cid:21) with a = − a ∗ and b = c T with respectto the Euclidean product in R n as a subspace of the Lorentz space R n +1 . The -form α takes then the form α = (cid:20) A BC (cid:21) with g ( A ( X ) Y, Z ) = − g ( Y, A ( X ) Z ) , g ( B ( X ) , Y ) = C ( X )( Y ) = C ( X, Y ) , where g isthe given hyperbolic metrix on M . The (1,1)-entry A of α is skew-symmetric. Now fromEquations (2.3), (2.2) and the fact that ∇ T = ∇ g for hyperbolic manifold, (1 , -entry ofthe condition d g α = 0 is(2.26) ( d g A )( X, Z ) + X ⊗ C ( Z ) − Z ⊗ C ( X ) + B ( X ) ⊗ Z ♯ − B ( Z ) ⊗ X ♯ = 0 as two form acting on ( X, Z ) . Here X ⊗ Z ♯ is the rank-one map Y g ( Y, Z ) X . We shallalso need some Hodge theory. Equip so ( n, with the SO ( n ) -invariant positive innerproduct induced from the standard Euclidean inner product in R n +1 , ( y, y ) E = k Y k g + λ , y = ( Y, λ ) . Take a harmonic one form representing α . Then the cohomology class α satisfies also the coboundary condition ∇ ∗ α = 0 . To write down the formula for ∇ ∗ weobserve that ∇ X = (cid:20) ∇ gX L X (cid:21) + (cid:20) XX ♯ (cid:21) is a sum of two terms, the first preserving the Euclidean inner product ( y, y ) E , whoseadjoint can be found by standard formula (see e.g. [11, p.2]), whereas the second part isself-adjoint. Thus −∇ ∗ α is given by X j ( δ X j α )( X j ) where δ X = (cid:18) ∇ gX − X − X ♯ L X (cid:19) . More precisely, ( δ X α )( Z ) is given, for any testing section y = ( Y, λ ) , by the Leibniz rule ( δ X α )( Z ) y = (cid:18) ∇ gX − X − X ♯ L X (cid:19) ( α ( Z ) y ) − α ( Z ) (cid:18)(cid:18) ∇ gX − X − X ♯ L X (cid:19) y (cid:19) − α ( ∇ gX Z ) y. (The sum P j ( δ X j α )( X j ) is well-defined but not the individual terms.) When acting onthe section y = (0 , we find(2.27) X j ( A ( X j ) X j + ( ∇ gX j B )( X j )) = 0 . It now follows from Equation (2.7), Lemmas 2.3 and 2.5 (keeping track of the changeof forms) that A ( X ) := A ( X ) + ∇ gX B + ( Xb ) I + f X is symmetric and trace-free andsatisfies the condition (c1-c4). Note here while performing computations as in Lemmas2.3-2.5 we use forms u with values in sl ( n +1) instead of in so ( n, , however all we need EIL’S LOCAL RIGIDITY THEOREM 11 is that d ∇ d ∇ = 0 , i.e. we will show that α vanishes identically. The trace free conditionand A ( X ) Y + ∇ gX B ( Y ) + ( Xb ) Y + f ( Y ) X = A ( Y ) X + ∇ gY B ( X ) + ( Y b ) X + f ( X ) Y imply that the map Z → A ( Z ) Y = A ( Z ) Y + ( ∇ Z B )( Y ) + ( Zb ) Y + f ( Y ) Z is tracefree. The symmetric relation implies g ( A ( Z ) Y + ( ∇ Z B )( Y ) + ( Zb ) Y + f ( Y ) Z, W )= g ( A ( Z ) W + ( ∇ Z B )( W ) + ( Zb ) W + f ( W ) Z, Y ) . We take { Z j } a local orthonormal frame and put Z = W = Z j in the above equation.Summing over j we find, in view of (2.27) that the right hand side is RHS = X j g ( A ( Z j ) Z j + ( ∇ Z j B )( Z j ) + ( Z j b ) Z j + f ( Z j ) Z j , Y ) = ( Y b ) + f ( Y ) and LHS = tr( A ( · ) Y ) = 0 . Namely the one-form
Y b + f ( Y ) = 0 . But f ( Y ) = ( db )( Y ) = Y b by (2.7), so f ( Y ) , and db = f = 0 . This implies in turn that A ( X ) = A ( X ) + ∇ gX B + ( Xb ) I + f X = A ( X ) + ∇ gX B is symmetric. We write B = B + B , the symmetric and respectively the skew symmetricpart of B . Since A is skew symmetric, the skew symmetric part of A ( X ) must vanish,that is(2.28) A ( X ) + ∇ gX B = 0 . This implies in turn A ( X ) = −∇ X B is exact. Thus d g A = 0 , and the relation (2.26)becomes X ⊗ C ( Z ) − Z ⊗ C ( X ) + B ( X ) ⊗ Z ♯ − B ( Z ) ⊗ X ♯ = 0 . Let { Y i } be an orthonormal basis, put Z = Y i and let the above act on Y j . Taking the sumand using B t = C we find as in the proof of Lemma 2.5, that X i g ( X ⊗ C ( Y i , Y i ) − Y i ⊗ C ( X, Y i ) + B ( X ) g ( Y i , Y i ) − B ( Y i ) g ( X, Y i ) , W ) = 0 i.e. tr g Cg ( X, W ) − C ( X, W ) + nC ( X, W ) − C ( X, W ) = 0 . But the same proof above implies that(2.29) tr g C = 0 , ( n − C ( X, W ) = 0 . Now let n > . Thus C ( X, W ) = 0 . Then B = C t = 0 and using the symmetriccondition on A = A we find that A is symmetric and thus A = 0 . This proves (1).Let n = 2 . We consider the so ( , ) -valued section u = (cid:18) B
00 0 (cid:19) . Using the formulas (2.6) and (2.28) we find α + ∇ u : X α ( X ) + ∇ X u = (cid:20) B C (cid:21) where C is the symmetric part of C and ( B ) t = C . So replacing α by α + ∇ u we mayassume that A = 0 , B is symmetric and B t = C . A direct calculation using (2.12) and y = ( Y, gives d ∇ α ( X, Z ) Y = (cid:20) ( C ( Z ) Y ) X − g ( X, Y ) B ( Z ) − ( C ( X ) Y ) Z + g ( Z, Y ) B ( X ) X ( C ( Z ) Y ) − C ( Z ) ∇ gX Y − Z ( C ( X ) Y ) + C ( X ) ∇ gZ Y − C ([ X, Z ]) Y (cid:21) . But by the formula (2.14) X ( C ( Z ) Y ) − C ( Z ) ∇ gX Y − Z ( C ( X ) Y ) + C ( X ) ∇ gZ Y − C ([ X, Z ]) Y = ( d g C )( X, Z ) Y Hence d ∇ α = 0 gives d g C = d g B = 0 . We have thus that B is symmetric, tr g B =tr g C = 0 by (2.29) and d g B = 0 , namely B satisfies (q1-q3).This completes the proof. (cid:3) Recall [5] that the Hitchin component, denoted by χ H ( π ( S ) , SL (3 , R )) , is a con-nected component in the character variety Hom( π ( S ) , SL (3 , R )) //SL (3 , R ) containingthe realization of π ( S ) as a subgroup of SL (2 , R ) composed with the irreducible repre-sentation of SL (2 , R ) on R . The tangent space of χ H ( π ( S ) , SL (3 , R )) at these specificFuchsian points can be obtained from the general theory in [5]. See also [7].R EFERENCES [1] Y. Benoist,
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Discrete subgroups of Lie groups, II , Ann. of Math (1962), 97-123.S CHOOL OF M ATHEMATICS , KIAS, H
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