spinor derivative r S with the projection p onto the kernel of the Cliord multiplication D :?(S)?! rs?(t M S) g p?(t M S)?!?(Ker ): The elememts of th

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1 Strictly pseudoconvex spin manifolds, Feerman spaces and Lorentzian twistor spinors Helga Baum January 1, 1997 Abstract We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties. Contents 1 Introduction 1 Algebraic prelimeries 3 3 Lorentzian twistor spinors 5 4 Pseudo-hermitian geometry 11 5 Feerman spaces 17 6 Spinor calculus for S 1 -bundles with isotropic bre over strictly pseudoconvex spin manifolds 1 7 Twistor spinors on Feerman spaces 30 1 Introduction In the present paper we study a relation between CR-geometry and the Lorentzian twistor equation. Besides the Dirac operator there is a second important conformally covariant dierential operator acting on the spinor elds?(s) of a semi-riemannian spin manifold (M; g), the so-called twistor operator D. The twistor operator is dened as the composition of the 1

2 spinor derivative r S with the projection p onto the kernel of the Cliord multiplication D :?(S)?! rs?(t M S) g p?(t M S)?!?(Ker ): The elememts of the kernel of D are called twistor spinors. A spinor eld ' is a twistor spinor if and only if it satises the twistor equation r S X' + 1 n X D' = 0 for each vector eld X, where D is the Dirac operator. Each twistor spinor ' denes a conformal vector eld V ' on M by where k is the index of g. g(v ' ; X) = i k+1 hx '; 'i ; Twistor spinors were introduced by R.Penrose in General Relativity (see [Pen67], [PR86], [NW84]). In mathematics the twistor equation appeared as an integrability condition for the canonical almost complex structure of the twistor space of an oriented four-dimensional Riemannian manifold (see [AHS78]). In the second half of the 80th Lichnerowicz and Friedrich started the systematic investigation of twistor spinors on Riemannian spin manifolds from the view point of conformal dierential geometry. Nowadays one has a lot of structure results and examples for manifolds with twistor spinors in the Riemannian setting (see [Lic88b], [Lic88a], [Lic89], [Fri89] [Lic90], [FP91], [BFGK91], [Hab90], [Hab9], [Hab93], [KR94], [KR95], [KR96b], [KR96a]). Crucial results were obtained by studying the properties of the conformal vector eld V ' of a twistor spinor '. In particular, twistor operators turned out to be a usefull tool in proving sharp eigenvalue estimates for coupled Dirac operators on compact Riemannian manifolds (see eg [Bau94]). In opposite to this, there is not much known about solutions of the twistor equation in the general Lorentzian setting. In 1991 Lewandowski studied local solutions of the twistor equation on 4-dimensional space-times, ([Lew91]). In particular, he proved that a 4-dimensional space-time admitting a twistor spinor ' without zeros and with twisting conformal vector eld V ' is locally conformal equivalent to a Feerman space. On the other hand, on 4-dimensional Feerman spaces there exist local solutions of the twistor equation. The aim of the present paper is the generalisation of this result. Feerman spaces were dened by Feerman ([Fef76]) in case of strictly pseudoconvex hypersurfaces incn, its denition was extended by Burns, Diederich, Shnider ([BDS77]), Farris ([Far86]) and Lee ([Lee86]) to general non-degenerate CR-manifolds. A Feerman space is the total space of the S 1 -principal bundle associated to the canonical bundle of a non-degenerate CR-manifold M equipped with a semi-riemannian metric dened by means of the Webster connection which is given by an additional pseudo-hemitian structure on M. By changing the topological type of the S 1 -bundle dening the Feerman space, we can

3 prove that there are global solutions of the twistor equation on the (modied) Feerman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension. These solutions have very special geometric properties which are only possible on Feerman spaces. More exactly, we prove (see Theorem 1, Theorem ): Let (M n+1 ; T 10 ; ) be a strictly pseudoconvex spin manifold and ( p F ; h ) its Feerman space. Then, on the Lorentzian spin manifold ( p F ; h ) there exist a non-trivial twistor spinor such that 1. The canonical vector eld V of is a regular isotropic Killing vector eld.. V = 0. In particular, is a pure or partially pure spinor eld. 3. r V = i c ; c = const R n f0g. 4. has constant length. On the other hand, if (B; h) is a Lorentzian spin manifold with a non-trivial twistor spinor satisfying , then B is an S 1 -principal bundle over a stricly pseudoconvex spin manifold (M; T 10 ; ) and (B; h) is locally isometric to the Feerman space ( p F ; h ) of (M; T 10 ; ). In particular, if (M n+1 ; T 10 ; ) is a compact strictly pseudoconvex spin manifold of constant Webster scalar curvature, then the Feerman space ( p F ; h ) of (M; T 10 ; ) is a (n+)-dimensional non-einsteinian Lorentzian spin manifold of constant scalar curvature R and the twistor spinor denes eigenspinors of the Dirac operator of ( p F ; h ) to the eigenvalues 1 q n+ n+1 R with constant length. After some algebraic prelimeries in section we introduce in section 3 the notion of Lorentzian twistor spinors and explain some of their basic properties. In order to dene the (modied) Feerman space we recall in section 4 the basic notions of pseudo-hermitian geometry. In particular, we explain the properties of the Webster connection of a non-degenerate pseudo-hermitian manifold, which are important for the spinor calculus on Feerman spaces. In section 5 the Feerman spaces are dened and in section 6 we derive a spinor calculus for Lorentzian metrics on S 1 -principal bundles with isotropic bre over strictly pseudoconvex spin manifolds. Finally, section 7 contains the proof of the Theorems 1 and which state the properties of the solutions of the twistor equation on Feerman spaces of strictly pseudoconvex spin manifolds. Algebraic prelimeries For concrete calculations we will use the following realization of the spinor representation. Let Cli n;k be the Cliord algebra of (Rn ;?h; i k ), where h; i k is the scalar product hx; yi k =?x 1 y 1? : : :? x k y k + x k+1 y k+1 + : : : + x n y n : 3

4 For the canonical basis (e 1 ; : : : ; e n ) ofrn one has the following relations in Cli n;k : e i e j + e j e i =?" j ij ; where Denote U = i 0 0?i! Then an isomorphism " j = ; V = 0 i i 0 is given by the Kronecker product! j = (?1 j k 1 j > k : ; E = ( i j k 1 j > k:! m;k : Cli C m;k?! M(m ;C ) ; T = 0?i i 0! and m;k (e j?1 ) = j?1 E : : : E U T : : : T m;k (e j ) = j E : : : E V T :{z : : T} j?1 : (1) Let Spin 0 (n; k) Cli n;k be the connected component of the identity of the spin group. The spinor representation is given by x n;k = n;k j Spin0 (n;k) : Spin 0(n; k)?! GL(C m ): We denote this representation by n;k. If n = m, m;k splits into the sum m;k = + m;k?, where m;k are the eigenspaces of the endomorphism m;k m;k(e 1 : : : e m ) to the eigenvalue i m+k. Let us denote by u(") C the vector and let Then (u(" 1 ; : : : ; " m ) j u(") = 1 p 1?"i! " = 1 u(" 1 ; : : : ; " m ) = u(" 1 ) : : : u(" m ) " j = 1: () mq j=1 the standard scalar product ofc m. " j = 1) is an orthonormal basis of m;k with respect to 4

5 3 Lorentzian twistor spinors Let (M n;1 ; g) be a connected space- and time oriented Lorentzian spin manifold with a xed time orientation?(t M), g(; ) =?1. We denote by S the spinor bundle of (M n;1 ; g), by r S :?(S)!?(T M S) the spinor derivative given by the Levi-Civita connection of (M n;1 ; g) and by D :?(S)!?(S) the Dirac operator on S. On S there exists an indenite scalar product h; i of index 1 dim S such that hx '; i = h'; X i (3) Xh'; i = hr S X'; i + h'; r S X i (4) for all vector elds X and all spinor elds ';?(S). Furthermore, there is a positive denite scalar product (; ) on S depending on the time orientation such that h'; i = ( '; ) (5) for all ';?(S) (see [Bau81], chap.1.5, ). Let p : T M S?! Ker denote the orthogonal projection onto the kernel of the Cliord multiplication (with respect to h; i). p is given by p(x ') = X ' + 1 n nx k=1 " k s k s k X '; where (s 1 ; : : : ; s n ) is a orthonormal basis of (M; g) and " k = g(s k ; s k ) = 1. Denition 1 The twistor operator D of (M n;1 ; g) is the operator given by the composition of the spinor derivative with the projection p D :?(S)?! rs?(t M S) g p?(t M S)?!?(Ker ): Locally, we have D' = nx k=1 " k s k (r S s k ' + 1 n s k D'): Denition A spinor eld '?(S) is called a twistor spinor, if D' = 0. Let us rst recall some properties of twistor spinors which are proved in the same way as in the Riemannian case. Proposition 1 ([BFGK91], Th.1.) For a spinor eld '?(S) the following conditions are equivalent: 1. ' is a twistor spinor.. ' satises the so-called twistor equation r S X ' + 1 X D' = 0 (6) n 5

6 for all vector elds X. 3. For all vector elds X and Y X r S Y ' + Y r S X' = g(x; Y ) D' (7) n holds. 4. There exists a spinor eld?(s) such that = g(x; X)X r S X' (8) for all vector elds X with jg(x; X)j = 1. Proposition ([BFGK91], Th.1.7) The twistor operator is conformally covariant: Let ~g = e g be a conformally equivalent metric to g and let ~ D be the twistor operator of (M; ~g). Then ~D ~' = e? 1 ^ D(e? 1 '); where : S?! ~ S denotes the canonical identication of the spinor bundles of (M; g) and (M; ~g). Proposition 3 ([BFGK91] Cor.1.) The dimension of the space of twistor spinors is conformally invariant and bounded by dim KerD [ n ]+1 : Proposition 4 ([BFGK91] Cor.1.3) Let '?(S) be a non-trivial twistor spinor and x 0 M. Then '(x 0 ) 6= 0 or D'(x 0 ) 6= 0. Let R be the scalar curvature and Ric the Ricci curvature of (M n;1 ; g). If dim M = n 3; K denotes the (,0) -Schouten tensor K(X; Y ) = 1 R n? (n? 1) g? Ric : We always identify T M with T M using the metric g. For a (; 0)-tensor eld B we denote by the same symbol B the corresponding (1; 1)-tensor eld B : T M?! T M, g(b(x); Y ) = B(X; Y ): Let C be the (,1)-Schouten-Weyl tensor C(X; Y ) = (r X K)(Y )? (r Y K)(X): Furthermore, let W be the (4,0)-Weyl tensor of (M; g) and let denote by the same symbol the corresponding (,)-tensor eld W : M?! M: Then we have 6

7 Proposition 5 ([BFGK91] Th.1.3, Th.1.5) Let '?(S) be a twistor spinor and = Y ^ Z M a two form. Then D ' = 1 4 n R' ; (9) n? 1 r S X D' = n K(X) ' ; (10) W () ' = 0 ; (11) W () D' = n C(Y; Z) ' ; (1) (r X W )() ' = X C(Y; Z) ' + n (X? W ()) D' : (13) If the scalar curvature R of (M n;1 ; g) is constant and non-zero, equation (9) shows that the spinor elds r := 1 n? 1 ' nr D' are formal eigenspinors of the Dirac operator D to the eigenvalue 1 nr A special class of twistor spinors are the so-called Killing spinors '?(S) dened by the condition r S X ' = X ' for all X?(T M); q n?1. where is a constant complex number, called the Killing number of '. Using the twistor equation and the properties (9) and (10) one obtains that for an Einstein space (M n;1 ; g) with constant scalar curvature R 6= 0 the spinor elds are Killing spinors to the Killing number = q 1 R. Hence, on this class of Lorentzian manifolds n(n?1) each twistor spinor is the sum of two Killing spinors. Therefore, we are specially interested in non-einsteinian Lorentzian manifolds which admit twistor spinors. To each spinor eld we associate a vector eld in the following way. Denition 3 Let '?(S). The vector led V ' denied by is called the canonical vector eld of '. g(v ' ; X) :=?hx '; 'i ; X?(T M) Because of (1), V ' is a real vector eld. By Zero(') and Zero(X) we denote the zero sets of a spinor eld ' or a vector eld X. Proposition 6 1. For each spinor eld '?(S) Zero(') = Zero(V ' ):. If n is even, n 6 and '?(S ) is a half spinor, then V ' ' = 0: In particular, V ' is an isotropic vector eld. 7

8 Proof: Let '?(S). From (5) follows for the time orientation g(v ' ; ) =?h '; 'i =?( '; ') =?('; ') : Since the scalar product (; ) is positive denite, this shows that Zero(V ' ) = Zero('). The second statement is proved by a direct calculation using a basis representation of ' and V ' and the formulas (1) and (). Remark: In the Riemannian case Proposition 6.1 is not true. There exist non-trivial spinor elds ' such that the canonical vector eld V ' is identically zero (see [KR95]). On the other hand, the zero set Zero(') of a Riemannian twistor spinor is discret ([BFGK91], Th..1). This is in the Lorentzian setting not the case. We call a subset A M isotropic, if each dierentiable curve in A is isotropic. Proposition 7 Let '?(S) be a twistor spinor. Then the zero set of ' is isotropic. Proof: Let : I?! Zero(') be a curve in Zero('). Then '((t)) 0 and therefore r _(t) ' 0. From the twistor equation (6) it follows _(t) D'((t)) 0. Since by Proposition 4 D'((t)) 6= 0, _(t) is isotropic for all t I. Proposition 8 Let '?(S) be a twistor spinor. Then V ' is a conformal vector eld and for the Lie derivative holds. L V 'g =? 4 Reh'; D'i g n Proof: Let V := V '. From the denition of V ' it follows (L V g)(x; Y ) = g(r X V; Y ) + g(x; r Y V ) Using (7) we obtain = X(g(V; Y ))? g(v; r X Y ) + Y (g(x; V ))? g(r Y X; V ) =?XhY '; 'i? Y hx '; 'i + hr X Y '; 'i + hr Y X '; 'i (4) =?hr X Y '; 'i? hy r S X '; 'i? hy '; rs X 'i? hr Y X '; 'i?hx r S Y '; 'i? hx '; rs Y 'i + hr XY '; 'i + hr Y X '; 'i (3) =?hy r S X ' + X rs Y '; 'i? h'; Y rs X ' + X rs Y 'i : (L V g) (X; Y ) =? 4 n g(x; Y ) Reh'; D'i: 8

9 From Proposition 8 follows that for each twistor spinor ' For the imaginary part of h'; D'i we have div(v ' ) =? Reh'; D'i. Proposition 9 Let '?(S) be a twistor spinor. Then the function C ' := Im h'; D'i is constant on M. Proof: Because of (3) the function hy ; i is real for each vector eld Y and each spinor eld. Furthermore, XhD'; 'i (4) = hr S XD'; 'i + hd'; r S X'i n hk(x) '; 'i? 1 hd'; X D'i: n (6);(10) = Hence XhD'; 'i is a real function. Therefore, C ' = Imh'; D'i is constant. Let us denote by C the (3; 0)-Schouten-Weyl tensor C(X; Y; Z) = g(x; C(Y; Z)): Proposition 10 Let '?(S) be a twistor spinor. Then 1. V '? C = 0:. If n = 4, then V '? W = 0: Proof: From (11) and (1) we obtain C(V ' ; X; Y ) = g(v ' ; C(X; Y )) =?hc(x; Y ) '; 'i =? 1 n hw (X ^ Y ) '; 'i = 1 h'; W (X ^ Y ) 'i = 0: n Let ' = au("; 1) + bu(?";?1)?(s " ) be a half spinor on a 4-dimensional manifold. Then by a direct calculation using (1) and () we obtain Hence, V ' = (jaj + jbj )s 1 + (jaj? jbj )s? Re(ia b)s 3? "Re(a b)s 4 : W (V ' ; s i ; s j ; s k ) = (jaj + jbj )W 1ijk + (jaj? jbj )W ijk?re(ia b)w 3ijk? "Re(a b)w 4ijk : (14) On the other hand, from the basis representation of 0 = W (s j ^ s k ) ' = X r<l " r " l W rljk s r s l ' 9

10 result the equations 0 = (W 1jk? "iw 34jk )a + (iw 13jk? "W 4jk? "W 14jk + iw 14jk ) b (15) 0 = (?W 1jk + "iw 34jk )b + (?iw 13jk + "W 4jk? "W 14jk + iw 3jk )a : (16) Then looking at the real and imaginary part of the equations (15)a(16) b and (15) b (16)a one obtains W (V ' ; s i ; s j ; s k ) = 0. Propositions 6 and 10 show that 4-dimensional Lorentzian spin manifolds with twistor spinor ' are of Petrov type N outside the zero zet of ' and W (see [ON95] chap. 5). Examples of Lorentzian manifolds admitting twistor spinors: 1. Minkowski space A spinor eld ' C 1 (Rn;1 ; n;1 ) on the Minkowski space is a twistor spinor if and only if '(x) = u + x v; where u; v n;1 are constant spinors.. Simply connected Lorentzian space forms Let M n;1 (K) be the simply connected Lorentzian space form of constant sectional C curvature K 6= 0. On M n;1 (K) there exist [ n ] linearly independent Killing spinors to the Killing number with = K (see [CGLS86]). Each Killing 4 spinor is a twistor spinor dimensional pp-manifolds A 4-dimensional Lorentzian spin manifold M 4;1 admits a parallel spinor eld if and only if M 4;1 is a pp-manifold. (see eg [Bau81] chap 4, [Gun88] chap 8, cor.4.4). Each parallel spinor eld is a twistor spinor dimensional Feerman spaces In [Lew91] and [Spi95] the following theorem was proved: Let ' be a twistor spinor on a 4-dimensional Lorentzian spin manifold (M 4;1 ; g) such that Zero(') = ; and rot(v ' ) is non-degenerate. Then (M 4;1 ; g) is locally conformal equivalent to a Feerman manifold. On each Feerman space there exists a local solution of the twistor equation. The aim of the present paper is the generalisation of this result of J. Lewandowski. We prove that there are global solutions of the twistor equation on the (modied) Feerman spaces of strictly preudoconvex spin manifolds of arbitrary dimension and study their properties. 10

11 4 Pseudo-hermitian geometry Before we are able to dene the Feerman spaces we need some notions from pseudohermitian geometry. Let M n+1 be a connected manifold of odd dimension n + 1. A complex CR-structure on M is a complex subbundle T 10 of T M C such that 1. dimc T 10 = n;. T 10 \ T 10 = f0g; 3. [?(T 10 );?(T 10 )]?(T 10 ) (integrability condition). A real CR-structure on M is a pair (H; J), where 1. H T M is a real n-dimensional subbundle,. J : H?! H is an almost complex structure on H : J =?id, 3. If X; Y?(H), then [JX; Y ] + [X; JY ]?(H) and N J (X; Y ) := J([JX; Y ] + [X; JY ])? [JX; JY ] + [X; Y ] 0 (integrability condition). Obviously the complex and real CR-structure correspond to each other: If T 10 T M C is a complex CR-structure, then H := Re (T 10 T 10 )), J(U + U) := i(u? U) denes a real CR-structure. If (H; J) is a real CR-structure, then the eigenspace of the complex extension of J on H C to the eigenvalue i is a complex CR-structure. CR-structures one has for example on real hypersurfaces of complex manifolds, on Sasaki-manifolds or on Heisenberg manifolds. A CR-manifold is an odd-dimensional manifold equipped with a (real or complex) CRstructure. Let (M; T 10 ) be a CR-manifold. The hermitian form on T 10 L : T 10 T 10?! E := T M C = T10 T 10 L(U; V ) := i[u; V ]E ; where X E denotes the projection of X T M C onto E, is called the Levi-form of (M; T 10 ). The CR-manifold is called non-degenerate, if its Levi-form L is non-degenerate. An nowhere vanishing 1-form 1 (M) is called a pseudo-hermitian structure on (M; T 10 ), if j H 0. (M; T 10 ; ) is called a pseudo-hermitian manifold. There exists a pseudo-hermitian structure on (M; T 10 ) if and only if M is orientable. Two pseudohermitian structures ; ~ diers by a real nowhere vanishing function f C 1 (M) : ~ = f. If M is oriented, we always choose such that a basis of the form (X 1 ; JX 1 ; : : : ; X n ; JX n ; T ) is positive oriented on M. C Let (M; T 10 ; ) be a pseudo-hermitian manifold. The hermitian form L : T 10 T 10?! L (U; V ) :=?id(u; V ) is called the Levi-form of (M; T 10 ; ). Obviously, we have (L(U; V )) = L (U; V ). The pseudo-hermitian manifold (M; T 10 ; ) is called strictly pseudoconvex, if the Levi-form L is positive denite. If the pseudo-hermitian manifold (M; T 10 ; ) is non-degenerate, then the pseudo-hermitian structure is a contact form. We denote by T?(T M) 11

12 the characteristic vector eld of this contact form, e.g. the vector eld uniquely dened by (T ) 1 and T? d 0: From now on we always suppose, that (M; T 10 ; ) is non-degenerate. On CR-manifolds we consider the following spaces of forms: Obviously, q;0 M := f! q M C j V?! = 0 8V T 10 g 0;q M := f! q M C j V?! = 0 8V T 10 g p;q M := spanf! ^ j! p;0 M; 0;q Mg: 1;0 M = f! 1 M C j!(jx) = i!(x) 8X Hg 0;1 M = f! 1 M C j!(jx) =?i!(x) 8X Hg 1;1 M = f! M C j!(jx; JY ) =!(X; Y ) 8X; Y Hg = f! M C j!(z; W ) =!( Z; W ) = 0 8Z; W T10 g: Let (M; T 10 ; ) be a non-degenerate pseudo-hermitian manifold and let T be the characteristic vector eld of. Then we dene and have p;q M := f! p;q M j T?! = 0g p;q M = p;q M (p?1;q ^C) ( p;q?1 M ^C): For example, by denition, d 1;1 M. Now, let us extend the Levi-form of (M; T 10 ; ) to T M C by L ( U; V ) := L (U; V ) = L (V; U) L (U; V ) := 0 U; V T10 L (T; ) := 0: Proposition 11 Let L : T M C T M C?!C be the Levi-form of (M; T 10 ; ) and let T be the characteristic vector eld of. Then [T; Z]?(T 10 T 10 ) if Z?(T 10 ) or Z?(T 10 ) ; (17) L ([T; U]; V ) + L (U; [T; V ]) = T (L (U; V )) 8 U; V?(T 10 ) ; (18) L ([T; U]; V ) = L ([T; V ]; U) 8 U; V?(T10 ) ; (19) L ([T; U]; V ) = L ([T; V ]; U) 8 U; V?(T10 ) ; (0) 1

13 Proof: Let Z?(T 10 ) or Z?(T 10 ). Then, by denition of and T ([T; Z]) =?d(t; Z)? T ((Z)) + Z((T )) = 0: Hence, [T; Z]?(H C ) =?(T 10 T 10 ). By denition of L and since d is an (1,1)-form, we obtain for U; V?(T 10 ) L ([T; U]; V ) + L (U; [T; V ]) =?i(d([t; U]; V ) + d(u; [T; V ])) Then, the Bianchi-identity gives L ([T; U]; V ) + L (U; [T; V ]) =?i([[u; V ]; T ]) (17) = i(([[t; U]; V ]) + ([U; [T; V ]])): = id([u; V ]; T ) + it (([U; V ]))? i[u; V ]((T )) =?it (d(u; V )) = T (L (U; V )): Since T 10 is integrable, we have [ V ; U]?(T10 ) and because of (17) [[ V ; U]; T ]?(H C ). Hence, Finally, L ([T; U]; V )? L ([T; V ]; U) =?id([t; U]; V ) + id([t; V ]; U) = i([[t; U]; V ])? i([[t; V ]; U]) = i([[ V ; U]; T ]) = 0: (19) L ([T; U]; V ) = L ([T; U]; V ) = L ([T; V ]; U) = L ([T; V ]; U): If we consider the Levi-form L as a bilinear form on the real tangent bundle, we obtain a symmetric bilinear form on T M which is non-degenerate on H. Proposition 1 Let (M n+1 ; T 10 ; ) be a non-degenerate pseudo-hermitian manifold and (H; J) the real CR-structure, dened by T 10. Let X and Y be two vector elds in H. Then the Levi-form L : T M T M?!R satises L (X; Y ) = d(x; JY ) ; (1) L (JX; JY ) = L (X; Y ) and L (JX; Y ) + L (X; JY ) = 0 ; () L ([T; X]; Y )? L ([T; Y ]; X) = L ([T; JX]; JY )? L ([T; JY ]; JX) : (3) Proof: By denition of H there are vector elds U; V?(T 10 ) such that X = U + U, Y = V + V. Therefore, L (X; Y ) = L (U + U; V + V ) = L (U; V ) + L ( U; V ) = Re L (U; V ): 13

14 Hence, L j T MT M is real, symmetric and on H non-degenerate. If X = U + U, then JX = i(u? U). Therefore, Furthermore, d(x; JY ) = d(u + U; i(v? V )) = id( U; V )? id(u; V ) =?Re (id(u; V )) = Re (L (U; V )) = L (X; Y ): L (JX; JY ) = d(jx; J Y ) =?d(jx; Y ) = L (Y; X) = L (X; Y ): The second formula of () follows since J =?id. Formula (3) is a direct corollary of (19) and (0). On non-degenerate pseudo-hermitian manifolds there exists a special covariant derivative, the so-called Webster connection. Proposition 13 Let (M; T 10 ; ) be a non-degenerate pseudo-hermitian manifold and let T be the characteristic vector eld of. Then there exists an uniquely determined covariant derivative r W :?(T 10 )?!?(T M C T 10 ) on T 10 such that 1. r W is metric with respect to L : X(L (U; V )) = L (r W X U; V ) + L (U; r W X V ) U; V?(T 10); X?(T M C ) (4). r W T U = pr 10[T; U], (5) 3. r W U = pr V 10[ V ; U], where pr 10 denotes the projection on T 10. Furthermore, r W satises (6) r W U V? r W V U = [U; V ]; U; V?(T 10 ): (7) Remark: This covariant derivative was introduced by Tanaka [Tan75] for strictly pseudoconvex manifolds and by Webster [Web78] in the pseudo-hermitian case. Proof: Let r be a covariant derivative satisfying (4)-(6). Then for U; V; Z?(T 10 ) L (r Z U; V ) = Z(L (U; V ))? L (U; r Z V ) = Z(L (U; V ))? L (U; [ Z; V ]): (8) Hence, r is uniquely determined by (5), (6), (8). On the other hand it is easy to check, that (5), (6), (8) dene a metric covariant derivative on T 10. If U; V; Z?(T 10 ), then L (U; [ V ; Z])? L (V; [ U; Z]) =?id(u; [V; Z]) + id(v; [U; Z]) = =?iu(([v; Z])) + i([u; [V; Z]]) + iv (([U; Z]))? i([v; [U; Z]]) = iu(d(v; Z))? iv (d(u; Z)) + i([[u; V ]; Z]) =?U(L (V; Z)) + V (L (U; Z))? id([u; V ]; Z) =?U(L (V; Z)) + V (L (U; Z)) + L ([U; V ]; Z): 14

15 Hence, L (r W U V? r W V U; Z) (8) = U(L (V; Z))? V (L (U; Z))? L (V; [ U; Z]) + L (U; [ V ; Z]) = L ([U; V ]; Z): Now, we extend the Webster connection to T M C by r W U := rw U and r W T := 0: Proposition 14 The torsion Tor W of the Webster connection r W :?(T M C )?!?(T M C T M C ) satises Tor W (U; V ) = Tor W ( U; V ) = 0 ; (9) Tor W (U; V ) = il (U; V ) T ; (30) Tor W (T; U) =?pr 01 [T; U] ; (31) Tor W (T; U) =?pr10 [T; U] ; (3) where pr 01 denotes the projection onto T 10, p 10 the projection onto T 10 and U; V?(T 10 ). Proof: By denition of r W on T M C follows Tor W (X; Y ) = Tor W ( X; Y ) for all X; Y?(T M C ). Then (9) follows from (7). Furthermore, Tor W (U; V ) = r W U V? r W V U? [U; V ] = r W U V? pr 10[ V ; U] + [ V ; U] = pr 10 [ U; V ] + pr01 [ V ; U] + ([ V ; U]) T = ([ V ; U]) T = d(u; V ) T = il (U; V ) T ; Tor W (T; U) = r W T U? r W U T? [T; U] = pr 10 [T; U]? [T; U] =?pr 01 [T; U] : Tor W (T; U) =?pr10 [T; U] follows, since pr10 [X; Y ] = pr 01 [ X; Y ] for all X; Y T M C. Let (M; T 10 ; ) be a non-degenerate pseudo-hermitian manifold and let (p; q) be the signature of (T 10 ; L ). Then g := L + denes a metric of signature (p; q+1) on M. 15

16 Proposition 15 Let (M; T 10 ; ) be a non-degenerate pseudo-hermitian manifold. Then the Webster connection r W :?(T M)?!?(T M T M) considered on the real tangent bundle is metric with respect to g and the torsion of r W is given by Tor W (X; Y ) = L (JX; Y ) T for X; Y?(H); (33) Tor W (T; X) =? 1 f[t; X] + J[T; JX]g for X?(H): (34) Furthermore, on?(h) holds. r W J = J r W (35) Proof: r W is metric with respect to g, since r W is metric on (T 10 ; L ) and r W T 0. If Z H C, then pr 01 (Z) = 1 (Z+iJZ) and pr 10(Z) = 1 (Z?iJZ). Now, let X = U+ U, Y = V + V?(H), U; V?(T10 ). Then from (9) - (31) follow Tor W (T; X) = Tor W (T; U + U) =?pr01 [T; U]? pr 10 [T; U] =? 1 =? 1? [T; U] + ij[t; U] + [T; U]? ij[t; U]? [T; U + U] + J[T; iu? i U] =? 1 ([T; X] + J[T; JX]) ; Tor W (X; Y ) = Tor W ( U; V ) + Tor W (U; V ) =?i L (U; V ) T + il (U; V ) T Furthermore, = (?il ( U; V + V ) + il (U; V + V )) T = L (JX; Y ) T: r W JX = r W (i(u? U)) = i(r W U? r W U) = J(r W U + r W U) = Jr W X: Now, let R rw?( M C End(T M C ; T M C )) be the curvature operator of r W Then the (4,0)-curvature tensor R W R rw (X; Y ) = [r W X ; rw Y ]? r W [X;Y ] : R W (X; Y; Z; V ) := g (R rw (X; Y )Z; W ); X; Y; Z; W T M C has the following properties 16

17 Proposition 16 Let X; Y; Z; V T M C, A; B; C; D T 10. Then 1) R W (X; Y; Z; V ) =?R W (Y; X; Z; V ) =?R W (X; Y; V; Z) ) R W (X; Y; Z; V ) = R W ( X; Y ; Z; V ) 3) R W (A; B; C; D) = RW (C; B; A; D) 4) R W (A; B; ; ) = 0 Proof: This follows by a direct calculation from the denition of r W on T M C. Let! M C be a complex -form and ~! : T 10?! T 10 the uniquely determinedc - linear map with!(u; V ) = L (~!U; V ), U; V T 10. Then the -trace of! is dened by Tr! := Tr(~!): If (Z 1 ; : : : ; Z n ) is an orthonormal basis of (T 10 ; L ), " k = L (Z k ; Z k ), then The (,0)-tensor eld Tr! = nx =1 Ric W := Tr (3;4) R W = "!(Z ; Z ): nx =1 is called the Webster-Ricci-tensor, the function R W := Tr Ric W " R W (; ; Z ; Z ) is the Webster scalar curvature. Proposition 16 shows that 1. Ric W 1;1 M,. Ric W (X; Y ) ir; X; Y T M, 3. R W is a real function. Geometric properties of pseudo-hermitian manifolds (M n+1 ; T 10 ; ; r W ) were studied for example in [Web78], [Lee88], [JL87], [BGS84], ([Gre85]. 5 Feerman spaces Let (M n+1 ; T 10 ) be a CR-manifold. The complex line bundle K := n+1;0 M of (n+1; 0)-forms is called the canonical bundle of (M n+1 ; T 10 ).R+ acts on K = Knf0g by multiplication. Let F := K =R+. Then (F; ; M) is the S 1 -principal bundle over M associated to K. We call (F; ; M) the canonical S 1 -bundle of (M; T 10 ). Now, 17

18 let (M; T 10 ; ) be a non-degenerate pseudo-hermitian manifold and r W :?(T 10 )?!?(T M C T 10 ) its Webster-connection. r W allows us to dene a connection A W on the canonical S 1 -bundle F in the following way: Let s = (Z 1 ; : : : ; Z n ) be a local orthonormal basis of (T 10 ; L ) over U M and let us denote by! s := (! ) the matrix of connection forms of r W with respect to s r W Z = X! Z : (Z 1 ; : : : ; Z n ; Z1 ; : : : ; Zn ; T ) is a local basis of T M C over U. Let ( 1 ; : : : ; n ; 1 ; : : : ; n ; ) be the corresponding dual basis. Then ^ s := ^ 1 ^ : : : ^ n : U?! K is a local section in K. We denote by s := [^ s ] the corresponding local section in F = K =R+. The Webster connection r W denes in the standard way a covariant derivative r K in the canonical line bundle K such that r K ^ s =? X! ^ s =? Tr! s ^ s : Since r W is metric with respect to L, the trace Tr! s is purely imaginary. Hence r K is induced by a connection A W on the associated S 1 -principal bundle (F; ; M; S 1 ) with the local connection forms s A W =? Tr! s : Let W be the curvature form of the connection A W on F. Since W is tensionell and right-invariant, it can be considered as -form on M with values in ir. Over U M holds. On the other hand, W = da s =?Tr d! s : (36) Hence, Ric W (X; Y ) = X = X " L (([r W X ; rw Y ]? r W [X;Y ] )Z ; Z ) d!? X ;! ^! (X; Y ): From (36) it follows Ric W = Tr d! s? Tr (! s ^! s ) = Tr d! s : W =?Ric W (37) The connection A W on the canonical S 1 -bundle (F; ; M) is called the Webster-connection on F. Two connections on an S 1 -principal bundle over M dier by an 1-form on M with values in ir. The connection A := A W? i (n + 1) RW 18

19 on the canonical S 1 -bundle (F; ; M) is called the Feerman connection on F. Let us consider the following right-invariant metric on F : h := 4 L? i n + A ; where denotes the symmetric tensor product. h is called the Feerman metric on F. If (T 10 ; L ) is of signature (p; q), then h has signature (p + 1; q + 1). In particular, if (M; T 10 ; ) is strictly pseudoconvex, h is a Lorentzian metric. The semi-riemannian manifold (F; h ) is called the Feerman space of (M; T 10 ; ). The bres of the canonical S 1 -bundle F are isotropic submanifolds of (F; h ). From the special choise of the Feerman connection A in the denition of h results that the conformal class [h ] of the metric h is an invariant of the oriented CR-manifold (M; T 10 ), e.g. if ~ = f, f > 0, is a further pseudo-hermitian structure on (M; T 10 ), then h ~ = f h (see [Lee86], Th ). We remark that Feerman spaces are never Einsteinian. Feerman spaces were rst dened by Feerman ([Fef76]) in case of strictly pseudoconvex hypersurfaces incn, its denition was extended by Burns, Diederich, Shnider ([BDS77]), Farris ([Far86]), and Lee ([Lee86]) to general CR-manifolds. Sparling ([Spa85]), Lee ([Lee86]), Graham ([Gra87]) and Koch ([Koc88]) studied geometric properties of Feerman spaces. In the following we always assume that (M; T 10 ; ) is strictly pseudoconvex. In order to nd global solutions of the Lorentzian twistor equation on Feerman spaces it is necessary to change the topological type of the canonical S 1 -bundle. Proposition 17 Let (M n+1 ; T 10 ; ) be a strictly pseudoconvex spin manifold. Then each spinor structure of the Riemann manifold (M; g ) denes a square root p F of the canonical S 1 -bundle F. (e.g. p F is an S 1 -bundle over M such that the associated line bundle L := p F S 1C satises L L = K ). Proof: Let U(n)?! SO(n)?! SO(n + 1) A + ib 7?! A := A B?B A! 7?! A be the canonical embedding of U(n) in SO(n + 1). The bundle P M of SO(n + 1)- frames of (M; g ) reduces to the group U(n): P H := f(x 1 ; JX 1 ; : : : ; X n ; JX n ; T ) j (X 1 ; JX 1 ; : : : ; X n ; JX n ) on-basis of (H; L )g is an U(n)-reduction of P M. Let (Q M ; f M ) be a spinor structure of (M; g ) and let us denote by (Q H ; f H ) the reduced spinor structure Q H := f?1 M (P H); f H := f M j Q H : 19!

20 Now, the proof of Proposition 17 is a repetition of Hitchin's proof of the fact that each spinor structure on a Kahler manifold denes a square root of the canonical bundle (see [Hit74]). Since we need some notation later on, we repeat the idea of the proof. Let ` : U(n)?! Spin(n) C = Spin(n) Z S 1 be dened by `(A) = ny k=1 cos k + sin k f k J 0 (f k ) i e np k=1 k ; (38) where (f 1 ; : : : ; f n ) is an unitary basis ofcn such that Af k = e i kf k and J :Cn 0!Cn is the standard complex structure of Cn. Then we have the following commutative diagram S 1 - j Spin(n) C j1 Spin(n) z? S 1 z - # det ` 6 Q QQ pr 1QQQs i U(n) -? SO(n) where i; j 1 ; j denote the canonical embeddings and : Spin(n)! SO(n) is the universal covering of SO(n). Hence, for each A U(n) and each square root of det(a) one has?1 (A) := j 1?1 (i(a)) = `(A) Det(A)? 1 : Now, let f(u ; g : U!?1 (U(n)))g ; are the cocycles dening the reduced spinor structure (Q H ; f H ). Then on U we choose a square root h : U! S 1 of the determinant of (g )?1 such that h = Det((g ))?1 and g = `((g )) h : (39) f(u ; h )g are cocyles dening a square root ( p F ; ; M) of the canonical S 1 - bundle (F; ; M). Let ( p F ; ; M) be the square root of the canonical S 1 -bundle dened by the spinor structure of (M; g ). Then the Webster connection A W on F denes a corresponding connection A pw on p F : Let f~s : U! Q H g be a covering of Q H by local sections with the transition functions g ; ~s = ~s g. Let s = f H (~s ) P H and denote by p s : U! p F the local sections in p F with transition functions h p s = p s h ; dened by (39). Then the local connection forms of A pw are given by p p s A W = 1 s AW =? 1 Tr! s (40) 0

21 and the curvature of A p W is p The connection A p W = 1 W =? 1 RicW : (41) on p F dened by p p A := A W i? 4(n + 1) RW is called the Feerman connection on p F and the Lorentzian metric h := 8 L? i n + A is the Feerman metric on p F. As we will see in the next section, the spinor structure (Q M ; f M ) of (M; g ) denes a canonical spinor structure on ( p F ; h ). Denition 4 The Lorentzian spin manifold ( p F ; h ) is called the Feerman space of the strictly pseudoconvex spin manifold (M; T 10 ; ; (Q M ; f M )). p 6 Spinor calculus for S 1 -bundles with isotropic bre over strictly pseudoconvex spin manifolds Let (M n+1 ; T 10 ; ) be a strictly pseudoconvex manifold and let (Q M ; f M ) be a spinor structure of (M; g ). Furthermore, consider an S 1 -principle bundle (B; ; M; S 1 ) over M, a connection A on B and a constant c Rnf0g. Then h := h A;c := L? ic A is a Lorentzian metric on B. In this section we want to derive a suitable spinor calculus for the Lorentzian manifold (B; h). Let N?(T B) be the fundamental vector eld on B dened by the element c i ir of the Lie algebra ir of S 1 f N(b) = c i (b) := d b e it c j t=0 : dt Denote by T?(T B) the A-horizontal lift of the characteristic vector eld T of. Then N and T are global isotropic vector elds on B such that h(n; T ) = 1. Consider the global vector elds s 1 = 1 p (N? T ) and s = 1 p (N + T ): (4) 1

22 Then h(s 1 ; s 1 ) =?1; h(s ; s ) = 1; h(s 1 ; s ) = 0: Let the time orientation of (B; h) be given by s 1 and the space orientation by the vectors (s ; X ; 1 JX; : : : ; 1 X n; JXn)), where (X 1 ; JX 1 ; : : : ; X n ; JX n ) P H ; and X denotes the A-horizontal lift of a vector eld X on M. Now, let (Q H ; f H ) be the reduced spinor structure of (M; g ) dened in the previous section. Denote by S H := Q H?1 (U (n)) n;0 the corresponding spinor bundle of (H; L ). Obviously, the bundle ^P B := f(s 1 ; s ; X 1 ; JX 1 ; : : : ; X n ; JX n) j (X 1 ; JX 1 ; : : : ; X n ; JX n ) on-basis of (H; L )g is an U(n)-reduction of the frame bundle P B of (B; h) with respect to the embedding U(n)?! SO 0 (n + ; 1) A + ib 7?! A B?B A 1 C A : Since ^PB P H we have P B P H U (n) SO 0 (n + ; 1): Therefore, Q B := Q H?1 (U (n)) Spin 0 (n + ; 1) ; f B := [f H ; ] is a spinor structure of the Lorentzian manifold (B; h). bundle S on (B; h) is given by The corresponding spinor S = Q H?1 (U (n)) n+;1 : (43) Proposition 18 Let S H be the spinor bundle of (H; L ) over M. Then the spinor bundle S of (B; h) can be identied with the sum S S H S H ; where the Cliord multiplication is given by s 1 ('; ) = (? ;?') (44) s ('; ) = (? ; ') (45) X ('; ) = (?X '; X ); X H: (46)

23 In particular, Furthermore, the positive and negative parts of S are N ('; ) = (? p ; 0) (47) T ('; ) = (0; p '): (48) S + = S + H S? H ; S? = S? H S + H : (49) The indenite scalar product h; i in S is given by h('; ); ( ^'; ^)i =?( ; ^') S H? ('; ^) S H ; (50) where (; ) S H is the usual positive denite scalar product in S H. Proof: By denition of the spinor bundle S (see (43)) we have only to check, how the Spin(n)-modul n+;1 decomposes into Spin(n)-representations. Let the embedding i :Rn!Rn+;1 be given by i(x) = (0; 0; x) and let Spin(n),! Spin 0 (n + ; 1) be the corresponding embedding of the spin groups. Consider the following isomorphisms of the representation spaces : n+;1?! n;0 n;0 u u(1) + v u(?1) 7?! (u ; v) where we use the notation of section. Then formula (1) shows that (e 1 (u u(1) + v u(?1))) = (?u;?v) (e (u u(1) + v u(?1))) = (u;?v) (e k (u u(1) + v u(?1))) = (?e k? u; e k? v); k > : Therefore, is an isomorphism of the Spin(n)-representations and (44)-(46) and because of (4) also the formulas (47), (48) are valid. Let! n+ = e 1 : : : e n+ be the volume element of Cli n+;1 and! n = e 1 e n the volume element of Cli n;0. Then using the identication we obtain! n+ (u; v) = (?! n u ;! n v): According to the denition of S this shows (49). Because of (5) the scalar product satises h('; ); ( ^'; ^)i = (s 1 ('; ); ( ^'; ^)) s1 = ((? ;?'); ( ^'; ^)) s1 =?( ; ^') S H? ('; ^) S H : 3

24 In order to describe the spinor derivative in the spinor bundle S of B we need the connection forms of the Levi-Civita connection of (B; h). Let X; Y; Z be local vector elds on (B; h) of constant length and constant scalar products with each other. Then the Levi-Civita connection r of (B; h) satises h(r X Y; Z) = 1 fh([x; Y ]; Z) + h([z; Y ]; X) + h([z; X]; Y )g: (51) For a vector Z T b B we denote by Z h the projection on the horizontal tangent space and by Z v the projection on the vertical tangent space. If X T (b) M, then X T b B denotes the horizontal lift of X. Let A (M; ir) be the curvature form of the connection A. From the connection theory in principle bundles follows for vector elds X; Y on M Now, let X; Y?(H). Since [T; X]?(H) and [X ; N] = 0 ; (5) [X ; Y ] v = i c A (X; Y ) N ; (53) [X ; Y ] h = [X; Y ] : (54) [X; Y ] = pr H [X; Y ] + ([X; Y ]) T = pr H [X; Y ]? d(x; Y ) T we obtain from (53) and (54) [T ; X ] = [T; X] + i c A (T; X) N ; (55) [X ; Y ] = pr H [X; Y ]? d(x; Y ) T + i c A (X; Y ) N: (56) Proposition 19 Let X; Y; Z?(H) be vector elds of constant lenght and constant L -scalar products with each other. Then h(r X Y ; Z ) = L (r W X Y; Z) h(r N Y ; Z ) = 1 d(y; Z) h(r T Y ; Z ) = 1 fl ([T; Y ]; Z)? L ([T; Z]; Y )? i c A (Y; Z)g h(r X Y ; N) =? 1 d(x; Y ) h(r X Y ; T ) = 1 fl ([T; X]; Y ) + L ([T; Y ]; X) + i c A (X; Y )g h(r T T ; Z ) =?i c A (T; Z) h(rn; T ) = h(rt ; T ) = h(rn ; N ) = 0 h(r N N; Z ) = h(r N T ; Z ) = h(r T N; Z ) = 0: 4

25 Proof: From (51) and (56) it follows h(r X Y ; Z ) = h( pr H [X; Y ] ; Z ) + h( pr H [Z; Y ] ; X ) + h( pr H [Z; X] ; Y ) = L ([X; Y ]; Z) + L ([Z; Y ]; X) + L ([Z; X]; Y ): According to (33) Tor W (X; Y ) = L (JX; Y ) T. Hence, L ([X; Y ]; Z) = L (r W X Y? r W Y X? Tor W (X; Y ); Z) = L (r W X Y? r W Y X; Z): Therefore, using that r W is metric with respect to L we obtain h(r X Y ; Z ) = L (r W X Y; Z): The other formulas follow immediately from the denition of h and (51), (5), (55) and (56). By denition the spinor derivative on S is given by the following formula: Let ~s : U?! Q H be a local section in Q H and s = (X 1 ; : : : ; X n ) = f H (~s) P H the corresponding orthonormal basis in (H; L ). Consider a local spinor eld = [ ~s; u ] in S. Then r S = [ ~s; du ]? 1 h(rs 1; s ) s 1 s? nx k=1 h(rs ; X k) s X k + 1 X k<l nx k=1 h(rs 1 ; X k) s 1 X k h(rx k; X l ) X k X l : Using the denition of s 1 and s (see (4)) and Proposition 19 we obtain h(rs 1 ; s ) = 0. Furthermore, if we denote by a k (Z) the vector eld from Proposition 19 results a k (Z) := h(r Z s ; X k) s? h(r Z s 1 ; X k) s 1 ; a k (N) = 0 a k (T ) =?i c A (T; X k ) N a k (X j ) = 1 d(x j; X k ) T? 1 fl ([T; X j ]; X k ) + +L ([T; X k ]; X j ) + i c A (X j ; X k )gn: These formulas and Proposition 19 give the following formulas for the spinor derivative in the spinor bundle S of (B; h): 5

26 Proposition 0 Let ~s : U?! Q H be a local section in Q H, s = f H (~s) = (X 1 ; : : : ; X n ) and let = [ ~s; u ] be a local section in S. Then for the spinor derivative of holds: r S N = [ ~s; N(u) ] d r S T = [ ~s; T (u) ] + i c (T? A ) N? i c 8 (A ) X k<l fl ([T; X k ]; X l )? L ([T; X l ]; X k )gx k X l r S X = [ ~s; X (u) ]? 1 4 (X? d) T + i c 8 (X? A ) N nx k=1 X k<l fl ([T; X]; X k ) + L ([T; X k ]; X)gX k N L (r W X X k; X l ) X k X l ; where denotes the projection of a form p M onto p M, on B and the vector eld X belongs to the set fx 1 ; : : : ; X n g. is its horizontal lift Proposition 1 Let (X 1 ; : : : ; X n ) be a local orthonormal basis of (H; L ) with X = J(X?1 ). Denote by 1 ; : : : ; n the dual basis of (X 1 ; : : : ; X n ) and by s = (Z 1 ; : : : ; Z n ), Z = p 1 (X?1? ix ), the corresponding local unitary basis of (T 10 ; L ). Consider the -forms X b s := fl ([T; X k ]; X l )? L ([T; X l ]; X k )g k ^ l ; X k<l d s (X) := L (r W X X k; X l ) k ^ l ; X H: k<l Then 1) b s 1;1 (M) and Tr b s = Tr! s (T ) ) d s (X) 1;1 (M) and Tr d s (X) = Tr! s (X), where! s is the matrix of connection forms of the Webster connection r W with respect to s = (Z 1 ; : : : ; Z n ). Proof: A -form belongs to 1;1 M i (JX; JY ) = (X; Y ) for all X; Y H. From formula (3) of Proposition 1 follows for X; Y fx 1 ; : : : ; X n g b s (JX; JY ) = L ([T; JX]; JY )? L ([T; JY ]; JX) (3) = L ([T; X]; Y )? L ([T; Y ]; X) = b s (X; Y ): 6

27 Therefore, b s 1;1 M. Furthermore, Tr b s = i = i nx nx =1 =1 b s (X?1 ; X ) fl ([T; X?1 ]; X )? L ([T; X ]; X?1 )g: Inserting one obtains X?1 = 1 p (Z + Z ); X = i p (Z? Z ) Tr b s = nx =1 = i = i fl ([T; Z ]; Z )? L ([T; Z ]; Z )g nx nx =1 =1 Im fl (pr 10 [T; Z ]; Z )g Im L (r W T Z ; Z )) = i Im ( Tr! s (T )) : Since r W is metric with respect to L, we have! +! = 0. Hence,! (T ) is imaginary. Therefore, Tr b s = Tr w s (T ). According to formula (35) of Proposition 15 and formula () of Proposition 1 we have for Y; Z fx 1 ; : : : ; X n g and X H d s (X)(JY; JZ) = L (r W X JY; JZ) = L (Jr W X Y; JZ) = L (r W X Y; Z) = d s(x)(y; Z): This shows that d s (X) 1;1 (M). Furthermore, Tr d s (X) = i = 1 nx nx =1 =1 = i Im Tr! s (X) = Tr! s (X): L (r W X X?1 ; X ) fl (r W X Z ; Z )? L (r W X Z ; Z )g Next we proof a property of the spinor bundle S H of (H; L ), which is very similar to the properties of the spinor bundle of Kahler manifolds (see [Kir86]). 7

28 Proposition Let (M n+1 ; T 10 ; ) be a strictly pseudoconvex spin manifold and ( p F ; h ) its Feerman space. Then the spinor bundle S H of (H; L ) has the following properties: 1. S H decomposes into n + 1 subbundles S H = n n+k nm r=0 S (?n+r)i ; where S ki is the eigenspace of the endomorphism d : S H! S H to the eigenvalue ki. The dimension of S ki is. In particular, there are two 1-dimensional subbundles S "ni ; " = 1, of S H satisfying d j S "ni = "ni Id S"ni.. If 1;1 M, then j = S " Tr "ni () Id S : "ni 3. The induced bundles S n"i on the Feerman space p F are trivial. A global section "?( S n"i ) is given in the following way: Let ~s : U?! Q H be a local section in Q H, s the local unitary basis in (T 10 ; L ), corresponding to f H (~s) : U?! P H. Furthermore, let p s : U?! p F be the local section in p F dened by ~s and let ' s : p F j U?! S 1 be given by p = p s ((p)) ' s (p). Then denes a global section in?( S n"i ). "(p) := [ ~s((p)); ' s (p)?" u("; ; ") ] Proof: n;0 is a U(n)-representation, where U(n) acts by U(n) `?! Spin C (n) n;0?! GL( n;0 ): The element 0 = e 1 e + + e n?1 e n Cli C n;0 0 u(" 1 ; : : : ; " n ) = i ( nx k=1 acts on n;0 by " k ) u(" 1 ; : : : ; " n ): Hence, n;0 decomposes into U(n)-invariant eigenspaces E r( 0 ) of 0 to the eigenvalues r = (?n + r)i, r = 0; : : : ; n. In particular, the eigenspace to the eigenvalue "ni, " = 1 ; is 1-dimensional and given by E in" ( 0 ) =C u("; : : : ; "): By denition of ` (see (38)) we obtain for A = diag(e i 1 ; : : : e in) `(A)u("; : : : ; ") = ( u("; : : : ; ") " =?1 DetA u("; : : : ; ") " = 1: (57) 8

29 Hence, E?ni is the trivial U(n)-representation and E ni is isomorphic to the U(n)- representation n (Cn ). Since the subspaces E r( 0 ) of n;0 are invariant under the action of?1 (U(n)) we obtain the decomposition S H = nm r=0 S r; where S r := Q H?1 (U (n)) E r( 0 ). If ~s : U?! Q H is a local section in Q H, d acts on S H by d [ ~s ; v ] = [ ~s ; 0 v ]: Therefore, S r is the eigenspace of d to the eigenvalue r. Now, let = [ q ; u("; : : : ; ") ] S "ni, " = 1. Denote f H (q) = (X 1 ; : : : ; X n ) P H, X = JX?1 and s = (Z 1 ; : : : ; Z n ) the corresponding unitary basis in (T 10 ; L ) with Z = p 1 (X?1? ijx?1 ). Let ( 1 ; : : : ; n ) be the dual basis of (Z 1 ; : : : ; Z n ) and ( 1 ; : : : ; n ) the dual basis of (X 1 ; : : : ; X n ). If 1;1 M is a form of type (1,1), then = = 1 nx ;=1 X 6= ^ (?1 ^?1 + ^ ) + i X ; ( ^?1??1 ^ ): Hence, = [ q ; 1 + i P P 6= ; (e?1 e?1 + e e ) u("; : : : ; ") (e e?1? e?1 e ) u("; : : : ; ") ] where = (Z ; Z ). Using formula (1) we obtain (e?1 e?1 + e e ) u("; : : : ; ") = 0 6= (e e?1? e?1 e ) u("; : : : ; ") = 0 6= (e e?1? e?1 e ) u("; : : : ; ") =?"i u("; : : : ; "): Therefore, = [ q ; " X (Z ; Z ) u("; : : : ; ") ] = " Tr : Now, let us consider the section "?( S "ni ) dened by "(p) := [ ~s((p)) ; ' s (p)?" u("; : : : ; ") ]: 9

30 Let ~s; ~^s : U?! Q H be two local sections, ~s = ~^s g and let h : U?! S 1 be the function dened by (39): `((g)) h = g; h = Det((g))?1 : (58) Then '^s (p) = ' s (p) h((p)) and "(p) = [ ~^s g ; ' s (p)?" u("; : : : ; ") ] = [ ~^s ; ' s (p)?" g u("; : : : ; ") ] = [ ~^s ; '^s (p)?" h " g u("; : : : ; ") ] (58) = [ ~^s ; '^s (p)?" h "+1`((g)) u("; : : : ; ") ] (57);(58) = [ ~^s ; '^s (p)?" u("; : : : ; ") ]: Hence, " is a global section in the bundle S "ni on p F. 7 Twistor spinors on Feerman spaces Let (M n+1 ; T 10 ; ) be a strictly pseudoconvex spin manifold, ( p F ; ; M) the square root of the canonical S 1 -bundle corresponding to the spinor structure of (M; g ) and h the Feerman metric on p F. Denote by "?( S H ) the global sections in the bundles S "ni over p F dened in Proposition. Now, we are able to solve the twistor equation on the Lorentzian spin manifold ( p F ; h ). Theorem 1 Let S := S H S H be the spinor bundle of ( p F ; h ). Then the spinor elds " := ( " ; 0)?(S), " = 1, are solutions of the twistor equation on ( p F ; h ) with the following properties: 1. The canonical vector eld V " of " is a regular isotropic Killing vector eld.. V " " = 0 : 3. r S V " " =? 1 p " i " : 4. k " k 1. Remark: If n is even, then 1 and?1 are linearly independent spinor elds in S +. If n is odd then 1?(S + ) and?1?(s? ) (see Proposition 18). The second property of Theorem 1 shows that " is a pure or partially pure spinor eld (see [TT94]). A vector eld is called regular, if all of its integral curves are closed and of the same shortest period. 30

31 Proof of Theorem 1: We use the formulas for thepspinor derivative in S given in Proposition 0 for the Feerman connection A = A and the constant c = 8. n+ Let ~s : U?! Q be a local section and ' s : p F j U?! S 1 the corresponding transition function in p F (see Proposition ). Then for the fundamental vector eld N on p F N(' s ) = n + i ' s (59) 4 holds. If Y p is an A -horizontal lift of a vector eld Y on M, we obtain using standard formulas from connection theory Y (' s ) =?' s p s A p (Y ) = 1 ' s f Tr! s (Y ) + i (n + 1) RW (Y )g; (60) where! s is the matrix of connection forms of the Webster connection with respect to the unitary basis s in (T 10 ; L ) corresponding to f H (~s). According to Proposition 18 we have N " = 0. Therefore, from Proposition 0 and (59), (60) result r S N " = r S T " = r S X " =?" n + i " d " ; 0? 1 " ftr! i s(t ) + (n + 1) RW g " + 1 p 4 b 1 s "? i n + A " ; 0? 1 " Tr! s(x) " + 1 d s(x) " ; 0? 1 4 (X? d) T " ; where b s and d s (X) are the 1;1 -forms dened in Proposition 1. Since " is a section in S "ni, b s and d s (X) act on " by multiplication with "Tr b s and " Tr d s (X), respectively (Proposition ). Hence, according to Proposition 1, r S T " = p 1 i?i n + A "? " 4(n + 1) RW " ; 0 r S X " =? 1 4 (X? d) T " : Furthermore, " is an eigenspinor of the action of d on S H to the eigenvalue "ni. Therefore, r S N " =? " i " : (61) Because of p A =? 1 RicW? i 4(n + 1) d(rw ) =? 1 RicW? i 4(n + 1) RW d; 31

32 p the curvature A Therefore, we obtain of the Feerman connection is a form of type (1,1). Hence, p A " = " Tr ( A p ) " = (? 1 " i" RW? 4(n + 1) RW in) " =?" n + 4(n + 1) RW ": r S T " = 0: (6) According to Proposition 18, T " = ( 0; p " ). If X fx 1 ; : : : ; X n g, the 1-form X? d acts on the spinor bundle by Cliord multiplication with J(X). Hence, we have p! r S X " = 0 ;? 4 J(X) " : (63) Now, using s 1 = 1 p (N? T ), s = 1 p (N + T ), we obtain?s 1 r S s 1 " = s r S s " = X r S X " = 0;? 1 p " i " ; where X fx 1 ; : : : ; X n g. This shows, that " is a twistor spinor (see Proposition 1). From Proposition 18 it follows ( " ; " ) = hs 1 " ; " i = h (0;? " ); ( " ; 0) i = ( " ; " ) S H = 1 : Furthermore, we obtain for the canonical vector eld V " V " = hs 1 " ; " i s 1? hs " ; " i s? = s 1 + s = p N: nx k=1 hx k "; " ix k Therefore, V " is regular and isotropic and satises V " " = 0. Because of (61) we have r S V " " =? 1 p " i " : It remains to show, that the vertical vector eld N is a Killing vector eld. This follows directly from the formulas of Proposition 19: L N h (Y; Z) = h (r Y N; Z) + h (Y; r Z N) = 0 for all vector elds Y and Z on p F. 3

33 Conversely, we have Theorem Let (B n+ ; h) be a Lorentzian spin manifold and let '?(S) be a nontrivial twistor spinor on (B; h) such that 1. The canonical vector eld V ' of ' is a regular isotropic Killing vector eld.. V ' ' = 0 : 3. r S ' = i c ' ; c = const Rnf0g. V' Then B is an S 1 -principal bundle over a strictly pseudoconvex spin manifold (M n+1 ; T 10 ; ) and (B; h) is locally isometric to the Feerman space ( p F ; h ) of (M; T 10 ; ). Proof: Since V ' is regular, it denes an S 1 -action on B B S 1?! B (p; e it ) 7?! V t L (p) where t V (p) is the integral curve of V = V ' through p and L is the period of the integral curves. Then M := B= S 1 is an n + 1 -dimensional manifold ir and V is the fundamental vector eld dened by the element i of the Lie algebra of L S1 in the S 1 -principal bundle (B; ; M; S 1 ). Now we use Sparling's characterization of Feerman spaces, proved by Graham in [Gra87]. Let W denote the (4,0)-Weyl tensor, C the (3,0)- Schouten-Weyl tensor and K the (,0)-Schouten tensor of (B; h). Graham proved: If V is an isotropic Killing vector eld such that V? W = 0 (64) V? C = 0 (65) K(V; V ) = const < 0; (66) then there exists a pseudo-hermitian structure (T 10 ; ) on M such that (B; h) is locally isometric to the Feerman space (F; h ) of (M; T 10 ; ). The local isometry is given by S 1 -equivariant bundle maps U : Bj U?! F j U. We rst prove that V = V ' satises (64)-(66). Property (65) is valid for each twistor spinor (see Proposition 10). Using W (X ^ Y ) ' = 0 (see (11) of Proposition 5) and the assumption V ' ' = 0 we obtain 0 = fw (X ^ Y ) V? V W (X ^ Y )g ' = fv? W (X ^ Y )g ' = W (X; Y; V ) ' 33