COMPLEX ALMOST CONTACT STRUCTURES IN A COMPLEX CONTACT MANIFOLD

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1 S ISHIHARA AND M KONISHI KODAI MATH J 5 (1982) 3-37 COMPLEX ALMOST CONTACT STRUCTURES IN A COMPLEX CONTACT MANIFOLD BY SHIGERU ISHIHARA AND MARIKO KONISHI 1 Introduction Let M be a complex manifold of odd dimension 2?n+l (Ξ>3) covered by an open covering $l={oi} consisting of coordinate neighborhoods If there is a holomorphic 1-form ω t in each Oi^% in such a way that for any O lf (11 i) ωiλ(dωi) m Φ in O t> (11 ϋ) Wi ftjwj in where f τj is a holomorphic function in OiΓ\O Jf then the set {{ω if O l )\O i^sh} of local structures is called a complex contact structure and M a complex contact manifold where ω% is called the contact form in O % On the other hand suppose that there are given in each C^e^ί a 1-form u lt a vector field U v and a tensor field G x of type (11) satisfying the following condition (12 i) and (12 ii): for any O lf O^% (12 i) G ι t =-I+u i U ι +v t V ι G t F=-FG t / and F being respectively the identity tensor field of type (11) and the complex structure of M where v t and V\ are defined in O % respectively by In OiίλOjΦφ there are functions a and b in such a way that u ι =au J bvj G ι =ag j bhj (12 ii) in OiΓ\O 3 Vi=buj+avj H l bgj J rah 3 where H t is defined in O τ by H t =FG x Then the set {(u l} U t G t O ι )\O i^ψ! of local structures is called a complex Received June

2 COMPLEX ALMOST CONTACT STRUCTURES 31 almost contact structure and M a complex almost contact manifold It is the purpose of the present paper to discuss the relation between complex contact structures and complex almost contact structures and to prove THEOREM Let M be a complex contact manifold of odd complex dimension 2m+l (^3) Then M admits always a complex almost contact structure of class C This subject has been partially studied in [3] and proved that a complex almost contact manifold admits a complex contact structure if it is normal 2 Lemmas From now on M is assumed to be a complex contact manifold of complex dimension 2m+l (^3) with structure {(ω if O ι )\O i tξ ( %} Then for any O τ O Jt O k (21) Q)i=f tj <uj in and hence the cocycle condition fjkfktfτj=l in ι ΓΛ J Γ\ k holds Then there is a complex line bundle P over M with {f ιj \ as its transition functions Denote by P a circle bundle over M associated with P Then there is a non-vanishing complex-valued function τ % in C^e^ί such that the function (22) h ιj =τ ι - Ί f ιj τ J in is the transition function of P in OιΓ\Oj and satisfies the condition On putting we have by using (21) and (22) A O =1 π ι =τ ι ~ 1 ω ί in O t (23) π ι =h ιj π J \h τj \=l in O ι ίλθ J Let σ be a connection in the circle bundle P Then there is a local real 1-form σ % in each O ι^ς& such that (24) y/'-ia t =y/-ia 3 +-^- in The condition ω ι A(dω ι ) m Φθ implies (25) π ι A(dπ ι ) m Φθ in O x and hence

3 32 SHIGERU ISHIHARA AND MARIKO KONISHI (26) π t AΩ t m Φθ in O τ where Ω % is defined by (27) Ω ι =dπ i -V ZI ϊσ ι Aπ ι in O t Thus Ω % is a local 2-form of rank 2m in each O τ and (28) Ω ι =h ιj Ω J in <nj holds because of (23) and (24) Thus we have LEMMA 21 77ι<?π? zs in each ^% a 2-form Ω τ of rank 2m and of class C satisfying (28) We now put (29) u ι = (π(π ij rπ i rπ ι ) ι ) v x (π ι π ι ) Δ W I which are local real 1-forms in each O t e5ί (21) v^utof Then we obtain F being the complex structure of M Using (25) we get u ι =auj bvn (211) in OiΓ\O J} where h ιj =a + V^ϊb b\ The relation (25) implies or equivalently (π i +π ι )Λ(dπ i +dπ ι ) m Φθ (π ι -π ι )Λ(dπ ι -dπ ι ) m Φθ (212) u ι Λv i Λ(du ι ) 2m Φ u ι Λv i Λ(dv i ) 2πι Φ On the other hand we obtain from (27) and (29) (213) Therefore (212) and (213) imply ^/==r(ω ί Ω ι )dv i +σ ι Au ι Zv 1 (214) u ι Av i A(G i ) 2m Φ u ι Av ί A(H ι ) 2m Φθ where (215) Gi dui GtAVi H ι =dv i Jrσ ι Au ι Thus (28) (211) (212) and (213) imply

4 COMPLEX ALMOST CONTACT STRUCTURES 33 LEMMA 22 There are in each O*e5i skew-symmetric local tensor fields G % and H % of rank 4 m such that (216) 6i{X 9 Y)=6i(FX 9 Y) f G ί {XY)=-H ι {FXY) in O % for any vector fields X and Y and (217) Gi=aGj-bHj H ι =bg 3 +ah Jy in OiΠOj where h τj =a-}-v lb We now state two more lemmas which are essential in the proof of our theorem (cf [1] [2] [4]) Let O(n) be the orthogonal group acting on n variables H(n) be the space consisting of all positive definite symmetric (n n)- matrices and GL(n R) be the general linear group acting on n variables LEMMA 23 Any real non-singular (n n)-?natrιx p can be written in one and only one way as the product p=aβ with a^o(n) and β^h(n) The mapping φ:gl(n R) >O(n)xH(n) defined by this decomposition gives a homeomorphism Remark Since O(n) and H(n) are real analytic submanifolds of GL(n R) φ:gl(n R)-*O(n)xH(n) is a real analytic homeomorphism ie any p^ GL(n R) can be decomposed analytically in one and only one way as p=aβ where a^o(n) and β^h(n) (See Hatakeyama [2] for example) LEMMA 24 Let %={Oi} be an open covering of M by coordinate neighborhoods Suppose that there is in each O*e2i a local tensor fields a t of type (2) and of class C Choose a field of orthonormal frames in each O^^i and let γ %J be the transformations of these fields of orthonormal frames in OiΓλOj Then {at} defines globally a tensor field of class C in M if and only if Tτj a /ϊtj = a t holds in OiΓ\O 3 for any O ly Oj^% 3 Proof of theorem Let Gi and H % be the skew-symmetric tensors appearing in Lemma 22 If we put for any p^o z (eδl) D i (p)={ytξt p (O ι ):G i (Y X)= for any ZeT p (OJ} then we get in O % a local distribution Ό % : p^d z (p) Lemma 22 implies D t (p) = Dj(p) for any p^oiγλoj Hence the local distributions D % defined in O τ determines a global distribution D in M which is of real dimension 2 LEMMA 31 There is a unique local basis {U ΐf V z -} of the distribution D in each Oι^% such that u ι (U ι )=l u τ (V τ )= v l (U l )= v t (7)=l in G i {U u X)=G i (V ι X)=H i (U lf X)=H ι (V t X)= O ι Γ\O J

5 34 SHIGERU ISHIHARA AND MARIKO KONISH1 for any vector field X and U ι =au j -bv J V ι =bu j +av J in O τ Γ\O 3 Proof If we put for any p^o t (esί) A t (p)={yet p (O t ):v t (Y)=} then we get a distribution A ι :p^^a i (p) in O x The distribution DΓ\A Z is 1- dimensional in O z because of (214) Thus there is a unique vector field U ι in O x such that Ό % spans DΓ\A t and u ι (U ι )=l ie such that 6t(U t X)= u i (U ι )=l for any vector field X Lemma 22 implies in for any vector field X Putting in O % (31) V ι =-FU ι and using Lemma 22 and (21) we get t)=l u i (V l )= for any vector field X Therefore {U lf FJ is in O % a local basis of the distribution D As a consequence of (211) we have U t =auj-bvj V ι =bu j +av J in Thus Lemma 31 is proved We shall now prove the theorem stated in 1 Proof of theorem Let g be a Hermΐtian metric in M such that g(u lf X) = u t (X) and g(v t X)=v t (X) for any vector field X Take an orthonormal adapted frame {E lf FE U E 2m FE 2 m U t Vi) with respect to g in each Then by Lemma 31 Gι has components of the form (32) φ[ with respect to the frame {E af FE a U t V x } in O lf where Φ[ is a nonsingular real skew-symmetric (4m 4m)-matrix By Lemma 23 Φ[ can be written in the form (33) Φί=αί fl with αίeθ(4m) and βi^h(4tm) If we put

6 (34) COMPLEX ALMOST CONTACT STRUCTURES 35 a' % \ \ o : o then <Xι and β t define tensor fields of class C in O x Since Φ[ is skew-symmetric ft and hence As is easily seen αί 2 e(4m) ι a' x 'βi-ai^hiim) the decomposition we obtain Thus by the uniqueness of rv' 2 T 14m being the unit (4m 4m)-matrix Consequently we have (35) Φ ι =a ι -β i (36) α'=- o o =-/ 4 i n + a On the other hand the complex structure F has components of the form (37) f P -1 2 P= -1 1 o -1 1 = O(4m) with respect to the adapted frame {E a FE a U t FJ in O τ Hence Lemma 22 implies that H τ has components of the form Ψί (38) Ψί=PΦ'i=-Φ' t P Therefore Ψί can be decomposed as (39) Ψ[=(P'a / ι ) β[=δ/ ι 'β{ δ' t =P-a' t Denote by γ l3 the transformation of adapted frames {E a FE a U lf V t } and {E'a FE'a Uj Vj} in O ι Γ\O J Then Lemmas 22 and 24 imply

7 36 SHIGERU ISHIHARA AND MARIKO KONISHI Φ ι =γ ιj -{aφ j -bψ 3 )- t γ l} Substituting (35) and (39) into this we have where y' l3 denotes the element of {Am) such that a b -b a Since γ' tj e(4m) and the decomposition is unique we get /Q 11Λ R' r' R f t r f \OiL) pi -/ ιj pj j ij The equation (311) shows by means of Lemma 24 that {β ι } defines a global tensor field g of class C which is a Hermitian metric in M Denote by G τ the local tensor field of type (11) having components a τ with respect to the adapted frame {E a EE a U ιy Vi) in O x Thus (34) implies (312) G ί U x =G ι V ι = u x og x =v^g x = in O x and (36) implies (313) G t2 = I J ru τ U i -\-Vi V τ in O x Furthermore using (33) and (34) we have ( Y)=6 X (X Y) (314) in O % g(u % X)= Uί (X) g(v τ X)=ι for any vector fields X and Y Next we denote by H x the local tensor field of type (11) having components Γ-a x with respect to the adapted frame {E a FE a U τ V x ) in O τ Then (315) H ι =FG ι in α holds Then (31) implies (316) G x =agj-bh 3 H x = in O x Γ\Oj Summing up Lemma 31 (312) (313) (315) and (316) we see that {(u ι U t G lf O ι ): O ι^<$ί} is a complex almost contact structure in M Thus the Theorem is proved

8 COMPLEX ALMOST CONTACT STRUCTURES 37 BIBLIOGRAPHY [1] CHEVALLEY C Theory of Lie groups I Princeton Univ Press 1946 [2] HATAKEYAMA Y On the existence of Riemannian metrics associated with 2-form of rank 2r Tδhoku Math J 14 (1962) [3] ISHIHARA S and M Konishi Real contact 3-structure and complex contact structure Southeast Asian Bulletin of Math 3 (1979) [4] SHIBUYA Y On the existence of a complex almost contact structure Kodai Math J 1 (1978) TOKYO INSTITUTE OF TECHNOLOGY