✬ ✫ ✩ ✪ Triple structures on a manifold Fernando Etayo (Departamento de Matem´ aticas, Estad´ ıstica y Computaci´ on. Universidad de Cantabria), Rafael Santamar´ ıa (Departamento de Matem´ aticas. Universidad de Le´ on), and Uju´ e R. Tr´ ıas (Departamento de Matem´ aticas, Estad´ ıstica y Computaci´ on. Universidad de Cantabria). ✬ ✫ ✩ ✪ ✎ ✍ ☞ ✌ Abstract Many geometric concepts can be defined by a suitable algebraic formalism. This point of view has interest because one can compare different geometric structures ha- ving similar algebraic expressions. In the present paper we study manifolds endowed with three (1,1)-tensor fields F , P and J satisfying F 2 = ±I, P 2 = ±I, J = P ◦ F, PF ± FP = O. We analyse the geometries arising from the above algebraic conditions. According to the chosen signs there exist eight diffe- rent geometries. We show that, in fact, there are only four. In particular, hyper- complex manifolds and manifolds endowed with a 3-web fit to this construction. We study geometric objects associated to these manifolds (such as metrics, con- nections, etc.), restrictions on dimensions, etc., and we show significative examples. ✎ ✍ ☞ ✌ Structures of square ±I Definition A triple structure on a manifold M is given by three tensor fields of type (1,1), F , P and J satisfying the relations: • F 2 = ǫ 1 I, P 2 = ǫ 2 I, J 2 = ǫ 3 I , where ǫ 1 ,ǫ 2 ,ǫ 3 ∈ {−1, +1}, i.e., each one of them is an almost product or almost complex structure. • P ◦ F ± F ◦ P = O, i.e., they commute or anti-commute. • J = P ◦ F (the other compositions can be obtained according to the above relations). These relations allow to define four geometries: 1. F 2 = I, P 2 = I,P ◦ F + F ◦ P = O; in this case, J 2 = −I . 2. F 2 = I, P 2 = I,P ◦ F − F ◦ P = O; in this case, J 2 = I . 3. F 2 = −I, P 2 = −I,P ◦ F + F ◦ P = O; in this case, J 2 = −I . 4. F 2 = −I, P 2 = −I,P ◦ F − F ◦ P = O; in this case, J 2 = I . One can expect to define four other geometries 5. F 2 = I, P 2 = −I,P ◦ F + F ◦ P = O; in this case, J 2 = I . 6. F 2 = I, P 2 = −I,P ◦ F − F ◦ P = O; in this case, J 2 = −I . 7. F 2 = −I, P 2 = I,P ◦ F + F ◦ P = O; in this case, J 2 = −I . 8. F 2 = −I, P 2 = I,P ◦ F − F ◦ P = O; in this case, J 2 = I . One can prove that they coincide with the above ones: 1 ⇔ 5 ⇔ 8 and 4 ⇔ 6 ⇔ 7. Definition Let (F, P, J ) be a triple structure. • We say that (F, P, J ) is integrable if the Nijenhuis tensor of the three tensor fields vanishes, N F = N P = N J = 0. • We say that one linear connection ∇ is adapted to (F, P, J ) if ∇ parallelizes F , P and J , i.e., ∇F = ∇P = ∇J = 0. The classical definition of integrability of polynomial structures is defined in this way: one polynomial structure P is integrable if there exists one torsion free connection which parallelizing P . We proved (see [9]) that if a torsion free connection parallelizes one (1, 1) tensor field H satisfying H 2 = ±I then N H = 0. Notation If H 2 = I , we denote by T + H (M ) and T − H (M ) the distributions on M defined by the eigenvectors of H p on T p (M ) for every p ∈ M and also we denote by H + and H − the projections over these distributions. Also we denote by Tor ∇ and R ∇ the torsion and the curvature tensors of the connection ∇, by F (M ) the principal bundle of linear frames, and by g (1) the first prolongation of the Lie algebra g. ✎ ✍ ☞ ✌ Almost biparacomplex structures A manifold M has an almost biparacomplex structure if it is en- dowed with two tensor fields F , P satisfying F 2 = P 2 = I, P ◦ F + F ◦ P = O. In this case, J 2 = −I and the manifold has got three equidimensional supplementary distributions defined by: V 1 = T + F (M ), V 2 = T − F (M ), V 3 = T + P (M ), with J (V 1 )= V 2 , then dim M =2n, with dim V i = n, i =1, 2, 3. This structure defines the following subfibre of F (M ): B =  p∈M            (X 1 ,...X n ,Y 1 ,...,Y n ) {X 1 ,...,X n } basis of V 1 (p), {Y 1 ,...,Y n } basis of V 2 (p), {X 1 + Y 1 ,...,X n + Y n } basis of V 3 (p)            , which has the following structural group ∆GL(n; R)=  A 0 0 A  : A ∈ GL(n; R)  . The Lie algebra of the group ∆GL(n; R) is ∆ ∗ gl(n; R)=  A 0 0 A  : A ∈ gl(n; R)  . The Lie algebra ∆ ∗ gl(n; R) is invariant under matrix transpositions and her first prolongation ∆ ∗ gl(n; R) (1) vanishes, then the Lie group ∆GL(n; R) admits functorial connections. Two functorial connections attached to the Lie group ∆GL(n; R): • The canonical connection is characterized by ∇ c F = ∇ c P =0, Tor ∇ c (X, Y )=0, ∀X ∈ V 1 , ∀Y ∈ V 2 . • The well-adapted connection is provided by some results of the work [6]. The canonical connection has the following expression: ∇ X Y = F + ([F − X, F + Y ]+ P [F + X,PF + Y ]) + F − ([F + X, F − Y ]+ P [F − X,PF − Y ]), ∀X, Y ∈ X(M ). The expression of the well-adapted connection is ∇ w X Y = ∇ c X Y − T (X, Y ), ∀X, Y ∈ X(M ), T (X, Y ) = 1 3  F + Tor ∇ c (F + X, F + Y )+ PF + Tor ∇ c (F + X,PF − Y ) + PF − Tor ∇ c (F − X,PF + Y )+ F − Tor ∇ c (F − X, F − Y )  . Both connections characterize the integrability of the almost bi- paracomplex structure (F, P ) and the ∆GL(n; R)-structure: N F = N P = N J =0 ⇔ Tor ∇ =0, and, the ∆GL(n; R)-structure defined by (F, P ) is integrable if and only if Tor ∇ =0, R ∇ = 0, (with ∇ = ∇ c or ∇ = ∇ w ). Note that if the almost biparacomplex structure (F, P ) is integra- ble then both connections are the same and the three distributions V 1 ,V 2 and V 3 are integrable too; i.e., V 1 ,V 2 and V 3 determine three equidimensional foliations on the manifold M ; i.e., M is endowed with a 3-web. Thus, an almost biparacomplex structure characte- rize the existence of a 3-web on a manifold. Also we have that the canonical and the well adapted connections coincide with the Chern connection of the 3-web. One always can obtain the following local expressions F =  I n O O −I n  ,P =  O I n I n O  ,J =  O −I n I n O  , which provide elemental examples in R 4 . ✎ ✍ ☞ ✌ Functorial connections Definition ([6]) A functorial connection is an assignment σ →∇(σ), that asso- ciates a linear connection ∇(σ) over M to each section σ of the bundle F (M )/G, satisfying : i) ∇(σ) is reducible to the subbundle P σ . ii) For every diffeomorphism f of M , ∇(f · σ)= f ·∇(σ), where f ·∇(σ) is the connection image of ∇(σ) by f . iii) There exists an integer r ≥ 0 such that ∇(σ)(x) only depends on j r p σ, for every point p ∈ M . ✎ ✍ ☞ ✌ Almost hyperproduct structures A manifold M has an almost hyperproduct structure if it is endowed with two tensor fields F , P satisfying F 2 = P 2 = I, P ◦ F − F ◦ P = O. In this case, J 2 = I . The manifold has got six distributions defined by the three almost product structures: V 1 = T + F (M ), V 2 = T − F (M ), V 3 = T + P (M ), V 4 = T − P (M ), V 5 = T + J (M ), V 6 = T − J (M ). Let us define the following four distributions: V 13 = V 1 ∩ V 3 ,V 14 = V 1 ∩ V 4 ,V 23 = V 2 ∩ V 3 ,V 24 = V 2 ∩ V 4 . One almost hyperproduct structure (F, P ) defines the following sub- bundle of F (M ): H =  p∈M                (X 1 ,...,X n 13 ,Y 1 ,...,Y n 14 ,U 1 ,...,U n 23 ,W 1 ,...,W n 24 ) {X 1 ,...,X n 13 } basis of V 13 (p), {Y 1 ,...,Y n 14 } basis of V 14 (p), {U 1 ,...,U n 23 } basis of V 23 (p) {W 1 ,...,W n 24 } basis of V 24 (p)                , with dim V 13 = n 13 , dim V 14 = n 14 , dim V 23 = n 23 , dim V 24 = n 24 . Therefore dim M = n 13 + n 14 + n 23 + n 24 = n and there are no restrictions on the dimension of the manifold M . The structural group of this subfibre bundle of F (M ) is the image of the product group GL(n 13 ; R) × GL(n 14 ; R) × GL(n 23 ; R) × GL(n 24 ; R) by the diagonal inmersion in GL(n; R):                  A 0 0 0 0 B 0 0 0 0 C 0 0 0 0 D       : A ∈ GL(n 13 ; R),B ∈ GL(n 14 ; R), C ∈ GL(n 23 ; R),D ∈ GL(n 24 ; R),            . We denote by ∆(n 13 ,n 14 ,n 23 ,n 24 ; R) to this Lie group, whose Lie algebra, ∆ ∗ (n 13 ,n 14 ,n 23 ,n 24 ; R), is                  A 0 0 0 0 B 0 0 0 0 C 0 0 0 0 D       : A ∈ GL(n 13 ; R),B ∈ GL(n 14 ; R), C ∈ GL(n 23 ; R),D ∈ GL(n 24 ; R),            . The first prolongation of ∆ ∗ (n 13 ,n 14 ,n 23 ,n 24 ; R) not vanishes. For example, if n 13 = n 14 =2,n 23 = n 24 = 0, the homomorphism defined by T : R 4 → ∆ ∗ (2, 2, 0, 0; R) e 1 →       0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0       , e 2 →       1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0       , e 3 →       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       , e 4 →       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       , is a non null element of ∆ ∗ (2, 2, 0, 0; R) (1) . Therefore, the Lie group ∆(n 13 ,n 14 ,n 23 ,n 24 ; R) does not admit functorial connections. The previous results can us obtain the following local expression for the three almost product structures: F =       I n 13 O O O O I n 14 O O O O −I n 23 O O O O −I n 24       ,P =       I n 13 O O O O −I n 14 O O O O I n 23 O O O O −I n 24       . which provide elemental examples in R 4 . ✎ ✍ ☞ ✌ Functorial connections The vanishing g (1) provides an obstruction to the existence of functorial con- nections attached to G. Theorem ([6]) If the Lie algebra g of G is invariant under transpositions and g (1) = 0 then G admits functorial connections. In these conditions, for every G-structure P → M , there exists a unique connec- tion ∇ adapted to the G-structure such that Trace (S ◦ i X ◦ Tor ∇ )=0, ∀S ∈ Γ(ad P ), ∀X ∈ X(M ). This connection is called the well-adapted connection of the G-structure and allow construct a functorial connection attached to G. ✎ ✍ ☞ ✌ Almost bicomplex structures A manifold M has an almost bicomplex structure if it is endowed with two tensor fields F , P satisfying F 2 = P 2 = I, P ◦ F − F ◦ P = O. In this case, J 2 = I . Then the manifold M has got two distributions defined by the almost product structure J : V 1 = T + J (M ), V 2 = T − J (M ). One can prove that J p (V 1 (p)) = V 1 (p),J p (V 2 (p)) = V 2 (p), for every p ∈ M , then, the two distributions are even-dimensional (not neces- sarily equals). Therefore M is also even-dimensional: dim M = 2(r + s), dim V 1 =2r, dim V 2 =2s. An almost bicomplex structure on a manifold M defines the following subfibre of F (M ): BC =  p∈M      (X 1 ,...X r ,PX 1 ,...PX r ,Y 1 ,...,Y s ,PY 1 ,...,PY s ) {X 1 ,...X r ,PX 1 ,...PX r } basis of V 1 (p), {Y 1 ,...,Y s ,PY 1 ,...,PY s } basis of V 2 (p),      , which has the following structural group ∆(r, s; C)=  A 0 0 B  : A ∈ GL(r ; C),B ∈ GL(s; C)  . The Lie algebra of this group, ∆ ∗ (r, s; C), is a subalgebra of gl(r + s; C) whose real representation is the Lie subalgebra of gl(2(r + s); R)                  A 1 A 2 0 0 −A 2 A 1 0 0 0 0 B 1 B 2 0 0 −B 2 B 1       :A 1 ,A 2 ∈ gl(2r ; R),B 1 ,B 2 ∈ gl(2s; R)            . The first prolongation of ∆ ∗ (r, s; C) does not vanish. For example, if r = s = 2 the homomorphism defined by T : R 4 → ∆ ∗ (2, 2; C) e 1 →       1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0       , e 2 →       0 −1 0 0 1 0 0 0 0 0 0 0 0 0 0 0       , e 3 →       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       , e 4 →       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       , is an non null element of ∆(r, s; C) (1) . Therefore, the Lie group ∆(r, s; C) does not admit functorial connections. This triple structure appears naturally on the product of manifolds M and N , M × N , endowed with an almost complex structure J M and J N respectively. We prove that the tensor fields defined by F (X, O)=(−J M (X ),O), P (X, O)=(J M (X ),O), ∀X ∈ X(M ), F (O, Y )=(O, J N (Y )), P (O, Y )=(O, J N (Y )), ∀Y ∈ X(N ). In this case, J = P ◦ F is the canonical almost product structure of M × N . Recently, this structures appeared in the study of bi- Hamiltonian systems by means of Hermitian structures (see [5]). ✎ ✍ ☞ ✌ Adapted metrics to a triple structure Let M be a manifold endowed with a triple structure. We can define metrics structures adapted to the triple structure (F, P, J ), according to the relations g (FX,FY )= ±g (X, Y ); g (PX,PY )= ±g (X, Y ), for all vector fields X, Y on M . For example, in [8], we study the case of the adapted metrics to an almost biparacomplex structure. Then g is an isometry or anti-isometry for the three tensor fields F , P and J . A triple metric manifold admits a functorial connections. In these cases, a functorial connection is a distinguished connection on M adapted to the triple structure and parallelizing the metric g . This connection is better adapted to the geometry of the manifold than the Levi-Civita connection of the metric. A particular case of this situation was studied in [4]: a para-Hermitian manifold (M,F,g ) is a case of a metric adapted to the almost hyperproduct structure (F, I ). ✎ ✍ ☞ ✌ Almost hypercomplex structures A manifold M has an almost hypercomplex structure if it is endowed with two tensor fields F , P satisfying F 2 = P 2 = I, P ◦ F + F ◦ P = O. In this case, J 2 = −I . Then dim M =4n, n ∈ N. An almost hipercomplex structure on a manifold M defines the following subfibre of F (M ): HC =  p∈M  (X 1 ,FX 1 ,PX 1 ,JX 1 ,...X n ,FX n ,PX n ,JX n ) {X 1 ,...X n } linearly independent in T p (M )  , whose structural group is the linear general group of order n over the quaternionic numbers, GL(n; H). This group can be considered as subgroup of GL(4n; R) by mean of the real representation GL(n; H) → GL(4n; R) A + iB + jC + kD →       A −B −C −D B A −D C C D A −B D −C B A       with i, j, k are the quaternionic imaginary unities and A,B,C,D ∈ GL(n; R). The Lie algebra of the group GL(n; H), is gl(n; H)=                  A −B −C −D B A −D C C D A −B D −C B A       : A,B,C,D ∈ gl(n; R)            , One has that gl(n; H) (1) = 0 and it is invariant under matrix trans- position, then the Lie group admits functorial connections. Alek- seevsky and Marchiafava attached to one GL(n; H)-structure one D-connection (see [1]), which is a particular case of functorial con- nection with a condition over the torsion tensor, whose expression is the following: ∇ H X Y = 1 12    (α,β,γ ) J α ([J β X, J γ Y ]+[J β Y,J γ X ])   + 1 12  2  α J α ([J α X, Y ]+[J α Y,X ])  + 1 12  − 1 3  α [J α X, J α Y ]+ 1 2 [X, Y ]  , ∀X, Y ∈ X(M ), with J 1 = F, J 2 = P, J 3 = J . This connection is called the Obata connection of the almost hypercomplex structure. This connection allows characterize the integrability of the almost hypercomplex structure and the GL(n; H)-structure associated. One has: • One almost hypercomplex structure is integrable if and only if the torsion tensor of the Obata connection vanishes. • One GL(n; H)-structure is integrable if and only if the Obata connection is locally flat; i.e., Tor ∇ H =0, R ∇ H = 0. A lot of examples, such as K3-surfaces and K¨ahlerian complex 2-dimensional torus, can be found, e.g., in [2] and [7]. ✎ ✍ ☞ ✌ References [1] D. V. Alekseevsky, S. Marchiafava: Quaternionic-like structures on a manifold. I and II. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 4 (1993), no. 1, 43–52, 53–61. [2] A. L. Besse: Einstein manifolds. Springer-Verlag, Berlin, (1987). [3] V. Cruceanu: Almost Hyperproduct structures on manifolds. An. S ¸tiint ¸. Univ. Al. I. Cuza Ia¸ si. Mat. (N. S.) 48 (2002), no. 2, 337–354. [4] F. Etayo, R. Santamar´ ıa: Functorial connections on almost para-Hermitian manifold. Proceed. Fourth Int. Workshop on Differential Geometry, Brasov (Romania), (1999), 243-256. [5] G. Marmo, A. 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