WSGP 23. Lenka Lakomá; Marek Jukl The decomposition of tensor spaces with almost complex structure

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1 WSGP 23 Lenka Lakomá; Marek Jukl The decomposition of tensor spaces with almost complex structure In: Jan Slovák and Martin Čadek (eds.): Proceedings of the 23rd Winter School "Geometry and Physics". Circolo Matematico di Palermo, Palermo, Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 72. pp. [145] Persistent URL: Terms of use: Circolo Matematico di Palermo, 2004 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 72 (2004), pp THE DECOMPOSITION OF TENSOR SPACES WITH ALMOST COMPLEX STRUCTURE LENKA LAKOMA, MAREK JUKL* ABSTRACT. Decomposition of tensor spaces with almost complex structures is a standard task in representation theory and thus in differential geometry. Our aim is to deduce explicit formulae by an elementary and straightforward approach. This decomposition is computed for tensors of the type (1,3) with symmetries of certain curvature tensors, providing an illustration of the general method on this well known example. 1. INTRODUCTION Let E be a real n-dimensional vector space and E? the tensor space of tensors of the type (p,<l). A fixed basis of E determines a unique basis of E%. The components of any tensor A with respect to this basis will be denoted AV "'V. Now let Fj be an arbitrary tensor of the type (1,1) such that F = 0. A tensor A E? is called F-traceless if the following conditions hold Vk )' Vr 1... n- pa A-ik-ifiik+V" _ n A~-ik-iaik+i'~ _ n The following theorem was proved in [5] and it shows F-decomposition for e-structures. e-structures are structures where the condition F^F* = ecjj, e = ±1 is fulfilled. Theorem 1. Let A be a tensor of the type (p, q). Ifn>2(p unique decomposition of A in the form + q) then there exists a W A j\h-j q = B j\hx + min{p,q} i i m - 2^ L Qi xw") x B 2000 Mathematics Subject Classification: 53C15, 53D15. Key words and phrases: decomposition of tensor spaces, almost complex structure, F-decomposition. * Supported by grant No. 201/02/0616 of the Grant Agency of Czech Republic. The paper is in final form and no version of it will be submitted elsewhere.

3 146 LENKA LAKOMÁ - MAREK JUKL where * l p\ l p\'" l Pt {TI}- {TS}- /~\ TI */>1 T-l *i Q P ''- p *«<zk {0} {!}.- 3<y\3tT2'"3<rt" r r Ío x 3c 2 ~F>; F) = S^ ғ) = Fj = Pl,P2,'" 5Pt^{l,2,---,p} (/>i <0j<---< A), 0i, 0*2, *, o~ t E {1,2,,q} (a,- are mutually different) ri,r ,r t E {0,1} Pl,P2," *,P- * = {ri,т 2,-",rj, o= < <7l,0"2," *,<7ť ri,r 2,-- -r* 2. DECOMPOSITION OF TENSORS OF THE TYPE (1,3). In this section we will compute F- decompositon of tensors of the type (1,3) for e-structures with e = 1. This structure is called almost complex structure. It was proved in [4] that we get 48 algebraic equations in 48 unknowns in generally and this system is not easy solvable. Therefore in [4] the contents of the theorem 1 was extended. We compute the decomposition of tensors of the type (1,3) which have following properties. (2) (a) 4, + A? kj = 0; (b) 4* + 4«+ A h kij = 0; (c)a ajk = 0; (d).4*- =.4* t. We will use next notation. (3) A;;,... = F?A:: a..., A? = F a A:: - It follows from this notation A'" = = -A":. If we denote (4) we can also deduce following properties (5) 4-. = Å a AІJ AJІ. For example the Riemannian tensors of Kahlerian space, K-space, CR-space have these properties ([6], [7]). We have the next theorem. Theorem 2. Let A be a tensor of the type (1,3) with properties (2), (5). Let F be almost complex structure. If n > 4 then there exists unique F-decomposition of the tensor A in the form (6) 4, = B? jk + 6?C sk + 6$ D* + SlEij + FfG jk + FfH* + íf/y,

4 THE DECOMPOSITION OF TENSOR SPACES WITH ALMOST COMPLEX STRUCTURE 147 where tensors B h j k,cjk>djk,ejk,gjk,hjkiijk have following form C jk =0; D jk - n+2 Л J*> (7) Ejk ~~ n+2 Л J fc > Gjk 2_A - Hjk - A - ~" n+2 ;7'A;> 1* - ---A - "" n+2 ;?'*;> 4* = A m + -& (*:M* - *M«+ 2 i?%+ F i A n - ~M«j) and the tensor B is F-traceless. The aim of the following text is to prove the main Theorem 2. We will suppose, that the tensor A can be expressed in a form (6) where B h jk is F-traceless tensor and Cjk, Dij, Eij, Gjk, H ik, Iij are certain tensors. We will suppose that the tensor B h has jk algebraic properties analogous to algebraic properties of the tensor A h, i.e. jk (8) B h jk + Bh kj = 0; Bh jk + Bh ki + Bh kij = 0; Bh Tk = Bh k. Let us alternate the expression (6) in j, k. Using we can write (9) 4* + ^ = ; 4* + I4; = 0 6 h (C ik + Ckj) + F, h (G jk + G kj ) + S) (D ik + Ei k ) + 6 h (D tj + E {j ) + Ff (^ + Iik) + Ft (H tj + 7 y ) = 0 Suppose that Cjk + Ckj 7-= 0. Then there exists a tensor e- 7 such that Contracting (9) by e j e k, we obtain e j e k (Cjk + C k j) = ±l. (10) ±6$ + a8± + e h Q { +e h Q { = 0, 1 2 where Q { = 2e a (D ia + E ia ) and Q { = 2e a (H ia + I ia ). After contraction (10) by i we have (11) ±5% - a# + e* 4 +c* 4= 0 Let's substitute S h from (11) in (10) then we get the following condition (12) ±6 h (1 + a 2 ) + e h Ua Qj + 4) + ^ (+* Qj + 4) = 0 Since Rank u^ < 2 it contradicts the assumption n > 2. We have following lemma.

5 148 LENKA LAKOMA - MAREK JUKL Lemma 1. The condition (13) Cjk + C kj = 0 holds for coefficients Cj k. We can use the previous arguments for the coefficients Gj k and we get Lemma 2. The condition (14) G jk + G kj = 0 holds for coefficients Cj k. The equation (9) now has a form (15) 6$ (D ik + E ik ) + Si (D {j + Ed + Ff (H ik + I ik ) + F* (H {j + I i5 ) = 0. Suppose that D ik + E ik ^ 0. Similarly to the previous cases we get the existence of a bivector e % n k such that e i rj k (D ik + E ik ) = \. When we contract (15) by e x n k we obtain the equation (16) ^ + o^ + ij fc Q i +^4= > 1 2 where Q { = e" (D aj + E aj ) and Q { = e a (H aj + I aj ). Contracting (16) we express <$5, then we replace it in equation (16): (17) i) (1 + a 2 ) + T?" (4 -a Q^J + r, h (Q 5 -a %) = 0. The equation (17) has no solution for n > 2. Lemma 3. The condition (18) D jk + E kj = 0. holds for coefficients Dj k, Ej k. In a similar way we obtain Lemma 4. The condition (19) H jk + I kj = 0. holds for coefficients Hj k, Ij k. When we apply lemmas to the Theorem 2 we have Lemma 5. When the condition (2(a)) is fulfilled then for n > 2 the tensor A may be expressed in a form (20) 4* = B$ k + 6 h C jk + 6$D ik - 6 h D {j + F, h G jk + F?H ik - F h H {j, where Gj k + C k j = 0, Gj k + Gkj = 0.

6 THE DECOMPOSITION OF TENSOR SPACES WITH ALMOST COMPLEX STRUCTURE 149 Using properties A^jk + A h jki + A h kij = 0; B k + B$ ki + B^ = 0 we get the equation (21) S h n jk + Sfiki + SliUj + F h U jk + FfU ki + FilUj = 0, where fyk = Cjifc - I^jifc + Dkj; n# = Gjfjfe - Hjjfc + Hfcj. But Sljk = 0; Cljk = 0 for n > 4, i.e. (22) Cjk = Djk - D k j ; Gjk = H jk - H k j. Let us replace Cj* and Gjk in (20) by (22). Then we get Lemma 6. If conditions (2 (a), (b)) are fulfilled then forn>a the tensor A may be expressed in a form A h jk = B h jk + 5 h (D jk - D kj ) + 5$D ik - 5 h D tj (23) + I 7. (IIj* - II*i) + I} II«* - I 7 * IIy The condition A 1 = A ijk gives,5? ( 7 ^ - / ^ - % + %) (24) + 6 h (D ik + H ik )-6 h {D ij + H il ) + IT (IIj* - II*i - H J + H kj) Equation (24) implies + Ff (H ik - D ik ) - F h (H {j - D i3 ) = 0. (25) H ik = D ik ; Dj k -D k j = D Tk -D rj. Using conditions A" jk = B jk = 0 and conditions (25) in the equation (23) we have after contraction by 6 (26) (n + l)(o j *-o* i ) + %-o jej = 0. It follows from (26) (27) D jk = D kj. Substitute (27) to (22). We obtain (28) C jk = 0; G jk = 2D jk. We can rewrite the equation (23) in a form (29) 4, = B h jk + 6 h 5D ik - ftdii + 2F i D fk + F h D ik - F h D rj. Contract (29) by 6fr then (30) y. y = -(n + 2)Ai and therefore (3D ^ = -nt2^-

7 150 LENKA LAKOMA - MAREK IUKL Substituting (18) to (19), (25), (28), (31) we get coefficients C jk, D jk, E jk, H jki I jk in the form mentioned in the Theorem 2. Now the tensor B^k has a form (32) B* k = 4* + --i-- (6$ A* - SlAq + 2F?A fk + F*A ik - I^AQ). All computed tensors are F-traceless and the proof is complete. When Ai jk is the Riemannian tensor then B^jk (32) is well known tensor of the holomorphically-proj ective curvature. Allow us to express our thanks to Prof. Mikes for his advices and ideas. REFERENCES [1] Krupka, D., The 7Vnce Decomposition ProЫem, Beitгäge zuг Algebгa und Geometгie 36, N. 2 (1995), [2] Lakomá, L., Projecłions of Tensor Spaces, Acta Univ. Palacki. Olomuc, Fac. Reг. Nat., Math. 38 (1999), [3] Lakomá, L., Mikeš, J., On the special trace decomposition problem on quaternionic structure, Pгoc. of the Тhiгd International Workshop oh Differential Geometгy and its Applications; Тhe fiгst Geгman-Romanian seminaг on geometry; Sibiu / Romania, Septembeг 18-23,1997, [4] Lakomá, L., Mikeš, J., Mikušová, L., The Decomposition of Tensor Spaces, Diffeгential Geometry and Applications; Satelite Conf. of ICM in Beгlin, Aug , 1998, Bгno, Masaгyk Univ. Brno, Czech Rep., 1999, [5] Mikeš, J., On general trace decompositon problem, Pгoc. Conf., Aug. 28-Sept. 1,1995, Bгno, Czech Rep., Masaгyk Univ., Brno (1996), [6] Mikeš, J., Holomorphically projective mappings and their generalizations, J. Math. Sci. 89(3) (1998), [7] Yano, K., Differential geometry on complex and almost complex spaces, Oxfoгd-London-New York- Paris-Frankŕuгt: Peгgamon Pгess. XII, 1965, 323p. DEPARГMENТ oғ ALGEBRA AND GEOMEТRY, FACULТY OF SCIENCE PALACКY UNIVERSIТY ТOMКOVA 40, OLOMOUC, CZECH REPUBLIC lakomaöprfnw.upol.cz juklfiaix.upol.cz