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Òåîðèÿ Õîäæà íà ìíîãîîáðàçèÿõ Ãðýÿ Ì. Âåðáèöêèé, 27 îêòÿáðÿ 2005 Øàã 3: Î÷åâèäíî, {{D ∗ , L a }, L b } ∗ = D1, ∗ ãäå D 1 := {{D, Λ a }, Λ b }. Èç Óòâåðæäåíèÿ 6.6 è (6.2) ñëåäóåò, ÷òî D 1 àëãåáðàè÷åñêèé äèôôåðåíöèàëüíûé îïåðàòîð ïåðâîãî ïîðÿäêà (ýòî êîììóòàòîð íåñêîëüêèõ äèôôåðåíöèàëüíûõ îïåðàòîðîâ ïåðâîãî ïîðÿäêà). Åñëè a, b ∈ Λ 1 (M), òî D 1 ïîíèæàåò ñòåïåíü íà 1. Â ñèëó Óòâåðæäåíèÿ 6.3, D 1 äèôôåðåíöèðîâàíèå. ßñíî, ÷òî äèôôåðåíöèðîâàíèå, êîòîðîå çàíóëÿåòñÿ íà Λ 0 (M), C ∞ (M)-ëèíåéíî. Ïîýòîìó â ýòîé ñèòóàöèè D 1 C ∞ (M)-ëèíåéíî. Èç Ëåììû 6.9 âûòåêàåò, ÷òî êîììóòàòîð D 1 ñ Λ c çàíóëÿåòñÿ, äëÿ âñåõ c ∈ Λ 1 (M): {D 1 , Λ c } = 0 (6.5) Îïåðàòîð D1 ∗ = {{D ∗ , L a }, L b } C ∞ (M)-ëèíåéíûé (áóäó÷è ñîïðÿæåííûì ê D 1 ), è êîììóòèðóåò ñ ëþáûì L a , êàê ñëåäóåò èç (6.5). Ïîýòîìó D1 ∗ Λ ∗ (M)-ëèíåéíûé. Ýòî äîêàçûâàåò (6.4) äëÿ a, b ∈ Λ 1 (M). Øàã 4: Î÷åâèäíî, L a = Λ a , åñëè a ∈ Λ 0 (M). Òîãäà {{D ∗ , L a }, L b } = {{D ∗ , Λ a }, Λ b } = {{D, L a }, L b } ∗ = 0, (6.6) ïîñêîëüêó D äèôôåðåíöèàëüíûé îïåðàòîð ïåðâîãî ïîðÿäêà. Ìû äîêàçàëè (6.4) äëÿ a, b ∈ Λ 0 (M). Øàã 5: Ïîñêîëüêó àëãåáðà Λ ∗ (M) ñóïåðêîììóòàòèâíà, {L a , L b } = 0 äëÿ âñåõ a, b ∈ Λ ∗ (M). Èñïîëüçóÿ ñóïåðêîììóòàòèâíóþ âåðñèþ ñîîòíîøåíèÿ ßêîáè, ìû ïîëó÷àåì {{D ∗ , L a }, L b } = (−1)ã˜b{{D ∗ , L b }, L a }, (6.7) äëÿ âñåõ a, b. Íà øàãå 3 è 4 ìû äîêàçàëè (6.4) äëÿ âñåõ a, b ∈ Λ 1 (M), a, b ∈ Λ 0 (M). Â ñèëó (6.7), ÷òîáû äîêàçàòü 6.8 îñòàåòñÿ óáåäèòüñÿ, ÷òî îïåðàòîð {{D ∗ , L a }, L b } Λ ∗ (M)-ëèíååí äëÿ a ∈ Λ 0 (M), b ∈ Λ 1 (M). Øàã 6: Â ýòîì ñëó÷àå, {D ∗ , L a } = {D ∗ , Λ a } = {D, L a } ∗ = L ∗ D(a) = Λ D(a). Îáîçíà÷èì çà g òåíçîð ðèìàíîâîé ìåòðèêè íà M. Òîãäà {{D ∗ , L a }, L b } = {Λ D(a) , L b } = g(D(a), b), ïîñêîëüêó Λ D(a) ïîäñòàíîâêà äâîéñòâåííîãî âåêòîðíîãî ïîëÿ D(a) ♯ . Ïîýòîìó {{D ∗ , L a }, L b } ñêàëÿðíàÿ ôóíêöèÿ, à ñëåäîâàòåëüíî, ýòîò îïåðàòîð Λ ∗ (M)- ëèíåéíûé. Ïðåäëîæåíèå 6.8 äîêàçàíî. Áëàãîäàðíîñòè: Àâòîð ïðèçíàòåëåí Ðîáåðòó Áðàéàíòó è Ïîëó-Àíäè Íàäþ çà ïîó÷èòåëüíóþ ïåðåïèñêó. Ï.-À. Íàäþ ïðèíàäëåæèò èäåÿ äîáàâèòü â òåêñò ñòàòüè Çàìå÷àíèå 5.4. 20
Òåîðèÿ Õîäæà íà ìíîãîîáðàçèÿõ Ãðýÿ Ì. Âåðáèöêèé, 27 îêòÿáðÿ 2005 Ñïèñîê ëèòåðàòóðû [AFHS] BS Acharya, JM Figueroa-O'Farrill, CM Hull, B Spence, Branes at conical singularities and holography, hep-th/9808014, Adv.Theor.Math.Phys. 2 (1999) 1249-1286 [AcW] Bobby Acharya, Edward Witten, Chiral Fermions from Manifolds of G 2 Holonomy, hep-th/0109152 [AtW] Michael Atiyah, Edward Witten, M-Theory Dynamics On A Manifold Of G 2 Holonomy, Adv. Theor. Math. Phys. 6 (2003) 1-106, also hep-th/0107177. [BFGK] H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistor and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik, 124, B. G. Teubner, Stuttgart, 1991. [CS] Cleyton, R., Swann, A., Einstein Metrics via Intrinsic or Parallel Torsion, math.DG/0211446, also in: Math. Z. 247 (2004), no. 3, 513-528. [ES] J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), 589-640. [F] Thomas Friedrich, Nearly K¤ahler and nearly parallel G 2-structures on spheres, 2 pages, math.DG/0509146. [FI] Friedrich, Thomas; Ivanov, Stefan, Parallel spinors and connections with skewsymmetric torsion in string theory, math.DG/0102142, Asian J. Math. 6 (2002), no. 2, 303-335. [Gr1] Gray, Alfred, Minimal varieties and almost Hermitian submanifolds, Michigan Math. J. 12 (1965), 273-287. [Gr2] Gray, Alfred, Nearly K¤ahler manifolds, J. Dierential Geometry 4 (1970), 283- 309. [Gr3] Gray, Alfred, Weak holonomy groups, Math. Z. 123 (1971), 290-300. [Gr4] Gray, Alfred, The structure of nearly Kahler manifolds, Math. Ann. 223 (1976), 233-248 [GH] Griths, Ph., Harris, J., Principles of Algebraic Geometry, New York, 1978. [Gru] R. Grunewald, Six-Dimensional Riemannian manifolds with real Killing spinors, Ann. Global Anal. Geom. 8 (1990), 43-59. [K] V. F. Kirichenko, K-spaces of maximal rank, Mat. Zametki 22 (1977), 465-476 [MNS1] Moroianu, Andrei; Nagy, Paul-Andi; Semmelmann, Uwe, Unit Killing vector elds on nearly K¤ahler manifolds, math.DG/0406492, also in Internat. J. Math. 16 (2005), no. 3, 281-301. [MNS2] Moroianu, Andrei; Nagy, Paul-Andi; Semmelmann, Uwe, Deformations of Nearly K¤ahler Structures, 15 pages, math.DG/0611223 [N] Paul-Andi Nagy, Nearly K¤ahler geometry and Riemannian foliations, Asian J. Math. 3 vol. 6, (2002), 481-504, math.DG/0203038. [V1] [V2] Verbitsky, M., Hyperk¤ahler manifolds with torsion, supersymmetry and Hodge theory, math.AG/0112215, 47 pages (Asian J. of Math., Vol. 6 (4), December 2002). M. Verbitsky, Vanishing theorems for locally conformal hyperk¤ahler manifolds, 2003, math.DG/0302219, also in Proc. of Steklov Institute, 246 (2004), 54-79. 21
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