The Srní lectures on non-integrable geometries with torsion

Ilka Agricola

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 5, page 5-84
  • ISSN: 0044-8753

Abstract

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This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a G -structure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory. They also provide the link to Kostant’s cubic Dirac operator.

How to cite

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Agricola, Ilka. "The Srní lectures on non-integrable geometries with torsion." Archivum Mathematicum 042.5 (2006): 5-84. <http://eudml.org/doc/18858>.

@article{Agricola2006,
abstract = {This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a $G$-structure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory. They also provide the link to Kostant’s cubic Dirac operator.},
author = {Agricola, Ilka},
journal = {Archivum Mathematicum},
keywords = {metric connection with torsion; intrinsic torsion; $G$-structure; characteristic connection; superstring theory; Strominger model; parallel spinor; non-integrable geometry; integrable geometry; Berger’s holonomy theorem; naturally reductive space; hyper-Kähler manifold with torsion; almost metric contact structure; $G_2$-manifold; $(7)$-manifold; $(3)$-structure; $3$-Sasakian manifold; metric connection with torsion; intrinsic torsion; -structure; characteristic connection; superstring theory},
language = {eng},
number = {5},
pages = {5-84},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The Srní lectures on non-integrable geometries with torsion},
url = {http://eudml.org/doc/18858},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Agricola, Ilka
TI - The Srní lectures on non-integrable geometries with torsion
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 5
SP - 5
EP - 84
AB - This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a $G$-structure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory. They also provide the link to Kostant’s cubic Dirac operator.
LA - eng
KW - metric connection with torsion; intrinsic torsion; $G$-structure; characteristic connection; superstring theory; Strominger model; parallel spinor; non-integrable geometry; integrable geometry; Berger’s holonomy theorem; naturally reductive space; hyper-Kähler manifold with torsion; almost metric contact structure; $G_2$-manifold; $(7)$-manifold; $(3)$-structure; $3$-Sasakian manifold; metric connection with torsion; intrinsic torsion; -structure; characteristic connection; superstring theory
UR - http://eudml.org/doc/18858
ER -

References

top
  1. E. Abbena, An example of an almost Kähler manifold which is not Kählerian, Bolletino U. M. I. (6) 3 A (1984), 383–392. (1984) Zbl0559.53023MR0769169
  2. E. Abbena S. Gabiero S. Salamon, Almost Hermitian geometry on six dimensional nilmanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXX (2001), 147–170. MR1882028
  3. I. Agricola, Connexions sur les espaces homogènes naturellement réductifs et leurs opérateurs de Dirac, C. R. Acad. Sci. Paris Sér. I 335 (2002), 43–46. Zbl1010.53024MR1920993
  4. I. Agricola, Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. Math. Phys. 232 (2003), 535–563. Zbl1032.53041MR1952476
  5. I. Agricola S. Chiossi A. Fino, Solvmanifolds with integrable and non-integrable G 2 -structures, math.DG/0510300, to appear in Differential Geom. Appl. MR2311729
  6. I. Agricola, Th. Friedrich, Global Analysis – Differential forms in Calculus, Geometry and Physics, Graduate Studies in Mathematics, Publications of the AMS 2002, Providence, Rhode Island 2002. MR1998826
  7. I. Agricola, Th. Friedrich, Killing spinors in supergravity with 4 -fluxes, Classical Quantum Gravity 20 (2003), 4707–4717. Zbl1045.83045MR2019441
  8. I. Agricola, Th. Friedrich, On the holonomy of connections with skew-symmetric torsion, Math. Ann. 328 (2004), 711–748. Zbl1055.53031MR2047649
  9. I. Agricola, Th. Friedrich, The Casimir operator of a metric connection with totally skew-symmetric torsion, J. Geom. Phys. 50 (2004), 188–204. MR2078225
  10. I. Agricola, Th. Friedrich, Geometric structures of vectorial type, math.DG/0509147, to appear in J. Geom. Phys. Zbl1106.53033MR2252869
  11. I. Agricola T. Friedrich P.-A. Nagy C. Puhle, On the Ricci tensor in the common sector of type II string theory, Classical Quantum Gravity 22 (2005), 2569–2577. MR2153698
  12. I. Agricola, Chr. Thier, The geodesics of metric connections with vectorial torsion, Ann. Global Anal. Geom. 26 (2004), 321–332. Zbl1130.53029MR2103403
  13. D. V. Alekseevski, Riemannian spaces with exceptional holonomy groups, Func. Anal. Prilozh. 2 (1968), 1–10. (1968) MR0231313
  14. D. V. Alekseevsky S. Marchiafava M. Pontecorvo, Compatible almost complex structures on quaternion Kähler manifolds, Ann. Global Anal. Geom. 16 (1998), 419–444. (1998) MR1648844
  15. D. V. Alekseevsky V. Cortés, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type, Vinberg, Ernest (ed.), Lie groups and invariant theory. Providence, RI: American Mathematical Society 213 (AMS). Translations. Series 2. Adv. Math. Sci. 56 (2005), 33–62. MR2140713
  16. V. Aleksiev G. Ganchev, On the classification of the almost contact metric manifolds, Mathematics and education in mathematics, Proc. 15th Spring Conf., Sunny Beach/Bulg. 1986, 155–161. (1986) MR0872914
  17. B. Alexandrov, S p ( n ) U ( 1 ) -connections with parallel totally skew-symmetric torsion, J. Geom. Phys. 57 (2006), 323–337, math.DG/0311248. Zbl1107.53012MR2265474
  18. B. Alexandrov, On weak holonomy, Math. Scand. 96 (2005), 169–189. Zbl1079.53071MR2153409
  19. B. Alexandrov, Th. Friedrich N. Schoemann, Almost Hermitian 6 -manifolds revisited, J. Geom. Phys. 53 (2005), 1–30. MR2102047
  20. B. Alexandrov S. Ivanov, Dirac operators on Hermitian spin surfaces, Ann. Global Anal. Geom. 18 (2000), 529–539. MR1800590
  21. T. Ali, M -theory on seven manifolds with G -fluxes, hep-th/0111220. 
  22. W. Ambrose, I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443. (1953) Zbl0052.18002MR0063739
  23. W. Ambrose, I. M. Singer, On homogeneous Riemannian manifolds, Duke Math. J. 25 (1958), 647–669. (1958) Zbl0134.17802MR0102842
  24. V. Apostolov T. Drăghici A. Moroianu, A splitting theorem for Kähler manifolds whose Ricci tensors have constant eigenvalues, Internat. J. Math. 12 (2001), 769–789. MR1850671
  25. V. Apostolov J. Armstrong T. Drăghici, Local rigidity of certain classes of almost Kähler 4 -manifolds, Ann. Global Anal. Geom. 21 (2002), 151–176. MR1894944
  26. J. Armstrong, Almost Kähler geometry, Ph. D. Thesis, Oxford University, 1998. (1998) 
  27. M. Atiyah, W. Schmid, A geometric construction for the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. (1977) MR0463358
  28. M. Atiyah, E. Witten, M -theory dynamics on a manifold of G 2 holonomy, Adv. Theor. Math. Phys. 6 (2002), 1–106. Zbl1033.81065MR1992874
  29. J. E. D’Atri, Geodesic spheres and symmetries in naturally reductive spaces, Michigan Math. J. 22 (1975), 71–76. (1975) MR0372786
  30. J. E. D’Atri, W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979). (1979) MR0519928
  31. Chr. Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509–521. (1993) MR1224089
  32. B. Banos A. F. Swann, Potentials for hyper-Kähler metrics with torsion, Classical Quantum Gravity 21 (2004), 3127–3135. MR2072130
  33. H. Baum, Th. Friedrich R. Grunewald I. Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik, Band 124, Teubner-Verlag Stuttgart/Leipzig, 1991. (1991) MR1164864
  34. K. Behrndt C. Jeschek, Fluxes in M -theory on 7 -manifolds and G -structures, hep-th/0302047. 
  35. F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40. Zbl0988.32017MR1760667
  36. F. A. Belgun, A. Moroianu, Nearly Kähler 6 -manifolds with reduced holonomy, Ann. Global Anal. Geom. 19 (2001), 307–319. Zbl0992.53037MR1842572
  37. M. Berger, Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279–330. (1955) MR0079806
  38. M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Sc. Norm. Sup. Pisa 15 (1961), 179–246. (1961) Zbl0101.14201MR0133083
  39. J. Berndt F. Tricerri L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, LNM 1598, Springer, 1995. (1995) MR1340192
  40. A. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete Bd. 10, Springer-Verlag Berlin-Heidelberg 1987. (1987) Zbl0613.53001MR0867684
  41. A. Bilal J.-P. Derendinger, K. Sfetsos, Weak G 2 -holonomy from self-duality, flux and supersymmetry, Nuclear Phys. B 628 (2002), 112–132. MR1901225
  42. J. M. Bismut, A local index theorem for non-Kählerian manifolds, Math. Ann. 284 (1989), 681–699. (1989) MR1006380
  43. D. E. Blair, Contact manifolds in Riemannian geometry, LNM 509 (1976), Springer. (1976) Zbl0319.53026MR0467588
  44. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics vol. 203, Birkhäuser, 2002. Zbl1011.53001MR1874240
  45. D. E. Blair, L. Vanhecke, New characterization of ϕ -symmetric spaces, Kodai Math. J. 10 (1987), 102–107. (1987) MR0879387
  46. M. Bobieński, The topological obstructions to the existence of an irreducible SO ( 3 ) -structure on a five manifold, math.DG/0601066. 
  47. M. Bobieński, P. Nurowski, Irreducible SO ( 3 ) -geometries in dimension five, to appear in J. Reine Angew. Math.; math.DG/0507152. MR2338127
  48. E. Bonan, Sur les variétés riemanniennes à groupe d’holonomie G 2 ou Spin ( 7 ) , C. R. Acad. Sc. Paris 262 (1966), 127–129. (1966) MR0196668
  49. C. P. Boyer, K. Galicki, 3 -Sasakian manifolds, in Essays on Einstein manifolds, (ed. by C. LeBrun and M. Wang), International Press 1999. (1999) Zbl1008.53047MR1798609
  50. C. P. Boyer, K. Galicki, Einstein manifolds and contact geometry, Proc. Amer. Math. Soc. 129 (2001), 2419–2430. Zbl0981.53027MR1823927
  51. C. P. Boyer, K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, to appear 2007. Zbl1155.53002MR2382957
  52. C. P. Boyer K. Galicki B. M. Mann, The geometry and topology of 3 -Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183–220. (1994) MR1293878
  53. L. Brink P. Ramond, X. Xiong, Supersymmetry and Euler multiplets, hep-th/0207253. 
  54. R. B. Brown, A. Gray, Riemannian manifolds with holonomy group Spin ( 7 ) , Differential Geometry in honor of K. Yano, Kinokiniya, Tokyo, 1972, 41–59. (1972) MR0328817
  55. R. L. Bryant, Metrics with exceptional holonomy, Ann. of Math. 126 (1987), 525–576. (1987) Zbl0637.53042MR0916718
  56. R. L. Bryant, Classical, exceptional, and exotic holonomies: a status report, Actes de la Table ronde de Géométrie Différentielle en l’honneur de M. Berger. Collection SMF Séminaires et Congrès 1 (1996), 93–166. (1996) Zbl0882.53014MR1427757
  57. R. L. Bryant, Some remarks on G 2 -structures, in Proceeding of the 2004 Gokova Conference on Geometry and Topology (May, 2003), math.DG/0305124. Zbl1115.53018
  58. R. L. Bryant, and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), 829–850. (1989) MR1016448
  59. J. Bureš, Multisymplectic structures of degree three of product type on 6 -dimensional manifolds, Suppl. Rend. Circ. Mat. Palermo II, Ser. bf 72 (2004), 91–98. MR2069397
  60. J. Bureš, J. Vanžura, Multisymplectic forms of degree three in dimension seven, Suppl. Rend. Circ. Mat. Palermo II, Ser. 71 (2003), 73–91. MR1982435
  61. J. B. Butruille, Classification des variétés approximativement kähleriennes homogènes, Ann. Global Anal. Geom. 27 (2005), 201–225. MR2158165
  62. D. M. J. Calderbank, H. Pedersen, Einstein-Weyl geometry, Surveys in differential geometry: Essays on Einstein manifolds. Lectures on geometry and topology, J. Diff. Geom. Suppl. 6 (1999), 387–423. (1999) Zbl0996.53030MR1798617
  63. E. Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Ac. Sc. 174 (1922), 593–595. (1922) 
  64. E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. Ec. Norm. Sup. 40 (1923), 325–412, part one. (1923) MR1509253
  65. E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie, suite), Ann. Ec. Norm. Sup. 41 (1924), 1–25, part one (continuation). (1924) MR1509255
  66. E. Cartan, Les récentes généralisations de la notion d’espace, Bull. Sc. Math. 48 (1924), 294–320. (1924) 
  67. E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (deuxième partie), Ann. Ec. Norm. Sup. 42 (1925), 17–88, part two. English transl. of both parts by A. Magnon and A. Ashtekar, On manifolds with an affine connection and the theory of general relativity. Napoli: Bibliopolis (1986). (1925) MR1509263
  68. I. Chavel, A class of Riemannian homogeneous spaces, J. Differential Geom. 4 (1970), 13–20. (1970) Zbl0197.18302MR0270295
  69. D. Chinea, G. Gonzales, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl. 156 (1990), 15–36. (1990) MR1080209
  70. D. Chinea, J. C. Marrero, Classifications of almost contact metric structures, Rev. Roumaine Math. Pures Appl. 37 (1992), 581–599. (1992) MR1172273
  71. S. G. Chiossi A. Fino, Conformally parallel G 2 -structures on a class of solvmanifolds, Math. Z. 252 (2006), 825–848. MR2206629
  72. S. Chiossi, S. Salamon, The intrinsic torsion of S U ( 3 ) and G 2 structures, in O. Gil-Medrano et. al. (eds.), Proc. Intern. Conf. Valencia, Spain, July 8-14, 2001, Singapore, World Scientific, 115–133 (2002). Zbl1024.53018MR1922042
  73. R. Cleyton, S. Ivanov, On the geometry of closed G 2 -structures, math.DG/0306362. Zbl1122.53026
  74. R. Cleyton, S. Ivanov, Curvature decomposition of G 2 manifolds, to appear. 
  75. R. Cleyton, S. Ivanov, Conformal equivalence between certain geometries in dimension 6 and 7 , math.DG/0607487. 
  76. R. Cleyton, A. Swann, Cohomogeneity-one G 2 -structures, J. Geom. Phys. 44 (2002), 202–220. Zbl1025.53024MR1969782
  77. R. Cleyton, A. Swann, Einstein metrics via intrinsic or parallel torsion, Math. Z. 247 (2004), 513–528. Zbl1069.53041MR2114426
  78. G. Curio B. Körs, D. Lüst, Fluxes and branes in type II vacua and M-theory geometry with G 2 and S p i n ( 7 ) holonomy, hep-th/0111165. 
  79. P. Dalakov, S. Ivanov, Harmonic spinors of Dirac operators of connections with torsion in dimension 4 , Classical Quantum Gravity 18 (2001), 253–265. MR1807617
  80. B. de Witt H. Nicolai, N. P. Warner, The embedding of gauged n = 8 supergravity into d = 11 supergravity, Nuclear Phys. B 255 (1985), 29. (1985) MR0792244
  81. B. de Witt D. J. Smit, and N. D. Hari Dass, Residual supersymmetry of compactified D = 10 Supergravity, Nuclear Phys. B 283 (1987), 165. (1987) 
  82. D. Ž. Djoković, Classification of trivectors of an eight-dimensional real vector space, Linear and Multilinear Algebra 13 (1983), 3–39. (1983) MR0691457
  83. I. G. Dotti, A. Fino, HyperKähler torsion structures invariant by nilpotent Lie groups, Classical Quantum Gravity 19 (2002), 551–562. Zbl1001.53031MR1889760
  84. S. Dragomir L. Ornea, Locally conformal Kähler geometry, Progr. Math. vol. 155, Birkhäuser Verlag, 1998. (1998) MR1481969
  85. M. J. Duff, M -theory on manifolds of G 2 -holonomy: the first twenty years, hep-th/0201062. 
  86. M. Fernández, A classification of Riemannian manifolds with structure group Spin ( 7 ) , Ann. Mat. Pura Appl. 143 (1986), 101–122. (1986) MR0859598
  87. M. Fernández, An example of a compact calibrated manifold associated with the exceptional Lie group G 2 , J. Differential Geom. 26 (1987), 367–370. (1987) MR0906398
  88. M. Fernández, A. Gray, Riemannian manifolds with structure group G 2 , Ann. Mat. Pura Appl. 132 (1982), 19–45. (1982) MR0696037
  89. J. Figueroa-O’Farrill G. Papadopoulos, Maximally supersymmetric solutions of ten- and eleven-dimensional supergravities, hep-th/0211089. 
  90. A. Fino, Almost contact homogeneous manifolds, Riv. Mat. Univ. Parma (5) 3 (1994), 321–332. (1994) Zbl0847.53036MR1342063
  91. A. Fino, Almost contact homogeneous structures, Boll. Un. Mat. Ital. A 9 (1995), 299–311. (1995) Zbl0835.53039MR1336238
  92. A. Fino, Intrinsic torsion and weak holonomy, Math. J. Toyama Univ. 21 (1998), 1–22. (1998) Zbl0980.53060MR1684209
  93. A. Fino, Almost Kähler 4 -dimensional Lie groups with J -invariant Ricci tensor, Differential Geom. Appl. 23 (2005), 26-37. Zbl1084.53025MR2148908
  94. A. Fino, G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2004), 439–450. Zbl1114.53043MR2101226
  95. A. Fino M. Parton S. Salamon, Families of strong KT structures in six dimensons, Comment. Math. Helv. 79 (2004), 317–340. MR2059435
  96. Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980), 117–146. (1980) Zbl0462.53027MR0600828
  97. Th. Friedrich, Dirac operators in Riemannian geometry, Grad. Stud. Math. vol. 25, 2000. Zbl0949.58032MR1777332
  98. Th. Friedrich, Weak Spin ( 9 ) -structures on 16 -dimensional Riemannian manifolds, Asian Math. J. 5 (2001), 129–160. MR1868168
  99. Th. Friedrich, S p i n ( 9 ) -structures and connections with totally skew-symmetric torsion, J. Geom. Phys. 47 (2003), 197–206. Zbl1039.53049MR1991473
  100. Th. Friedrich, On types of non-integrable geometries, Rend. Circ. Mat. Palermo (2) Suppl. 71 (2003), 99–113. Zbl1079.53041MR1982437
  101. Th. Friedrich, G 2 -manifolds with parallel characteristic torsion, math.DG/0604441, to appear in Differential Geom. Appl. Zbl1141.53019MR2373939
  102. Th. Friedrich, R. Grunewald, On the first eigenvalue of the Dirac operator on 6 -dimensional manifolds, Ann. Global Anal. Geom. 3 (1985), 265–273. (1985) Zbl0577.58034MR0813132
  103. Th. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002), 303–336. Zbl1127.53304MR1928632
  104. Th. Friedrich, S. Ivanov, Almost contact manifolds, connections with torsion and parallel spinors, J. Reine Angew. Math. 559 (2003), 217–236. Zbl1035.53058MR1989651
  105. Th. Friedrich, S. Ivanov, Killing spinor equations in dimension 7 and geometry of integrable G 2 -manifolds, J. Geom. Phys. 48 (2003), 1–11. MR2006222
  106. Th. Friedrich, I. Kath, Einstein manifolds of dimension five with small eigenvalues of the Dirac operator, J. Differential Geom. 19 (1989), 263–279. (1989) MR0982174
  107. Th. Friedrich, I. Kath, 7 -dimensional compact Riemannian manifolds with Killing spinors, Comm. Math. Phys. 133 (1990), 543–561. (1990) Zbl0722.53038MR1079795
  108. Th. Friedrich I. Kath A. Moroianu, U. Semmelmann, On nearly parallel G 2 -structures, J. Geom. Phys. 3 (1997), 256–286. (1997) MR1484591
  109. Th. Friedrich, E. C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), 128–172. Zbl0961.53023MR1738150
  110. Th. Friedrich, S. Sulanke, Ein Kriterium für die formale Selbstadjungiertheit des Dirac-Operators, Coll. Math. XL (1979), 239–247. (1979) Zbl0426.58023MR0547866
  111. J.-X. Fu, S.-T. Yau, Existence of supersymmetric Hermitian metrics with torsion on non-Kähler manifolds, hep-th/0509028. 
  112. A. Fujiki, M. Pontecorvo, On Hermitian geometry of complex surfaces, in O. Kowalski et al. (ed.), Complex, contact and symmetric manifolds. In honor of L. Vanhecke. Selected lectures from the international conference “Curvature in Geometry" held in Lecce, Italy, June 11-14, 2003. Birkhäuser, Progr. Math. 234 (2005), 153–163. Zbl1085.53065MR2105147
  113. T. Fukami, S. Ishihara, Almost Hermitian structure on S 6 , Hokkaido Math. J. 7 (1978), 206–213. (1978) MR0509406
  114. S. J. Gates C. M. Hull M. Rocek, Twisted multiplets and new supersymmetric nonlinear sigma models, Nuclear Phys. B 248 (1984), 157. (1984) MR0776369
  115. J. Gauntlett N. Kim D. Martelli D. Waldram, Fivebranes wrapped on SLAG three-cycles and related geometry, hep-th/0110034. 
  116. J. P. Gauntlett D. Martelli, D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D (3) 69 (2004), 086002. MR2095098
  117. P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1 × S 3 , J. Reine Angew. Math. 469 (1995), 1–50. (1995) MR1363825
  118. P. Gauduchon, Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. Ser. VII 2 (1997), 257–289. (1997) Zbl0876.53015MR1456265
  119. P. Gauduchon, K. P. Tod, Hyper-Hermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291–304. (1998) Zbl0945.53042MR1619847
  120. P. B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geom. 10 (1975), 601–618. (1975) Zbl0316.53035MR0400315
  121. G. W. Gibbons H. Lü C. N. Pope, and K. S. Stelle, Supersymmetric domain walls from metrics of special holonomy, Nuclear Phys. B 623 (2002), 3–46. MR1883449
  122. M. Godlinski W. Kopczynski P. Nurowski, Locally Sasakian manifolds, Classical Quantum Gravity 17 (2000), L105–L115. MR1791091
  123. S. Goette, Equivariant η -invariants on homogeneous spaces, Math. Z. 232 (1999), 1–42. (1999) Zbl0941.58016MR1714278
  124. S. I. Goldberg, Integrabilty of almost Kähler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100. (1969) MR0238238
  125. E. Goldstein S. Prokushkin, Geometric model for complex non-Kähler manifolds with SU ( 3 ) -structure, Comm. Math. Phys. 251 (2004), 65–78. MR2096734
  126. C. Gordon, W. Ziller, Naturally reductive metrics of nonpositive Ricci curvature, Proc. Amer. Math. Soc. 91 (1984), 287–290. (1984) Zbl0513.53049MR0740188
  127. G. Grantcharov, Y. S. Poon, Geometry of hyper-Kähler connections with torsion, Comm. Math. Phys. 213 (2000), 19–37. Zbl0993.53016MR1782143
  128. A. Gray, Nearly Kähler manifolds, J. Differential Geom. 4 (1970), 283–309. (1970) Zbl0201.54401MR0267502
  129. A. Gray, Weak holonomy groups, Math. Z. 123 (1971), 290–300. (1971) Zbl0222.53043MR0293537
  130. A. Gray, The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233–248. (1976) Zbl0345.53019MR0417965
  131. A. Gray, L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35–58. (1980) Zbl0444.53032MR0581924
  132. M. B. Green J. H. Schwarz, and E. Witten, Superstring theory. Volume 2: Loop amplitudes, anomalies and phenomenology, Cambridge Monogr. Math. Phys. 1987. (1987) MR0878144
  133. B. H. Gross B. Kostant P. Ramond, and S. Sternberg, The Weyl character formula, the half-spin representations, and equal rank subgroups, Proc. Natl. Acad. Sci. USA 95 (1998), no. 15, 8441–8442. (1998) MR1639139
  134. R. Grunewald, Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. Glob. Anal. Geom. 8 (1990), 43–59. (1990) Zbl0704.53050MR1075238
  135. G. B. Gurevich, Classification of trivectors of rank 8 , (in Russian), Dokl. Akad. Nauk SSSR 2 (1935), 353–355. (1935) 
  136. G. B. Gurevich, Algebra of trivectors II, (in Russian), Trudy Sem. Vektor. Tenzor. Anal. 6 (1948), 28–124. (1948) MR0057861
  137. F. W. Hehl P. Von Der Heyde G. D. Kerlick J. M. Nester, General relativity with spin and torsion: Foundations and prospects, Rev. Modern Phys. 48 (1976), 393–416. (1976) MR0439001
  138. F. W. Hehl J. D. McCrea E. W. Mielke Y. Ne’eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rep. 258 (1995), 1–171. (1995) MR1340371
  139. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure Appl. Math. vol. 80, Acad. Press, New York, 1978. (1978) Zbl0451.53038MR0514561
  140. N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55. (1974) Zbl0284.58016MR0358873
  141. N. Hitchin, The geometry of three-forms in six and seven dimensions, J. Differential Geom. 55 (2000), 547–576. MR1863733
  142. N. Hitchin, Stable forms and special metrics, math.DG/0107101; Contemp. Math. 288 (2001), 70–89. Zbl1004.53034MR1871001
  143. P. S. Howe, G. Papadopoulos, Ultraviolet behavior of two-dimensional supersymmetric nonlinear sigma models, Nuclear Phys. B 289 (1987), 264–276. (1987) 
  144. P. S. Howe, G. Papadopoulos, Finitness and anomalies in ( 4 , 0 ) supersymmetric sigma models, Nuclear Phys. B 381 (1992), 360. (1992) 
  145. P. S. Howe, G. Papadopoulos, Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B 379 (1996), 80–86. (1996) MR1396267
  146. J.-S. Huang, P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185–202. MR1862801
  147. C. M. Hull, Lectures On Nonlinear Sigma Models And Strings, PRINT-87-0480(Cambridge); Lectures given at Super Field Theories Workshop, Vancouver, Canada, July 25-Aug 6, 1986. (1986) MR1102925
  148. J.-I. Igusa,, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028. (1970) Zbl0217.36203MR0277558
  149. A. Ikeda, Formally self adjointness for the Dirac operator on homogeneous spaces, Osaka J. Math. 12 (1975), 173–185. (1975) Zbl0317.58019MR0376962
  150. L. Infeld (volume dedicated to), Recent developments in General Relativity, Oxford, Pergamon Press & Warszawa, PWN, 1962. (1962) MR0164694
  151. S. Ivanov, Connections with torsion, parallel spinors and geometry of Spin ( 7 ) -manifolds, Math. Res. Lett. 11 (2004), 171–186. MR2067465
  152. S. Ivanov, I. Minchev, Quaternionic Kähler and hyperKähler manifolds with torsion and twistor spaces, J. Reine Angew. Math. 567 (2004), 215–233. MR2038309
  153. S. Ivanov, G. Papadopoulos, Vanishing theorems and string background, Classical Quantum Gravity 18 (2001), 1089–1110. MR1822270
  154. S. Ivanov M. Parton, P. Piccinni, Locally conformal parallel G 2 - and Spin ( 7 ) -structures, math.DG/0509038, to appear in Math. Res. Lett. 13 (2006). MR2231110
  155. W. Jelonek, Some simple examples of almost Kähler non-Kähler structures, Math. Ann. 305 (1996), 639–649. (1996) Zbl0858.53027MR1399708
  156. G. Jensen, Imbeddings of Stiefel manifolds into Grassmannians, Duke Math. J. 42 (1975), 397–407. (1975) Zbl0335.53042MR0375164
  157. D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Differential Geom. 35 (1992), 743–761. (1992) Zbl0735.53050MR1163458
  158. D. Joyce, Compact Riemannian 7 -manifolds with holonomy G 2 . I, J. Differential Geom. 43 (1996), 291–328. (1996) MR1424428
  159. D. Joyce, Compact Riemannian 7 -manifolds with holonomy G 2 . II, J. Differential Geom. 43 (1996), 329–375. (1996) MR1424428
  160. D. Joyce, Compact 8 -manifolds with holonomy Spin ( 7 ) , Invent. Math. 123 (1996), 507–552. (1996) MR1383960
  161. D. Joyce, Compact manifolds with special holonomy, Oxford Science Publ., 2000. Zbl1027.53052MR1787733
  162. A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35–42. (1983) Zbl0521.53048MR0686346
  163. T. Kashiwada, A note on a Riemannian space with Sasakian 3 -structure, Natur. Sci. Rep. Ochanomizu Univ. 22 (1971), 1–2. (1971) Zbl0228.53033MR0303449
  164. T. Kashiwada, On a contact 3-structure, Math. Z. 238 (2001), 829–832. Zbl1004.53058MR1872576
  165. G. Ketsetzis, S. Salamon, Complex structures on the Iwasawa manifold, Adv. Geom. 4 (2004), 165–179. Zbl1059.22012MR2055676
  166. T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961), 212–221. (1961) Zbl0095.22903MR0127952
  167. F. Klein, Das Erlanger Programm, Ostwalds Klassiker der exakten Wissenschaften Band 253, Verlag H. Deutsch, Frankfurt a. M., 1995. (1995) Zbl0833.01037
  168. K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Global Anal. Geom. 4 (1986), 291–325. (1986) Zbl0629.53058MR0910548
  169. K.-D. Kirchberg, Killing spinors on Kähler manifolds, Ann. Global Anal. Geom. 11 (1993), 141–164. (1993) Zbl0810.53033MR1225435
  170. K.-D. Kirchberg, Integrability conditions for almost Hermitian and almost Kähler 4 -manifolds, math.DG/0605611. 
  171. V. F. Kirichenko, K -spaces of maximal rank, Mat. Zametki 22 (1977), 465–476. (1977) MR0474103
  172. V. F. Kirichenko A. R. Rustanov, Differential geometry of quasi-Sasakian manifolds, Sb. Math. 193 (2002), 1173-1201; translation from Mat. Sb. 193 (2002), 71–100. (193) MR1934545
  173. S. Kobayashi, K. Nomizu, Foundations of differential geometry I, Wiley Classics Library, Wiley Inc., Princeton, 1963, 1991. (1963) Zbl0119.37502
  174. S. Kobayashi, K. Nomizu, Foundations of differential geometry II, Wiley Classics Library, Wiley Inc., Princeton, 1969, 1996. (1969) 
  175. W. Kopczyński, An anisotropic universe with torsion, Phys. Lett. A 43 (1973), 63–64. (1973) 
  176. B. Kostant, On differential geometry and homogeneous spaces II, Proc. N. A. S. 42 (1956), 354–357. (1956) Zbl0075.31603MR0088017
  177. B. Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), 447–501. (1999) Zbl0952.17005MR1719734
  178. B. Kostant, Dirac cohomology for the Cubic Dirac operator, in: Joseph, Anthony (ed.) et al., Studies in memory of Issai Schur. Basel: Birkhäuser. Progr. Math. 210 (2003), 69–93. Zbl1165.17301MR1985723
  179. B. Kostant, P. Michor, The generalized Cayley map from an algebraic group to its Lie algebra, preprint (arXiv:math.RT/0109066v1, 10 Sep 2001), to appear in The Orbit Method in Geometry and Physics (A. A. Kirillov Festschrift), Progr. Math. (2003). Zbl1072.20051MR1995382
  180. A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003), 125–160. Zbl1043.53041MR2024648
  181. O. Kowalski, L. Vanhecke, Four-dimensional naturally reductive homogeneous spaces, Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Sem. Mat., Torino, Fasc. Spec. (1983), 223-232. (1983) Zbl0631.53039MR0829007
  182. O. Kowalski, L. Vanhecke, A generalization of a theorem on naturally reductive homogeneous spaces, Proc. Amer. Math. Soc. 91 (1984), 433–435. (1984) Zbl0542.53029MR0744644
  183. O. Kowalski, L. Vanhecke, Classification of five-dimensional naturally reductive spaces, Math. Proc. Cambridge Philos. Soc. 97 (1985), 445–463. (1985) Zbl0555.53024MR0778679
  184. O. Kowalski, S. Wegrzynowski, A classification of 5 -dimensional ϕ -symmetric spaces, Tensor, N. S. 46 (1987), 379–386. (1987) 
  185. E. Kreyszig, Differential geometry, Dover Publ., inc., New York, 1991, unabridged republication of the 1963 printing. (1991) MR1118149
  186. G. Landweber, Harmonic spinors on homogeneous spaces, Represent. Theory 4 (2000), 466–473. Zbl0972.22008MR1780719
  187. J.-L. Li, S.-T. Yau, Existence of supersymmetric string theory with torsion, J. Differential Geom. 70 (2005), 143–182 and hep-th/0411136. Zbl1102.53052MR2192064
  188. A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9. (1963) Zbl0136.18401MR0156292
  189. A. Lichnerowicz, Spin manifolds, Killing spinors and universality of the Hijazi inequality, Lett. Math. Phys. 13 (1987), 331–344. (1987) Zbl0624.53034MR0895296
  190. A. Lichnerowicz, Les spineurs-twisteurs sur une variété spinorielle compacte, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 381–385. (1988) Zbl0641.53014MR0934624
  191. D. Lüst, S. Theisen, Lectures on String Theory, Springer-Verlag, 1989. (1989) MR1028064
  192. D. Martelli J. Sparks S.-T. Yau, Sasaki-Einstein Manifolds and Volume Minimisation, hep-th/0603021. 
  193. F. Martín Cabrera, Special almost Hermitian geometry, J. Geom. Phys. 55 (2005), 450–470. MR2162420
  194. F. Martín Cabrera M. D. Monar Hernandez A. F. Swann, Classification of G 2 -structures, J. London Math. Soc. II. Ser. 53 (1996), 407–416. (1996) MR1373070
  195. F. Martín Cabrera, A. F. Swann, Almost Hermitian structures and quaternionic geometries, Differential Geom. Appl. 21 (2004), 199–214. MR2073825
  196. Y. McKenzie Wang, Parallel spinors and parallel forms, Ann. Global Anal. Geom. 7 (1989), 59–68. (1989) MR1029845
  197. S. Mehdi, R. Zierau, Principal Series Representations and Harmonic Spinors, to appear in Adv. Math. (preprint at http://www.math.okstate.edu/~zierau/papers.html). Zbl1085.22011MR2186917
  198. J. Michelson A. Strominger, The geometry of (super) conformal quantum mechanics, Comm. Math. Phys. 213 (2000), 1–17. MR1782142
  199. V. Miquel, The volume of small geodesic balls for a metric connection, Compositio Math. 46 (1982), 121–132. (1982) MR0660156
  200. V. Miquel, Volumes of geodesic balls and spheres associated to a metric connection with torsion, Contemp. Math. 288 (2001), 119–128. Zbl1005.53012MR1871004
  201. S. Nagai, Naturally reductive Riemannian homogeneous structure on a homogeneous real hypersurface in a complex space form, Boll. Un. Mat. Ital. A (7) 9 (1995), 391–400. (1995) Zbl0835.53068MR1336245
  202. S. Nagai, Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms, Geom. Dedicata 62 (1996), 253–268. (1996) Zbl0860.53032MR1406440
  203. S. Nagai, The classification of naturally reductive homogeneous real hypersurfaces in complex projective space, Arch. Math. 69 (1997), 523–528. (1997) Zbl0901.53037MR1480520
  204. P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations, Asian J. Math. 6 (2002) 481–504. Zbl1041.53021MR1946344
  205. P.-A. Nagy, On nearly-Kähler geometry, Ann. Global Anal. Geom. 22 (2002), 167–178. Zbl1020.53030MR1923275
  206. P. Nurowski M. Przanowski, A four-dimensional example of Ricci-flat metric admitting almost-Kähler non-Kähler structure, ESI preprint 477, 1997; Classical Quantum Gravity 16 (1999), L9–L16. (1997) MR1682582
  207. P. Nurowski, Distinguished dimensions for special Riemannian geometries, math.DG/0601020. 
  208. R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1–30. (1972) Zbl0249.22004MR0318398
  209. R. Penrose, Spinors and torsion in general relativity, Found. of Phys. 13 (1983), 325-339. (1983) MR0838841
  210. Y. S. Poon, A. F. Swann, Potential functions of HKT spaces, Classical Quantum Gravity 18 (2001), 4711–4714. Zbl1007.53038MR1894924
  211. Y. S. Poon, A. F. Swann, Superconformal symmetry and HyperKähler manifolds with torsion, Comm. Math. Phys. 241 (2003), 177–189. MR2013757
  212. Chr. Puhle, The Killing equation with higher order potentials, Ph. D. Thesis, Humboldt-Universität zu Berlin, 2006/07. 
  213. W. Reichel, Über trilineare alternierende Formen in sechs und sieben Veränderlichen und die durch sie definierten geometrischen Gebilde, Druck von B. G. Teubner in Leipzig 1907, Dissertation an der Universität Greifswald. (1907) 
  214. M. Rocek, Modified Calabi-Yau manifolds with torsion, in: Yau, Shing-Tung (ed.), Essays on mirror manifolds. Cambridge, MA: International Press. 1992, 480–488. (1992) Zbl0859.53050MR1191438
  215. M. L. Ruggiero, A. Tartaglia, Einstein–Cartan theory as a theory of defects in space-time, Amer. J. Phys. 71 (2003), 1303–1313. MR2016766
  216. S. Salamon, Riemannian geometry and holonomy groups, Pitman Res. Notes Math. Ser. 201, Jon Wiley & Sons, 1989. (1989) Zbl0685.53001MR1004008
  217. S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311–333. MR1812058
  218. S. Salamon, A tour of exceptional geometry, Milan J. Math. 71 (2003), 59–94. Zbl1055.53039MR2120916
  219. K. Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 39 (1987), 677–684. (1987) Zbl0637.53053MR0905633
  220. N. Schoemann, Almost hermitian structures with parallel torsion, PhD thesis, Humboldt-Universität zu Berlin, 2006. Zbl1137.53014MR2360237
  221. J. A. Schouten, Der Ricci-Kalkül, Grundlehren Math. Wiss. 10, Springer-Verlag Berlin, 1924. (1924) MR0516659
  222. J. A. Schouten, Klassifizierung der alternierenden Größen dritten Grades in 7 Dimensionen, Rend. Circ. Mat. Palermo 55 (1931), 137–156. (1931) 
  223. E. Schrödinger, Diracsches Elektron im Schwerefeld I, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-Math. Klasse 1932, Verlag der Akademie der Wissenschaften Berlin, 1932, 436–460. (1932) 
  224. J. Simons, On the transitivity of holonomy systems, Ann. of Math. 76 (1962), 213–234. (1962) Zbl0106.15201MR0148010
  225. S. Slebarski, The Dirac operator on homogeneous spaces and representations of reductive Lie groups I, Amer. J. Math. 109 (1987), 283–301. (1987) Zbl0649.58031MR0882424
  226. S. Slebarski, The Dirac operator on homogeneous spaces and representations of reductive Lie groups II, Amer. J. Math. 109 (1987), 499–520. (1987) Zbl0669.22003MR0892596
  227. P. Spindel A. Sevrin W. Troost, and A. van Proeyen, Extended supersymmetric σ -models on group manifolds, Nuclear Phys. B 308 (1988), 662–698. (1988) MR0967938
  228. S. Sternberg, Lie algebras, Lecture Notes in Math. 1999. (1999) 
  229. A. Strominger, Superstrings with torsion, Nuclear Phys. B 274 (1986), 253–284. (1986) MR0851702
  230. K. Strubecker, Differentialgeometrie. II: Theorie der Flächenmetrik, Sammlung Göschen, W. de Gruyter, Berlin, 1969. (1969) Zbl0169.23501MR0239514
  231. A. F. Swann, Aspects symplectiques de la géométrie quaternionique, C. R. Acad. Sci. Paris, Sér. I 308 (1989), 225–228. (1989) Zbl0661.53023MR0986384
  232. A. F. Swann, HyperKähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421–450. (1991) Zbl0711.53051MR1096180
  233. A. F. Swann, Weakening holonomy, ESI preprint No. 816 (2000); in S. Marchiafava et. al. (eds.), Proc. of the Second Meeting on Quaternionic Structures in Mathematics and Physics, Roma 6-10 September 1999, World Scientific, Singapore 2001, 405–415. (1999) MR1848678
  234. J. Tafel, A class of cosmological models with torsion and spin, Acta Phys. Polon. B 6 (1975), 537–554. (1975) 
  235. S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349–379. (1989) Zbl0677.53043MR1000553
  236. W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467–468. (1976) Zbl0324.53031MR0402764
  237. A. Trautman, On the structure of the Einstein-Cartan equations, Sympos. Math. 12 (1973), 139–162. (1973) Zbl0273.53021MR0376097
  238. A. Trautman, Spin and torsion may avert gravitational singularities, Nature Phys. Sci. 242 (1973) 7. (1973) 
  239. A. Trautman, Gauge and optical aspects of gravitation, Classical Quantum Gravity 16 (1999), 157–175. (1999) Zbl0948.83010MR1728438
  240. F. Tricerri, L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Notes Series, vol. 83, Cambridge Univ. Press, Cambridge, 1983. (1983) Zbl0509.53043MR0712664
  241. F. Tricerri, L. Vanhecke, Geodesic spheres and naturally reductive homogeneous spaces, Riv. Mat. Univ. Parma 10 (1984), 123–131. (1984) Zbl0563.53040MR0777319
  242. F. Tricerri, L. Vanhecke, Naturally reductive homogeneous spaces and generalized Heisenberg groups, Compositio Math. 52 (1984), 389–408. (1984) Zbl0551.53028MR0756730
  243. I. Vaisman, On locally conformal almost kähler manifolds, Israel J. Math. 24 (1976), 338–351. (1976) Zbl0335.53055MR0418003
  244. I. Vaisman, Locally conformal Kähler manifolds with parallel Lee form, Rend. Math. Roma 12 (1979), 263–284. (1979) Zbl0447.53032MR0557668
  245. P. Van Nieuwenhuizen, Supergravity, Phys. Rep. 68 (1981), 189–398. (1981) Zbl0465.53041MR0615178
  246. J. Vanžura, One kind of multisymplectic structures on 6 -manifolds, Proceedings of the Colloquium on Differential Geometry, Debrecen, 2000, 375–391 (2001). MR1859316
  247. M. Verbitsky, HyperKähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math. 6 (2002), 679–712. MR1958088
  248. A. B. Vinberg, A. G. Ehlahvili, Classification of trivectors of a 9 -dimensional space, Sel. Math. Sov. 7 (1988), 63–98. Translated from Tr. Semin. Vektorn. Tensorm. Anal. Prilozh. Geom. Mekh. Fiz. 18 (1978), 197–233. (1988) Zbl0648.15021MR0504529
  249. M. Y. Wang, Parallel spinors and parallel forms, Ann. Global Anal. Geom. 7 (1989), 59–68. (1989) Zbl0688.53007MR1029845
  250. M. Y. Wang, W. Ziller, On normal homogeneous Einstein manifolds, Ann. Sci. Éc. Norm. Sup., série 18 (1985), 563–633. (1985) Zbl0598.53049MR0839687
  251. R. Westwick, Real trivectors of rank seven, Linear and Multilinear Algebra 10 (1981), 183–204. (1981) Zbl0464.15001MR0630147
  252. F. Witt, Generalised G 2 -manifolds, Comm. Math. Phys. 265 (2006), 275–303, math.DG/0411642. Zbl1154.53014MR2231673
  253. F. Witt, Special metrics and Triality, math.DG/0602414. 
  254. J. A. Wolf, Partially harmonic spinors and representations of reductive Lie groups, J. Funct. Anal. 15 (1974), 117–154. (1974) Zbl0279.22009MR0393351
  255. S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampŕe equations. I, Comm. Pure Appl. Math. 31 (1978), 339–411. (1978) MR0480350
  256. W. Ziller, The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces, Comment. Math. Helv. 52 (1977), 573–590. (1977) Zbl0368.53033MR0474145
  257. W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann. 259 (1982), 351–358. (1982) Zbl0469.53043MR0661203

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