The Srní lectures on non-integrable geometries with torsion. (English) Zbl 1164.53300

Agricola, Ilka

Arch. Math., Brno 42, No. 5, 5-84 (2006).

Summary: 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 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.

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53D15	Almost contact and almost symplectic manifolds
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory

Keywords:

metric connection with torsion; intrinsic torsion; \(G\)-structure; characteristic connection; superstring theory

References:

[1]	E. Abbena, An example of an almost Kähler manifold which is not Kählerian, Bolletino U. M. I. (6) 3 A (1984), 383-392.
[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), p. 147-170.
[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.
[4]	I. Agricola, Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. Math. Phys. 232 (2003), 535-563.
[5]	I. Agricola, S. Chiossi, A. Fino, Solvmanifolds with integrable and non-integrable \(G_2\)-structures, math.DG/0510300, to appear in Differ. Geom. Appl.
[6]	I. Agricola and Th. Friedrich, Global Analysis – Differential forms in Calculus, Geometry and Physics, Graduate Studies in Mathematics, Publications of the AMS 2002, Providence, Rhode Island 2002.
[7]	I. Agricola, Killing spinors in supergravity with \(4\)-fluxes, Class. Quant. Grav. 20 (2003), 4707-4717.
[8]	I. Agricola, On the holonomy of connections with skew-symmetric torsion, Math. Ann. 328 (2004), 711-748.
[9]	I. Agricola, The Casimir operator of a metric connection with totally skew-symmetric torsion, Journ. Geom. Phys. 50 (2004), 188-204.
[10]	I. Agricola, Geometric structures of vectorial type, math.DG/0509147, to appear in J. Geom. Phys.
[11]	I. Agricola, T. Friedrich, P.-A. Nagy, C. Puhle, On the Ricci tensor in the common sector of type II string theory, Class. Quant. Grav. 22 (2005) 2569-2577.
[12]	I.Agricola and Chr. Thier, The geodesics of metric connections with vectorial torsion, Ann. Global Anal. Geom. 26 (2004), 321-332.
[13]	D. V. Alekseevski, Riemannian spaces with exceptional holonomy groups, Func. Anal. Prilozh. 2 (1968), 1-10.
[14]	D. V. Alekseevsky, S. Marchiafava, M. Pontecorvo, Compatible almost complex structures on quaternion Kähler manifolds, Ann. Global Anal. Geom. 16 (1998), 419-444.
[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. Advances in the Mathematical Sciences 56 (2005), 33-62.
[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.
[17]	B. Alexandrov, \(Sp(n)U(1)\)-connections with parallel totally skew-symmetric torsion, math.DG/0311248, to appear in Journ. Geom. Phys.
[18]	I. Agricola, On weak holonomy, Math. Scand. 96 (2005), 169-189.
[19]	B. Alexandrov, Th. Friedrich, Nils Schoemann, Almost Hermitian \(6\)-manifolds revisited, J. Geom. Phys. 53 (2005), 1-30.
[20]	B. Alexandrov, S. Ivanov, Dirac operators on Hermitian spin surfaces, Ann. Global Anal. Geom. 18 (2000), 529-539.
[21]	T. Ali, \(M\)-theory on seven manifolds with \(G\)-fluxes, hep-th/0111220.
[22]	W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Am. Math. Soc. 75 (1953), 428-443.
[23]	I. Agricola, On homogeneous Riemannian manifolds, Duke Math. J. 25 (1958), 647-669.
[24]	V. Apostolov, T. Drăghici, A. Moroianu, A splitting theorem for Kähler manifolds whose Ricci tensors have constant eigenvalues, Intern. J. Math. 12 (2001), 769-789.
[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.
[26]	J. Armstrong, Almost Kähler geometry, Ph. D. Thesis, Oxford University, 1998.
[27]	M. Atiyah and W. Schmid, A geometric construction for the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1-62.
[28]	M. Atiyah and E. Witten, \(M\)-theory dynamics on a manifold of \(G_2\) holonomy, Adv. Theor. Math. Phys. 6 (2002), 1-106.
[29]	J. E. D’Atri, Geodesic spheres and symmetries in naturally reductive spaces, Michigan Math. J. 22 (1975), 71-76.
[30]	J. E. D’Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Memoirs of Amer. Math. Soc. 18 (1979). [Bär93]{Baer93} Chr. Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509-521.
[31]	B. Banos, A. F. Swann, Potentials for hyper-Kähler metrics with torsion, Class. Quant. Grav. 21 (2004), 3127-3135.
[32]	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.
[33]	K. Behrndt, C. Jeschek, Fluxes in \(M\)-theory on \(7\)-manifolds and \(G\)-structures, hep-th/0302047.
[34]	F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1-40.
[35]	F. A. Belgun and A. Moroianu, Nearly Kähler \(6\)-manifolds with reduced holonomy, Ann. Global Anal. Geom. 19 (2001), 307-319.
[36]	M. Berger, Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279-330.
[37]	M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Sc. Norm. Sup. Pisa 15 (1961), 179-246.
[38]	J. Berndt, F. Tricerri, L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, LNM 1598, Springer, 1995.
[39]	A. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete Bd. 10, Springer-Verlag Berlin-Heidelberg 1987.
[40]	A. Bilal, J.-P. Derendinger and K. Sfetsos, Weak \(G_2\)-holonomy from self-duality, flux and supersymmetry, Nucl. Phys. B 628 (2002), 112-132.
[41]	J. M. Bismut, A local index theorem for non-Kählerian manifolds, Math. Ann. 284 (1989), 681-699.
[42]	D. E. Blair, Contact manifoldsin Riemannian geometry, LNM 509 (1976), Springer.
[43]	D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics vol. 203, Birkhäuser, 2002.
[44]	D. E. Blair and L. Vanhecke, New characterization of \(\vphi\)-symmetric spaces, Kodai Math. J. 10 (1987), 102-107.
[45]	M. Bobieński, The topological obstructions to the existence of an irreducible \(\SO(3)\)-structure on a five manifold, math.DG/0601066.
[46]	M. Bobieński and P. Nurowski, Irreducible \(\SO(3)\)-geometries in dimension five, to appear in J. Reine Angew. Math.; math.DG/0507152.
[47]	E. Bonan, Sur les variétés riemanniennes à groupe d’holonomie \(G_2\) ou \(\Spin(7)\), C. R. Acad. Sc. Paris 262 (1966), 127-129.
[48]	C. P. Boyer and K. Galicki, \(3\)-Sasakian manifolds, in Essays on Einstein manifolds, (ed. by C. LeBrun and M. Wang), International Press 1999.
[49]	I. Agricola, Einstein manifolds and contact geometry, Proc. Amer. Math. Soc. 129 (2001), 2419-2430.
[50]	I. Agricola, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, to appear 2007.
[51]	C. P. Boyer, K. Galicki, B. M. Mann, The geometry and topology of \(3\)-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183-220.
[52]	L. Brink, P. Ramond and X. Xiong, Supersymmetry and Euler multiplets, hep-th/0207253.
[53]	R. B. Brown and A. Gray, Riemannian manifolds with holonomy group \(\Spin(7)\), Differential Geometry in honor of K. Yano, Kinokiniya, Tokyo, 1972, 41-59.
[54]	R. L. Bryant, Metrics with exceptional holonomy, Ann Math. 126 (1987), 525-576.
[55]	I. Agricola, 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.
[56]	I. Agricola, Some remarks on \(G_2\)-structures, in Proceeding of the 2004 Gokova Conference on Geometry and Topology (May, 2003), math.DG/0305124.
[57]	R. L. Bryant, and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), 829-850.
[58]	J. Bureš, Multisymplectic structures of degree three of product type on \(6\)-dimensional manifolds, Suppl. Rend. Circ. Mat. Palermo II, Ser. 72 (2004), 91-98.
[59]	J. Bureš and J. Vanžura, Multisymplectic forms of degree three in dimension seven, Suppl. Rend. Circ. Mat. Palermo II, Ser. 71 (2003), 73-91.
[60]	J. B. Butruille, Classification des variétés approximativement kähleriennes homogènes, Ann. Glob. Anal. Geom. 27 (2005), 201-225.
[61]	D.M.J. Calderbank and 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.
[62]	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.
[63]	I. Agricola, 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.
[64]	I. Agricola, 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).
[65]	I. Agricola, Les récentes généralisations de la notion d’espace, Bull. Sc. Math. 48 (1924), 294-320.
[66]	I. Agricola, 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).
[67]	I. Chavel, A class of {R}iemannian homogeneous spaces, J. Differ. Geom. 4 (1970), 13-20.
[68]	D. Chinea and G. Gonzales, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl. 156 (1990), 15-36.
[69]	D. Chinea and J. C. Marrero, Classifications of almost contact metric structures, Rev. Roumaine Math. Pures Appl. 37 (1992), 581-599.
[70]	S. G. Chiossi, A. Fino, Conformally parallel \(G_2\)-structures on a class of solvmanifolds, Math. Z. 252 (2006), 825-848.
[71]	S. Chiossi and S. Salamon, The intrinsic torsion of \(SU(3)\) 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).
[72]	R. Cleyton and S. Ivanov, On the geometry of closed \(G_2\)-structures, math.DG/0306362.
[73]	I. Agricola, Curvature decomposition of \(G_2\) manifolds, math.DG/0605xxx.
[74]	I. Agricola, Conformal equivalence for certain geometries in dimension \(6\) and \(7\), math.DG/0605xxx.
[75]	R. Cleyton and A. Swann, Cohomogeneity-one \(G_{2}\)-structures, J. Geom. Phys. 44 (2002), 202-220.
[76]	I. Agricola, Einstein metrics via intrinsic or parallel torsion, Math. Z. 247 (2004), 513-528.
[77]	G. Curio, B. Körs and D. Lüst, Fluxes and branes in type II vacua and M-theory geometry with \(G_2\) and \(Spin(7)\) holonomy, hep-th/0111165.
[78]	P. Dalakov and S. Ivanov, Harmonic spinors of Dirac operators of connections with torsion in dimension \(4\), Class. Quant. Grav. 18 (2001), 253-265.
[79]	B. de Wit, D.J. Smit, and N.D. Hari Dass, Residual supersymmetry of compactified \(D=10\) Supergravity, Nucl. Phys. B 283 (1987), 165.
[80]	D. Ž. Djoković, Classification of trivectors of an eight-dimensional real vector space, Linear Multilinear Algebra 13 (1983), 3-39.
[81]	I. G. Dotti and A. Fino, HyperKähler torsion structures invariant by nilpotent Lie groups, Class. Quant. Grav. 19 (2002), 551-562.
[82]	S. Dragomir, L. Ornea, Locally conformal Kähler geometry, Progress in Mathematics vol. 155, Birkhäuser Verlag, 1998.
[83]	M. J. Duff, \(M\)-theory on manifolds of \(G_2\)-holonomy: the first twenty years, hep-th/0201062.
[84]	M. Fernández, A classification of Riemannian manifolds with structure group \(\Spin(7)\), Ann. Mat. Pura Appl. 143 (1986), 101-122.
[85]	I. Agricola, An example of a compact calibrated manifold associated with the exceptional Lie group \(G_2\), Journ. Diff. Geom. 26 (1987), 367-370.
[86]	M. Fernández and A. Gray, Riemannian manifolds with structure group \(\mathrm{G}_2\), Ann. Mat. Pura Appl. 132 (1982), 19-45.
[87]	J. Figueroa-O’Farrill, G. Papadopoulos, Maximally supersymmetric solutions of ten- and eleven-dimensional supergravities, hep-th/0211089.
[88]	A. Fino, Almost contact homogeneous manifolds, Riv. Mat. Univ. Parma (5) 3 (1994), 321-332.
[89]	I. Agricola, Almost contact homogeneous structures, Boll. Unione Mat. Ital., VII. Ser., A 9 (1995), 299-311.
[90]	I. Agricola, Intrinsic torsion and weak holonomy, Math. J. Toyama Univ. 21 (1998), 1-22.
[91]	I. Agricola, Almost Kähler \(4\)-dimensional Lie groups with \(J\)-invariant Ricci tensor, Differ. Geom. Appl. 23 (2005), 26-37.
[92]	A. Fino and G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2004) 439-450.
[93]	A. Fino, M. Parton, S. Salamon, Families of strong KT structures in six dimensons, Comment. Math. Helv. 79 (2004), 317-340.
[94]	Th. Friedrich, Der erste {E}igenwert des {D}irac-{O}perators einer kompakten, {R}iemannschen {M}annigfaltigkeit nichtnegativer {S}kalarkrümmung, Math. Nachr. 97 (1980), 117-146.
[95]	I. Agricola, Dirac operators in {R}iemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Prov., 2000.
[96]	I. Agricola, Weak \(\mathrm{Spin}(9)\)-structures on \(16\)-dimensional Riemannian manifolds, Asian Math. Journ. 5 (2001), 129-160.
[97]	I. Agricola, \(Spin(9)\)-structures and connections with totally skew-symmetric torsion, Journ. Geom. Phys. 47 (2003), 197-206.
[98]	I. Agricola, On types of non-integrable geometries, Suppl. Rend. Circ. Mat. di Palermo Ser. II, 71 (2003), 99-113.
[99]	I. Agricola, \(G_2\)-manifolds with parallel characteristic torsion, math.DG/0604441, to appear in Differ. Geom. Appl.
[100]	Th. Friedrich and R. Grunewald, On the first eigenvalue of the Dirac operator on \(6\)-dimensional manifolds, Ann Glob. Anal. Geom. 3 (1985), 265-273.
[101]	Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian Journ. Math. 6 (2002), 303-336.
[102]	I. Agricola, Almost contact manifolds, connections with torsion and parallel spinors, J. Reine Angew. Math. 559 (2003), 217-236.
[103]	I. Agricola, Killing spinor equations in dimension \(7\) and geometry of integrable \(\mathrm{G}_2\)-manifolds, Journ. Geom. Phys. 48 (2003), 1-11.
[104]	Th. Friedrich and I. Kath, Einstein manifolds of dimension five with small eigenvalues of the Dirac operator, Journ. Diff. Geom. 19 (1989), 263-279.
[105]	I. Agricola, \(7\)-dimensional compact Riemannian manifolds with Killing spinors, Comm. Math. Phys. 133 (1990), 543-561.
[106]	Th. Friedrich, I. Kath, A. Moroianu and U. Semmelmann, On nearly parallel \(\mathrm{G}_2\)-structures, Journ. Geom. Phys. 23 (1997), 256-286.
[107]	Th. Friedrich and E. C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, Journ. Geom. Phys. 33 (2000), 128-172.
[108]	Th. Friedrich and S. Sulanke, Ein {K}riterium für die formale {S}elbstadjungiertheit des {D}irac-{O}perators, Coll. Math. XL (1979), 239-247.
[109]	J.-X. Fu and S.-T. Yau, Existence of supersymmetric Hermitian metrics with torsion on non-Kähler manifolds, hep-th/0509028.
[110]	A. Fujiki and 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, Progress in Mathematics 234, 153-163 (2005).
[111]	T. Fukami and S. Ishihara, Almost Hermitian structure on \(S^6\), Hokkaido Math. J. 7 (1978), 206-213.
[112]	S. J. Gates, C.M. Hull, M. Rocek, Twisted multiplets and new supersymmetric nonlinear sigma models, Nucl. Phys. B 248 (1984), 157.
[113]	J. Gauntlett, N. Kim, D. Martelli, D. Waldram, Fivebranes wrapped on SLAG three-cycles and related geometry, hep-th/0110034.
[114]	J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D (3) 69 (2004), 086002.
[115]	P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type \(S^1 \times S^3\), J. Reine Angew. Math. 469 (1995), 1-50.
[116]	I. Agricola, Hermitian connections and Dirac operators, Boll. Un. Mat. Ial. ser. VII 2 (1997), 257-289.
[117]	P. Gauduchon and K. P. Tod Hyper-Hermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291-304.
[118]	P. B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differ. Geom. 10 (1975), 601-618.
[119]	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.
[120]	M. Godlinski, W. Kopczynski, P. Nurowski, Locally Sasakian manifolds, Class. Quant. Grav. 17 (2000), L105-L115.
[121]	S. Goette, Equivariant \(\eta\)-invariants on homogeneous spaces, Math. Z. 232 (1999), 1-42.
[122]	S. I. Goldberg, Integrabilty of almost Kähler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96-100.
[123]	E. Goldstein, S. Prokushkin, Geometric model for complex non-Kähler manifolds with \(\SU(3)\)-structure, Comm. Math. Phys. 251 (2004), 65-78.
[124]	C. Gordon and W. Ziller, Naturally reductive metrics of nonpositive Ricci curvature, Proc. Amer. Math. Soc. 91 (1984), 287-290.
[125]	G. Grantcharov and Y. S. Poon, Geometry of hyper-Kähler connections with torsion, Commun. Math. Phys. 213 (2000), 19-37.
[126]	A. Gray, Nearly Kähler manifolds, Journ. Diff. Geom. 4 (1970), 283-309.
[127]	I. Agricola, Weak holonomy groups, Math. Z. 123 (1971), 290-300.
[128]	I. Agricola, The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233-248.
[129]	A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura e Appl. 123 (1980), 35-58.
[130]	M. B. Green, J. H. Schwarz, and E. Witten, Superstring theory. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1987.
[131]	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.
[132]	R. Grunewald, Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. Glob. Anal. Geom. 8 (1990), 43-59.
[133]	G. B. Gurevich, Classification of trivectors of rank \(8\) (in Russian), Dokl. Akad. Nauk SSSR 2 (1935), 353-355.
[134]	I. Agricola, Algebra of trivectors II (in Russian), Trudy Sem. Vektor. Tenzor. Anal. 6 (1948), 28-124.
[135]	F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, J. M. Nester General relativity with spin and torsion: Foundations and prospects, Rev. Mod. Phys. 48 (1976), 393-416.
[136]	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. Reports 258 (1995), 1-171.
[137]	S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied Mathematics, vol. 80, Acad. Press, New York, 1978.
[138]	N. Hitchin, Harmonic spinors, Adv. in Math. 14 (1974), 1-55.
[139]	I. Agricola, The geometry of three-forms in six and seven dimensions, Journ. Diff. Geom. 55 (2000), 547-576.
[140]	I. Agricola, Stable forms and special metrics, math.DG/0107101; Contemp. Math. 288 (2001), 70-89.
[141]	P. S. Howe and G. Papadopoulos, Ultraviolet behavior of two-dimensional supersymmetric nonlinear sigma models, Nucl. Phys. B 289 (1987), 264-276.
[142]	I. Agricola, Finitness and anomalies in \((4,0)\) supersymmetric sigma models, Nucl. Phys. B 381 (1992), 360.
[143]	I. Agricola, Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B 379 (1996), 80-86.
[144]	J.-S. Huang and P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185-202.
[145]	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.
[146]	J.-I. Igusa,A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997-1028.
[147]	A. Ikeda, Formally self adjointness for the {D}irac operator on homogeneous spaces, Osaka J. Math. 12 (1975), 173-185.
[148]	L. Infeld (volume dedicated to), Recent developments in General Relativity, Oxford, Pergamon Press & Warszawa, PWN, 1962.
[149]	S. Ivanov, Connections with torsion, parallel spinors and geometry of \(\Spin(7)\)-manifolds, Math. Res. Lett. 11 (2004). 171-186.
[150]	S. Ivanov and I. Minchev, Quaternionic Kähler and hyperKähler manifolds with torsion and twistor spaces, J. Reine Angew. Math. 567 (2004), 215-233.
[151]	S. Ivanov and G. Papadopoulos, Vanishing theorems and string background, Class. Quant. Grav. 18 (2001), 1089-1110.
[152]	S. Ivanov, M. Parton and P. Piccinni, Locally conformal parallel \(G_2\)- and \(\Spin(7)\)-structures, math.DG/0509038, to appear in Math. Res. Letters, 13 (2006).
[153]	W. Jelonek, Some simple examples of almost Kähler non-Kähler structures, Math. Ann. 305 (1996), 639-649.
[154]	G. Jensen, Imbeddings of {S}tiefel manifolds into {G}rassmannians, Duke Math. J. 42 (1975), 397-407.
[155]	D. Joyce, Compact hypercomplex and quaternionic manifolds, Journ. Diff. Geom. 35 (1992), 743-761.
[156]	I. Agricola, Compact Riemannian \(7\)-manifolds with holonomy \(G_2\). I, Journ. Diff. Geom. 43 (1996), 291-328.
[157]	I. Agricola, Compact Riemannian \(7\)-manifolds with holonomy \(G_2\). II, Journ. Diff. Geom. 43 (1996), 329-375.
[158]	I. Agricola, Compact \(8\)-manifolds with holonomy \(\Spin(7)\), Invent. Math. 123 (1996), 507-552.
[159]	I. Agricola, Compact manifolds with special holonomy, Oxford Science Publ., 2000.
[160]	A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35-42.
[161]	T. Kashiwada, A note on a Riemannian space with Sasakian \(3\)-structure, Natural Sci. Rep. Ochanomizu Univ. 22 (1971), 1-2.
[162]	I. Agricola, On a contact 3-structure, Math. Z. 238 (2001), 829-832.
[163]	G. Ketsetzis and S. Salamon, Complex structures on the Iwasawa manifold, Adv. Geom. 4 (2004), 165-179.
[164]	T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961), 212-221.
[165]	F. Klein, Das Erlanger Programm, Ostwalds Klassiker der exakten Wissenschaften Band 253, Verlag H. Deutsch, Frankfurt a. M., 1995.
[166]	K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Glob. Anal. Geom. 4 (1986), 291-325.
[167]	I. Agricola, Killing spinors on Kähler manifolds, Ann. Glob. Anal. Geom. 11 (1993), 141-164.
[168]	I. Agricola, Integrability conditions for almost Hermitian and almost Kähler \(4\)-manifolds, math.DG/0605611.
[169]	V. F. Kirichenko, \(K\)-spaces of maximal rank, Mat. Zam. 22 (1977), 465-476.
[170]	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.
[171]	S. Kobayashi and K. Nomizu, Foundations of differential geometry {I}, Wiley Classics Library, Wiley Inc., Princeton, 1963, 1991.
[172]	I. Agricola, Foundations of differential geometry {II}, Wiley Classics Library, Wiley Inc., Princeton, 1969, 1996.
[173]	W. Kopczyński, An anisotropic universe with torsion, Phys. Lett. 43 A (1973), 63-64.
[174]	B. Kostant, On differential geometry and homogeneous spaces {II}, Proc. N. A. S. 42 (1956), 354-357.
[175]	I. Agricola, A cubic {D}irac operator and the emergence of {E}uler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), 447-501.
[176]	I. Agricola, Dirac cohomology for the Cubic Dirac operator, in: Joseph, Anthony (ed.) et al., Studies in memory of Issai Schur. Basel: Birkhäuser. Prog. Math. 210 (2003), 69-93.
[177]	B. Kostant and 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), Progress in Mathematics, Birkhäuser, 2003.
[178]	A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003), 125-160.
[179]	O. Kowalski and L. Vanhecke, Four-dimensional naturally reductive homogeneous spaces, Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Semin. Mat., Torino, Fasc. Spec., 223-232 (1983).
[180]	I. Agricola, A generalization of a theorem on naturally reductive homogeneous spaces, Proc. Am. Math. Soc. 91 (1984), 433-435.
[181]	I. Agricola, Classification of five-dimensional naturally reductive spaces, Math. Proc. Camb. Philos. Soc. 97 (1985), 445-463.
[182]	O. Kowalski and S. Wegrzynowski, A classification of \(5\)-dimensional \(\vphi\)-symmetric spaces, Tensor, N. S. 46 (1987), 379-386.
[183]	E. Kreyszig, Differential geometry, Dover Publ., inc., New York, 1991, unabridged republication of the 1963 printing.
[184]	G. Landweber, Harmonic spinors on homogeneous spaces, Repr. Theory 4 (2000), 466–473.
[185]	J.-L. Li and S.-T. Yau, Existence of supersymmetric string theory with torsion, Journ. Diff. Geom. 70 (2005), 143-182 and hep-th/0411136.
[186]	A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9.
[187]	I. Agricola, Spin manifolds, Killing spinors and universality of the Hijazi inequality, Lett. Math. Phys. 13 (1987), 331-344.
[188]	I. Agricola, Les spineurs-twisteurs sur une variété spinorielle compacte, C. R. Acad. Sci. Paris Série I 306 (1988), 381-385.
[189]	D. Lüst and S. Theisen, Lectures on String Theory, Springer-Verlag, 1989.
[190]	D. Martelli, J. Sparks, S.-T. Yau, Sasaki-Einstein Manifolds and Volume Minimisation, hep-th/0603021.
[191]	F. Martín Cabrera, Special almost Hermitian geometry, J. Geom. Phys. 55 (2005), 450-470.
[192]	F. Martín Cabrera, M. D. Monar Hernandez, A. F. Swann, Classification of \(G_2\)-structures, Journ. Lond. Math. Soc. II. Ser. 53 (1996), 407-416.
[193]	F. Martín Cabrera and A. F. Swann, Almost Hermitian structures and quaternionic geometries, Differ. Geom. Appl. 21 (2004), 199-214.
[194]	Y. McKenzie Wang, Parallel spinors and parallel forms, Ann. Glob. Anal. Geom. 7 (1989), 59-68.
[195]	S. Mehdi and R. Zierau, Principal Series Representations and Harmonic Spinors, to appear in Adv. Math. (preprint at http://www.math.okstate.edu/~zierau/papers.html).
[196]	J. Michelson, A. Strominger, The geometry of (super) conformal quantum mechanics, Commun. Math. Phys. 213 (2000), 1-17.
[197]	V. Miquel, The volume of small geodesic balls for a metric connection, Compositio Math. 46 (1982), 121-132.
[198]	I. Agricola, Volumes of geodesic balls and spheres associated to a metric connection with torsion, Contemp. Math. 288 (2001), 119-128.
[199]	S. Nagai, Naturally reductive Riemannian homogeneous structure on a homogeneous real hypersurface in a complex space form, Boll. Unione Mat. Ital., VII. Ser., A 9 (1995) 391-400.
[200]	I. Agricola, Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms, Geom. Dedicata 62 (1996) 253-268.
[201]	I. Agricola, The classification of naturally reductive homogeneous real hypersurfaces in complex projective space, Arch. Math. 69 (1997), 523-528.
[202]	P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations, Asian J. Math. 6 (2002) 481-504.
[203]	I. Agricola, On nearly-Kähler geometry, Ann. Global Anal. Geom. 22 (2002), 167-178.
[204]	P. Nurowski, M. Przanowski, A four-dimensional example of Ricci-flat metric admitting almost-Kähler non-Kähler structure, ESI preprint 477, 1997; Class. Quant. Grav. 16 (1999), L9-L16.
[205]	P. Nurowski, Distinguished dimensions for special Riemannian geometries, math.DG/0601020.
[206]	R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30.
[207]	R. Penrose, Spinors and torsion in general relativity, Found. of Phys. 13 (1983), 325-339.
[208]	Y. S. Poon and A. F. Swann, Potential functions of HKT spaces, Class. Quant. Grav. 18 (2001), 4711-4714.
[209]	I. Agricola, Superconformal symmetry and HyperKähler manifolds with torsion, Commun. Math. Phys. 241 (2003), 177-189.
[210]	Chr. Puhle, The Killing equation with higher order potentials, Ph. D. Thesis, Humboldt-Universität zu Berlin, 2006/07.
[211]	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. · JFM 38.0674.03
[212]	M. Rocek, Modified Calabi-Yau manifolds with torsion, in: Yau, Shing-Tung (ed.), Essays on mirror manifolds. Cambridge, MA: International Press. 480-488 (1992).
[213]	M. L. Ruggiero and A. Tartaglia, Einstein–Cartan theory as a theory of defects in space-time, Amer. J. Phys. 71 (2003), 1303-1313.
[214]	S. Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematical Series, 201. Jon Wiley & Sons, 1989.
[215]	I. Agricola, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.
[216]	I. Agricola, A tour of exceptional geometry, Milan J. Math. 71 (2003), 59-94.
[217]	K. Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 39 (1987), 677-684.
[218]	N. Schoemann, Almost hermitian structures with parallel torsion, PhD thesis, Humboldt-Universität zu Berlin, 2006.
[219]	J. A. Schouten, Der Ricci-Kalkül, Grundl. der math. Wissensch. 10, Springer-Verlag Berlin, 1924.
[220]	I. Agricola, Klassifizierung der alternierenden Größen dritten Grades in \(7\) Dimensionen, Rend. Circ. Mat. Palermo 55 (1931), 137-156.
[221]	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.
[222]	J. Simons, On the transitivity of holonomy systems, Ann. Math. 76 (1962), 213-234.
[223]	S. Slebarski, The {D}irac operator on homogeneous spaces and representations of reductive {L}ie groups {I}, Amer. J. Math. 109 (1987), 283-301.
[224]	S. Slebarski, The {D}irac operator on homogeneous spaces and representations of reductive {L}ie groups {II}, Amer. J. Math. 109 (1987), 499-520.
[225]	P. Spindel, A. Sevrin, W. Troost, and A. van Proeyen, Extended supersymmetric \(\sigma\)-models on group manifolds, Nucl. Phys. B 308 (1988), 662-698.
[226]	S. Sternberg, Lie algebras, Lecture notes, 1999.
[227]	A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986), 253-284.
[228]	K. Strubecker, Differentialgeometrie. {II}: {T}heorie der {F}lächenmetrik, Sammlung Göschen, W. de Gruyter, Berlin, 1969.
[229]	A. F. Swann, Aspects symplectiques de la géométrie quaternionique, C. R. Acad. Sci., Paris, Sér. I 308 (1989), 225-228.
[230]	I. Agricola, HyperKähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421-450.
[231]	, I. Agricola, 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.
[232]	J. Tafel, A class of cosmological models with torsion and spin, Acta Phys. Polon. B6 (1975), 537-554.
[233]	S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349-379.
[234]	W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468.
[235]	A. Trautman, On the structure of the Einstein-Cartan equations, Symp. Math. 12 (1973), 139-162.
[236]	I. Agricola, Spin and torsion may avert gravitational singularities, Nature Phys. Sci. 242 (1973) 7.
[237]	I. Agricola, Gauge and optical aspects of gravitation, Class. Quant. Grav. 16 (1999), 157-175.
[238]	F. Tricerri and L. Vanhecke, Homogeneous structures on {R}iemannian manifolds, London Math. Soc. Lecture Notes Series, vol. 83, Cambridge Univ. Press, Cambridge, 1983.
[239]	I. Agricola, Geodesic spheres and naturally reductive homogeneous spaces, Riv. Mat. Univ. Parma 10 (1984), 123-131.
[240]	I. Agricola, Naturally reductive homogeneous spaces and generalized Heisenberg groups, Comp. Math. 52 (1984), 389-408.
[241]	I. Vaisman, On locally conformal almost kähler manifolds, Israel J. Math. 24 (1976), 338-351.
[242]	I. Agricola, Locally conformal Kähler manifolds with parallel Lee form, Rend. Math. Roma 12 (1979), 263-284.
[243]	P. Van Nieuwenhuizen, Supergravity, Phys. Rep. 68 (1981), 189-398.
[244]	J. Vanžura, One kind of multisymplectic structures on \(6\)-manifolds, Proceedings of the Colloquium on Differential Geometry, Debrecen, 2000, 375-391 (2001).
[245]	M. Verbitsky, HyperKähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math. 6 (2002), 679-712.
[246]	A. B. Vinberg and 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.
[247]	M. Y. Wang, Parallel spinors and parallel forms, Ann. Global Anal. Geom. 7 (1989), 59-68.
[248]	M. Y. Wang and W. Ziller, On normal homogeneous {E}instein manifolds, Ann. Sci. Éc. Norm. Sup., \(4^{e}\) série 18 (1985), 563-633.
[249]	R. Westwick, Real trivectors of rank seven, Linear Multilinear Algebra 10 (1981), 183-204.
[250]	F. Witt, Generalised \(G_2\)-manifolds, to appear in Comm. Math. Phys., math.DG/0411642.
[251]	I. Agricola, Special metrics and Triality, math.DG/0602414.
[252]	B. de Witt, H. Nicolai and N.P. Warner, The embedding of gauged \(n=8\) supergravity into \(d=11\) supergravity, Nucl. Phys. B 255 (1985), 29.
[253]	J. A. Wolf, Partially harmonic spinors and representations of reductive Lie groups, Journ. Funct. Anal. 15 (1974), 117-154.
[254]	S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp\`re equations. I, Comm. pure and appl. Math. 31 (1978), 339-411.
[255]	W. Ziller, The {J}acobi equation on naturally reductive compact {R}iemannian homogeneous spaces, Comment. Math. Helv. 52 (1977), 573-590.
[256]	W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann. 259 (1982), 351-358.