J. Geom. Phys. 91 (2015), 146–162. a Quaternion-Kähler Manifold Is a 4N

Citations From References: 0 From Reviews: 0

MR3327056 (Review) 53C26 Gambioli, A. [Gambioli, Andrea] (3-DAWS-M); Nagatomo, Y. [Nagatomo, Yasuyuki] (J-MEIJ2); Salamon, S. [Salamon, Simon M.] (4-LNDKC) Special geometries associated to quaternion-K¨ahler8-manifolds. (English summary) J. Geom. Phys. 91 (2015), 146–162.

A quaternion-Kählermanifold is a 4n-dimensional smooth manifold (n ≥ 2) endowed with a Riemannian metric g with holonomy contained in the subgroup Sp(n)Sp(1) of SO(4n). In this interesting paper under review, the authors develop a calculus of differential forms on a quaternion-Kählermanifold admitting an isometric circle action and establish the existence of direct links between quaternion-Kählergeometry in dimension eight, half-flat geometry in dimension six, and G2 geometry in dimension seven. Gabriel Eduard Vˆılcu

References

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Citations From References: 5 From Reviews: 0

MR3132080 (Review) 53C25 53C29 53C44 53D20 Madsen, Thomas Bruun (4-LNDKC); Salamon, Simon [Salamon, Simon M.] (4-LNDKC) Half-ﬂat structures on S3 × S3. (English summary) Ann. Global Anal. Geom. 44 (2013), no. 4, 369–390.

A half-flat SU(3)-structure on a 6-dimensional manifold M is defined by a pair consisting of a 3-form and a 4-form which satisfy some compatibility conditions. From a half-flat SU(3)-structure one can reconstruct a metric g with holonomy group G2 via Hitchin flow. In the simplest case when M is a nearly-Kählerspace, the metric g is the conical metric associated with M. The authors describe left-invariant half-flat SU(3)-structures on the Lie group S3 × S3 using the representation theory of the group SO(4) and matrix algebra. In particular, it is proven that on this group there exists unique left-invariant nearly-Kählerstructure. The authors give a description of the moduli space of left-invariant half-flat SU(3)- structures in terms of matrix algebra. It is proven that essentially the moduli space is a finite-dimensional symplectic quotient. The matrix algebra is used also to simplify and interpret the Hitchin flow equations for the associated cohomogeneity one Ricci-flat metrics with holonomy G2. In the final part of the paper, the authors present results of a numerical study of Hitchin’s evolution equations for S3 ×S3. They recover metrics that behave asymptotically locally conically. Dmitri˘ıVladimir Alekseevsky

References 1. Apostolov, V., Salamon, S.: Kählerreduction of metrics with holonomy G2. Comm. Math. Phys. 246(1), 43–61 (2004) MR2044890 2. Atiyah, M., Maldacena, J., Vafa, C.: An M-theory flop as a large N duality. Strings, branes, and M-theory. J. Math. Phys. 42(7), 3209–3220 (2001) MR1840340 3. Bar, C.: Real Killing spinors and holonomy. Comm. Math. Phys. 154(3), 509–521 (1993) MR1224089 4. Bedulli, L., Vezzoni, L,: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57(4), 1125–1146 (2007) MR2287296 5. Brandhuber, A.: G2 holonomy spaces; from invariant three-forms. Nuclear Phys. B 629(1–3), 393–416 (2002) MR1903163 6. Brandhuber, A., Gomis, J., Gubser, S., Gukov, S.: Gauge theory at large N and new G2 holonomy metrics. Nuclear Phys. B 611(1–3), 179–204 (2001) MR1857379 7. Bryant, R.: Non-embedding and non-extension results in special holonomy. The many facets of geometry. Oxford Cniversity Press, Oxford (2010) MR2681703 8. Bryant, R., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58(3), 829–850(1989) MR1016448 9. Butruille, J.-P.: Espacede twisteurs d’une variétépresque hermitienne de dimension 6. Ann. Inst. Fourier (Grenoble) 57(5), 1451–485 (2007) MR2364136 10. Butruille J.-P.: Homogeneous nearly Kahler manifolds. Handbook of pseudo- Riemannian geometry and supersymmetry. In: IRMA Lect. Math. Theor. Phys., vol 16, pp. 399–423. Eur. Math. Soc., Zürich (2010) MR2681596 11. Calabi, R: Construction and properties of some 6-dimensional almost complex manifolds. Trans. Amer. Math. Soc. 87, 407–438 (1958) MR0130698 12. Chiossi, S., Fino, A.: Conformally parallel G2 structures on a class of solvmanifolds. Math. Z. 252(4), 825–848 (2006) MR2206629 13. Chiossi, S., Salamon, S., The intrinsic torsion of G2 and G2 structures. Differential geometry, Valencia,: 115–133. World Sci. Publ. River Edge, NJ (2001). (2002) 14. Chong, Z., Cvetiˇc,M., Gibbons, G.: H. L., C. Pope, P. Wagner, General metrics of G2 Coholonomy and contraction limits. Nuclear Phys. B 638(3), 459–482 (2002) MR1922498 15. Conti, D.: SU(3)-holonomy metrics from nilpotent Lie groups. arXiv:l 108.2450 [math.DG] 16. Cvetiˇc,M., Gibbons, G., Lü,H., Pope. C.: Supersymmetric M3-branes and G2 manifolds. Nuclear Phys. B 620(1–2), 3–28 (2002) MR1873144 17. Cvetiˇc,M., Gibbons, G., Lü,H., Pope, C.: A G2 unification of the deformed and resolved conifolds. Phys. Lett. B 534(1–4), 172–180 (2002) MR1913118 18. Cvetiˇc,M., Gibbons, G.. Lü,H., Pope, C.: Cohomogeneity one manifolds of Spin(7) and G2 holonomy. Phys. Rev. D 65(3), 29 (2002) (no. 10, 106004) MR1919035 19. Cvetiˇc,M., Gibbons, G., Lü, H., Pope, C.: Orientifolds and slumps in G2 and Spin(7) metrics. Ann. Physics 310(2), 265–301 (2004) MR2044740 20. Dancer, A.: McKenzie Wang, Painlevéexpansions, cohomogeneity one metrics and exceptional holonomy. Comm. Anal. Geom. 12(4), 887–926 (2004) MR2104080 21. Dancer, A.: McKenzie Wang, Superpotentials and the cohomogeneity one Einstein equations. Comm. Math. Phys. 260(1), 75–115 (2005) MR2175990 22. Eschenburg, J.: McKenzie Wang, The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10(1), 109–137 (2000) MR1758585 23. Ferapontov, E., Huard, B., Zhang, A.: On the central quadric ansatz: integrable models and Painlevéreductions. J. Phys. A 45(10), 195–204 (2012) MR2924500 24. Fernández,M., Gray, A.: Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. (4) 132, 19–45 (1982) MR0696037 25. Hitchin, N.: Stable forms and special metrics. Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000). In: Contemp. Math., vol. 288, pp 70–89. Amer. Math. Soc., Providence (2001) MR1871001 26. E. Lerman, R. Montgomery, R. Sjamaar, Examples of singular reduction. Symplectic geometry, 127–155, London Math. Soc. Lecture Note Ser., 192, Cambridge Univ. Press, Cambridge, 1993. MR1297133 27. Reichel, W.: Uber¨ die Trilinearen Alternierenden Formen in 6 und 7 Veränderlichen. Dissertation, Greifswald (1907) 28. Reidegeld, F.: Exceptional holonomy and Einstein metrics constructed from Aloff- Wallach spaces. Proc. London Math. Soc. 102(6), 1127–1160 (2011) MR2806102 29. Reyes Carrión,R.: A Generalization Of The Notion Of Instanton. Differential Geom. Appl. 8(1), 1–20 (1998) MR1601546 30. Salamon, S.: Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics Series, 201. Longman Scientific & Technical, Harlow (1989) ISBN: 0-582-01767-X (copublished in the United States with Wiley, New York) MR1004008 31. Schulte-Hengesbach, F.: Half-flat structures on Lie groups. PhD thesis, Hamburg (2010) 32. Westwick, R.: Real trivectors of rank seven. Linear Multilinear Algebra 10(3), 183–204 (1981) MR0630147 Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 2016

Citations From References: 1 From Reviews: 0

MR3066792 (Review) 22E25 53C25 Abbena, Elsa (I-TRIN); Garbiero, Sergio (I-TRIN); Salamon, Simon [Salamon, Simon M.] (4-LNDKC) Bach-ﬂat Lie groups in dimension 4. (English, French summaries) C. R. Math. Acad. Sci. Paris 351 (2013), no. 7-8, 303–306.

Summary: “We establish the existence of solvable Lie groups of dimension 4 and left- invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally ﬂat.” Michael Roch Jablonski

References

1. V. Apostolov, D.M.J. Calderbank, P. Gauduchon, Ambitoric geometry I: Einstein metrics and extremal ambikählerstructures, arXiv:1302.6975. 2. R. Bach, Zur Weylschen Relativitätstheorieund der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Z. 9 (1921) 110–135. MR1544454 3. M.L. Barberis, Hypercomplex structures on four-dimensional Lie groups, Proc. Am. Math. Soc. 125 (1997) 1043–1054. MR1353375 4. L. BérardBergery, Les espaces homogènesriemanniens de dimension 4, in: Rie- mannian Geometry in Dimension 4, Paris, 1978/1979, in: Textes Math., vol. 3, CEDIC, Paris, 1981, pp. 40–60. MR0769130 5. A.L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1986. MR2371700 6. H.-D. Cao, G. Catino, Q. Chen, C. Mantegazza, L. Mazzieri, Bach-flat gradient steady Ricci solitons, arXiv:1107.4591. 7. A. Derziński,Self-dual Kählermanifolds and Einstein manifolds of dimension four, Compos. Math. 49 (1983) 405–433. MR0707181 8. V. De Smedt, S. Salamon, Anti-self-dual metrics on Lie groups, Contemp. Math. 308 (2002) 63–75. MR1955629 9. G.R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differ. Geom. 3 (1969) 309–349. MR0261487 10. C.N. Kozameh, E.T. Newman, K.P. Tod, Conformal Einstein spaces, Gen. Relativ. Gravit. 17 (1985) 343–352. MR0788800 11. M. Listing, Conformal Einstein spaces in N-dimensions, Ann. Glob. Anal. Geom. 20 (2001) 183–197. MR1857177 12. T.B. Madsen, A. Swann, Invariant strong KT geometry on four-dimensional solvable Lie groups, J. Lie Theory 21 (2011) 55–70. MR2797819 13. J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329. MR0425012 14. P. Nurowski, J.F. Plebański,Non-vacuum twisting type-N metrics, Class. Quantum Gravity 18 (2001) 341–351. MR1807623 15. H.-J. Schmidt, Non-trivial solutions of the Bach equation exist, Ann. Phys. 41 (1984) 435–436. MR0800322 16. G. Tian, J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005) 357–415. MR2138071 Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 2016

Citations From References: 5 From Reviews: 0

MR2264396 (2007m:53054) 53C29 53C10 Conti, Diego (I-SNS); Salamon, Simon (I-TRNP) Reduced holonomy, hypersurfaces and extensions. (English summary) Int. J. Geom. Methods Mod. Phys. 3 (2006), no. 5-6, 899–912.

The Lie groups Sp(n), SU(n),G2, Spin(7) are holonomy groups of Riemannian Ricci-flat manifolds (M n, g). By a result of M. Y. K. Wang [Ann. Global Anal. Geom. 7 (1989), no. 1, 59–68; MR1029845] they also stabilise a spinor field ψ, rendering it parallel. Interpreting the last condition on a hypersurface N of M gives rise to the notion of generalised Killing spinor [C. Bär,P. Gauduchon and A. Moroianu, Math. Z. 249 (2005), no. 3, 545–580; MR2121740]. This means that the derivative of the spinor is given by Clifford multiplication with the second fundamental form A of the submanifold 1 N: ∇X ψ = 2 A(X)ψ. The paper shows that a generalised Killing spinor defines a so-called ‘hypo’ G- structure, whose intrinsic torsion is A and hence a symmetric tensor. The authors investigate the cases where the ambient dimension is n = 8, 7, 6 and determine the explicit relations with the differential forms governing the special geometry induced on N. The latter are known as coclosed G2, half-flat and 5-dimensional hypo-structures respectively, Vice versa, starting from the hypersurface-to-be they construct a Ricci-flat manifold containing it, assuming A satisfies the Codazzi equations. The procedure works in the real analytic category. As examples, all five-dimensional nilpotent Lie algebras with an invariant hypo- structure are classified, for which see also [D. Conti and S. Salamon, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5319–5343]. Simon G. Chiossi c Copyright American Mathematical Society 2007, 2016

Citations From References: 20 From Reviews: 3

MR2044890 (2005e:53070) 53C29 53C55 53D20 Apostolov, Vestislav (3-QU); Salamon, Simon (I-TRNP) K¨ahlerreduction of metrics with holonomy G2. (English summary) Comm. Math. Phys. 246 (2004), no. 1, 43–61.

The techniques known to produce 7-dimensional Riemannian manifolds (Y, g) with holonomy equal to G2 are based on the inclusions of the subgroups SU(2) and SU(3) in the exceptional group. They can be broadly divided into two types: the first is associated to SU(2) geometry, the father of all constructions being the one that led to the complete metrics of [R. L. Bryant and S. M. Salamon, Duke Math. J. 58 (1989), no. 3, 829–850; MR1016448; see also G. W. Gibbons, D. N. Page and C. N. Pope, Comm. Math. Phys. 127 (1990), no. 3, 529–553; MR1040893]. The alternative approach originates from the properties of special Hermitian geometry in dimension six. A neat formalisation can be found in [N. J. Hitchin, in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70–89, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001; MR1871001]. There are fewer results in the opposite direction, that is, reducing the G2 data on Y to a lower dimensional set-up, as in the massive [M. F. Atiyah and E. Witten, Adv. Theor. Math. Phys. 6 (2002), no. 1, 1–106; MR1992874]. There, S1-quotients of the classical complete examples mentioned above are studied in great detail. This paper more generally investigates holonomy G2 manifolds Y with a Killing vector field acting freely. When the quotient N 6 = Y/S1 is Kähler,a second local isometry appears naturally, providing a means to further reduce to a manifold of dimension four. The latter is complex with a holomorphic symplectic form. Strikingly, it is also endowed with a one-parameter family of Kählerforms essentially depending on the fibres’ size in the original fibration Y → N, an important feature for string theory purposes [see, e.g., M. Cvetiˇcet al., Nuclear Phys. B 638 (2002), no. 1-2, 186–206; MR1924090]. The reader is then shown how the whole process is actually reversible: starting from a hyper-Kählerfour-manifold M with an additional Ricci-flat structure, one can recover explicitly irreducible metrics with exceptional holonomy. The method generalises a construction first studied by physicists [G. W. Gibbons et al., Nuclear Phys. B 623 (2002), no. 1-2, 3–46; MR1883449] when M is hyper-Kähler. Indeed, this case too is discussed by the authors and linked to the complex Monge–Ampèreoperator. Uniform and comprehensive presentation is given of some crucial examples, i.e. torus bundles corresponding to compact quotients of nilpotent Lie groups. Interestingly, there are reasons for arguing that these manifolds, seen as hypersurfaces inside Y , are in some sense dual to the quotients N obtained by Kählerreduction, as hinted in [S. Chiossi and S. M. Salamon, in Differential geometry, Valencia, 2001, 115–133, World Sci. Publishing, River Edge, NJ, 2002; MR1922042]. Simon G. Chiossi

References

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Citations From References: 9 From Reviews: 0

MR2120916 (2006b:53061) 53C29 53C26 Salamon, Simon (I-TRNP) A tour of exceptional geometry. (English summary) Milan J. Math. 71 (2003), 59–94.

This article discusses geometries in dimensions 6, 7 and 8 based on structure groups that give rise to irreducible Ricci-ﬂat metrics, in particular SU(3), G2 and Spin(7), respectively. The emphasis is on methods to describe explicit examples, mostly via Lie groups, but there is also some material on compact examples arising from the resolution of the Calabi conjecture [S. T. Yau, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411; MR0480350] and resolutions of singularities on ﬁnite quotients of tori [D. D. Joyce, Compact manifolds with special holonomy, Oxford Univ. Press, Oxford, 2000; MR1787733]. A series of non-compact examples are obtained from an explicit Spin(7) 8 metric on R . The article is based on a lecture of the author, and retains much of that style, with the emphasis on indicating methods and main ideas. Andrew Swann

References 1. E. Abbena, S. Garbiero, and S. Salamon, Almost Hermitian geometry of 6-dimensional nilmanifolds. Ann. Sc. Norm. Sup. 30 (2001), 147–170. MR1882028 2. V. Apostolov and S. Salamon, Kählerreduction of metrics with holonomy G2. To appear in Comm. Math. Phys.. cf. MR2044890 3. M. Atiyah and E. Witten, M-theory dynamics on a manifold of G2 holonomy. Adv. Theor. Math. Phys. 6 (2003), 1–106. MR1992874 4. M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London A 362 (1978), 425–461. MR0506229 5. C. Bär, Real Killing spinors and holonomy. Comm. Math. Phys. 154 (1993) 509–521. MR1224089 6. A. Beauville, VariétésKählériennesdont la première class de Chern est nulle. J. Differ. Geom. 18 (1983), 755–782. MR0730926 7. F. Belgun and A. Moroianu, Nearly Kähler6-manifolds with reduced holonomy. Ann. Global Anal. Geom. 19 (2001), 307–319. MR1842572 8. M. Berger, Sur les groupes d’holonomie homogènedes variétésàconnexion affine et des variétésriemanniennes. Bull.Soc.Math. France 83 (1955), 279–330. MR0079806 9. M. Berger, Les variétésriemanniennes homogènenormales simplement conexes àcourbure strictemente positive. Ann. Sc. Norm. Sup. Pisa 15 (1961), 179–246. MR0133083 10. A. Brandhuber, J. Gomis, S. S. Gubser, and S. Gukov, Gauge theory at large N and new G2 holonomy metrics. Nuclear Phys. B 611 (2001), 179–204. MR1857379 11. R. Bryant, Metrics with exceptional holonomy. Annals of Math. 126 (1987), 525–576. MR0916718 12. R. Bryant and R. Harvey, Submanifolds in hyper-Kählergeometry. J. Amer. Math. Soc. 2 (1989), 1–31. MR0953169 13. R. Bryant and S. Salamon, On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58 (1989), 829–850. MR1016448 14. R. L. Bryant, Submanifolds and special structures on the octonians. J. Differ. Geom. 17 (1982), 185–232. MR0664494 15. F. M. Cabrera, M. D. Monar, and A. F. Swann, Classification of G2-structures. London Math. Soc. 53 (1996), 407–416. MR1373070 16. E. Calabi, Métriqueskählérienneset fibrésholomorphes. Ann. Ec. Norm. Sup. 12 (1979), 269–294. MR0543218 17. G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis, and G. Zoupanos, Non-Kählerstring backgrounds and their five torsion classes. Nuclear Phys. B 652 (2003), 5–34. MR1959324 18. W. Chen and Y. Ruan, A new cohomology theory of orbifold. math.AG/0004129. 19. S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G2 structures. In Differential Geometry, Valencia 2001. World Scientific, 2002, 115–133. MR1922042 20. M. Cvetic,ˇ G. W. Gibbons, H. Lü,and C. N. Pope. New complete noncompact Spin (7) manifolds. Nuclear Phys. B 620 (2002), 29–54. MR1873145 21. I. G. Dotti and A. Fino, Abelian hypercomplex 8-dimensional nilmanifolds. Ann. Global Anal. Geom. 18 (2000), 47–59. MR1739524 22. B. Fantechi and L. Göttsche, Orbifold cohomology for global quotients. math.AG/0104207. 23. M. Fernándezand A. Gray, Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. 32 (1982), 19–45. MR0696037 24. M. Fernándezand L. Ugarte, Dolbeault cohomology for G2-structures. Geom. Dedi- cata 70 (1998), 57–86. MR1612346 25. T. Friedmann and E. Witten, Unification scale, proton decay, and manifolds of G2 holonomy. hep-th/0211269. 26. T. Friedrich, I. Kath, A. Moroianu, and U. Semmelmann, On nearly parallel G2 structures. J. Geom. Phys. 27 (1998), 155–177. 27. A. Fujiki, On primitively symplectic compact KählerV-manifolds of dimension four. In Classification of algebraic and analytic manifolds (Katata, 1982), volume 39 of Progr. Math.. Birkhäuser,1983, 71–250. MR0728609 28. G. W. Gibbons, H. Lü,C. N. Pope, and K. S. Stelle, Supersymmetric domain walls from metrics of special holonomy. hep-th/0108191. 29. G. W. Gibbons, D. N. Page, and C. N. Pope, Einstein metrics on S3,R3, and R4 bundles. Comm. Math. Phys. 127 (1990), 529–553. MR1040893 30. E. Goldstein and S. Prokushkin, Geometric model for complex non-Kähler manifolds with SU(3) structure. hep-th/0212307. 31. A. Gray, Weak holonomy groups. Math. Z. 123 (1971), 290–300. MR0293537 32. A. Gray, The structure of nearly Kählermanifolds. Math. Ann. 223 (1976), 233–248. MR0417965 33. A. Gray and L. Hervella, The sixteen classes of almost Hermitian manifolds. Ann. Mat. Pura Appl. 282 (1980), 1–21. 34. R. Harvey and H. B. Lawson, Calibrated geometries. Acta Math. 148 (1982), 47–157. MR0666108 35. N. Hitchin, The geometry of three-forms in six dimensions. J. Differ. Geom. 55 (2000), 547–576. MR1863733 36. N. Hitchin, Stable forms and special metrics. In Global Differential Geometry: The Mathematical Legacy of Alfred Gray, volume 288 of Contemp. Math.. American Math. Soc., 2001, 70–89. MR1871001 37. N. Hitchin, The Dirac operator. In M. R. Bridson and S. M. Salamon, editors, Invitations to Geometry and Topology. Oxford University Press, 2002. MR1967744 38. N. J. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55 (1989), 59–126. MR0887284 39. D. D. Joyce, Compact 8-manifolds with holonomy Spin (7). Invent. Math. 123 (1996), 507–552. MR1383960 40. D. D. Joyce, Compact Riemannian 7-manifolds with holonomy G2, I. J. Differ. Geom. 43 (1996), 291–328. MR1424428 41. D. D. Joyce, Compact manifolds with special holonomy. Oxford University Press, 2000. MR1787733 42. P. Z. Kobak and A. Swann, Quaternionic geometry of a nilpotent variety. Math. Ann. 297 (1993), 747–764. MR1245417 43. B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple lie group. Amer. J. Math. 81 (1959), 973–1032. MR0114875 44. A. Kovalov, Twisted connected sums and special Riemannian holonomy. math.DG/0012189. 45. J. Lauret, Geometric structures on nilpotent Lie groups: on their classification and a distinguished compatible metric. math.DG/0210143. 46. N. R. O’Brian and J. H. Rawnsley. Twistor spaces. Ann. Global Anal. Geom. 3 (1985), 29–58. MR0812312 47. M. Parton, Private communication. 48. R. Reyes-Carrion, A generalization of the notion of instanton. Differ. Geom. Appl. 8 (1998), 1–20. MR1601546 49. S. Salamon, Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics 201. Longman, 1989. MR1004008 50. S. Salamon, Quaternion-Kählergeometry. In C. LeBrun and M. Wang, editors, Essays on Einstein Manifolds, volume VI of Surveys in Differential Geometry. International Press, 1999, 83–121. MR1798608 51. S. Salamon, Almost product structures. In Global Differential Geometry: The Math- ematical Legacy of Alfred Gray, volume 288 of Contemp. Math. American Math. Soc., 2001, 162–181. MR1871007 52. S. M. Salamon, Spinors and cohomology. Rend. Mat. Univ. Torino 50 (1992), 393– 410. MR1261451 53. I. M. Singer and J. A. Thorpe, The curvature of 4-dimensional einstein spaces. In Global Analysis (Papers in Honor of K. Kodaira). Univ. Tokyo Press, 1969, 355–365. MR0256303 54. J. A. Wolf, The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math 120 (1968), 59–148. MR0223501 55. J. A. Wolf and A. Gray, Homogeneous spaces defined by lie group automorphisms I. J. Differ. Geom. 2 (1968), 77–114. MR0236328 Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 2006, 2016

Citations From References: 51 From Reviews: 10

MR1922042 (2003g:53030) 53C10 53C29 Chiossi, Simon (I-GENO); Salamon, Simon (I-TRNP) The intrinsic torsion of SU(3) and G2 structures. (English summary) Diﬀerential geometry, Valencia, 2001, 115–133, World Sci. Publ., River Edge, NJ, 2002.

In this article the authors analyse the relationship between the components of the intrinsic torsion τ1 of an SU(3)-structure on a 6-manifold and those of the intrinsic torsion τ2 of a G2-structure on a 7-manifold. Thus they begin with a quite detailed description of the torsion of an SU(3)-structure. This is done by refining the theory for U(3)-structures. Then they relate, in a purely algebraic sense, the torsion τ1 with the torsion τ2. The relations obtained in this way are enough to study situations in which the inclusion SU(3) ⊆ G2 is fixed. Such a case, called “static”, is given by a product of an interval or circle with a Riemannian 6-manifold endowed with an SU(3)-structure. They subsequently examine the case in which the 6-manifold M is equipped with a family of SU(3)-structures parametrized by t varying in some real interval I and the G2-structure is defined on a warped product manifold M × I fibring over I. This last situation is called “dynamic”, because the inclusion SU(3) ⊆ G2 varies from point to point. They introduce the notion of half-flat SU(3) structure to interpret the corresponding evolution equations discussed by N. J. Hitchin [in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70–89, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001; MR1871001]. Thus it is proved that, for a half- flat almost Hermitian 6-manifold M, there exists a metric with holonomy contained in G2 on M × I, for some interval I. The authors also provide additional examples of incomplete metrics with holonomy G2 of the type discovered by G. W. Gibbons et al. [Nuclear Phys. B 623 (2002), no. 1-2, 3–46; MR1883449] by considering warped product manifolds M × I, where M is one of the three 6-dimensional nilmanifolds with b1 = 4. Finally, the authors consider G2-structures defined on circle bundles over 6-manifolds endowed with an appropriate structure. Thus they provide an explicit description of τ2 as a function of τ1 and a curvature 2-form. In particular, they study the case of the canonical circle bundle over a Kähler3-fold. When the holonomy of the bundle space reduces to G2, the base is a symplectic manifold with a type of generalized Calabi-Yau geometry that is described in terms of τ1. {For the collection containing this paper see MR1919551} Francisco Mart´ınCabrera c Copyright American Mathematical Society 2003, 2016

Citations From References: 2 From Reviews: 0

MR1423174 (98a:53071) 53C35 53C25 57T15 Fino, A. [Fino, Anna] (I-TRIN); Salamon, S. (4-OX) Observations on the topology of symmetric spaces. Geometry and physics (Aarhus, 1995), 275–286, Lecture Notes in Pure and Appl. Math., 184, Dekker, New York, 1997.

Let M be a 2n-dimensional closed oriented manifold with Poincarépolynomial P (t). When the Euler characteristic χ 6= 0, the authors introduce an invariant P 00(−1) 1 φ := − n2, 2 2χ 2 which is additive with respect to products of manifolds. Its value is computed explicitly for homogeneous spaces G/H with compact and simple G and nonzero Euler characteristic. In particular, it is shown that φ2 is non-negative and is 0 precisely when the universal cover of G/H is a product of 2-spheres. The authors then specialize to the case when G/H is a 2n-dimensional irreducible Riemannian symmetric space. In this case, if it is either Hermitian symmetric or if all 1 roots of G have the same length, then φ2 = 6 (h − 2)n, where h is the Coxeter number of 1 2 G. In the case of quaternionic symmetric spaces of real dimension 4m, then φ2 = 3 m . The motivation for considering the invariant φ2 comes from studying the topology of closed hyper-Kählerand quaternionic Kählermanifolds by the second author [Topol- ogy 35 (1996), no. 1, 137–155; MR1367278; Invent. Math. 67 (1982), no. 1, 143–171; MR0664330] and by C. LeBrun and the second author [Invent. Math. 118 (1994), no. 1, 109–132; MR1288469]. The results in the paper under review therefore serve as a basis of comparison between topological invariants of the model spaces and those of general positive quaternionic Kählermanifolds. {For the collection containing this paper see MR1423152} McKenzie Y. Wang c Copyright American Mathematical Society 1998, 2016 Citations From References: 20 From Reviews: 4

MR1413621 (98e:53075) 53C25 53C30 Galicki, Krzysztof (1-NM-MS); Salamon, Simon (4-OX) Betti numbers of 3-Sasakian manifolds. (English summary) Geom. Dedicata 63 (1996), no. 1, 45–68.

3-Sasakian manifolds are certain Einstein manifolds of dimension 4n + 3 which have recently received renewed attention. For example, C. P. Boyer, Galicki and B. M. Mann [Bull. London Math. Soc. 28 (1996), no. 4, 401–408; MR1384830] showed that these give rise to infinitely many homotopy types of compact Einstein manifolds in each dimension congruent to 3 (mod 4). A 3-Sasakian manifold S admits a triple of orthonormal Killing fields defining a local action of SU(2). The quotient M is a quaternionic Kählerorbifold and S is said to be regular if M is a manifold. In the regular case, S is a bundle over M with fibre RP(3) or S3. There are several known restrictions on the Betti numbers of M and the paper under review examines how these are manifested on S. The main results are as follows. For any compact 3-Sasakian manifold of dimension 4n + 3, the odd Betti numbers bi vanish for i < 2n + 1. This corresponds to the vanishing of the odd Betti numbers on M when S is regular, but the proof for S is rather different. The remaining results are only for compact regular 3-Sasakian manifolds S: (1) there are only finitely many such S in any given dimension; (2) if b2(S) > 0 then S is determined Pn uniquely; (3) k=1 k(n + 1 − k)(n + 1 − 2k)b2k(S) = 0. These three results are closely tied to the results for M obtained by C. R. LeBrun and Salamon [Invent. Math. 118 (1994), no. 1, 109–132; MR1288469], but it is striking that relation (3) is simpler in the 3-Sasakian case. The Poincarépolynomials of the homogeneous 3-Sasakian manifolds are computed to illustrate these relations. The authors also show that a compact regular 3-Sasakian manifold in dimension 15 or 19 with b4 = 0 is a sphere or a real projective space. This follows from the strong result that a compact quaternionic Kählermanifold in dimension 12 or 16 with b4 = 0 is isometric to quaternionic projective space, which was announced by LeBrun and Salamon, but first proved in this paper. If S is 3-Sasakian, then S1 × S is a hypercomplex manifold which is locally conformal to hyper-Kählerwithout being hyper-Kähler,and any such manifold N which satisfies a regularity condition is of this form. The authors close the paper by indicating how the above results on the topology of S apply to N. Andrew Swann c Copyright American Mathematical Society 1998, 2016

Citations From References: 26 From Reviews: 6 MR1367278 (97f:32042) 32J27 19L10 53C25 Salamon, S. M. (4-OX) On the cohomology of K¨ahlerand hyper-K¨ahlermanifolds. Topology 35 (1996), no. 1, 137–155.

The main results of this paper are linear relations for the Hodge numbers of a compact Kählermanifold with zero first Chern class and, most interestingly, for the Betti numbers of a compact hyper-Kählermanifold. Riemann-Roch techniques are used to calculate the Chern number c1cn−1 of a compact Kählermanifold of real dimension 2n in terms of the Euler characteristics χp = P(−1)ihp,i, a result previously obtained by A. S. Libgober and J. W. Wood [J. Differ- ential Geom. 32 (1990), no. 1, 139–154; MR1064869]. When the Hodge numbers satisfy the mirror symmetry relation hp,q = hn−p,q, as is the case for a hyper-Kählermanifold, the expression for c1cn−1 may be written in terms of the Betti numbers. For a hyper-Kählermanifold, one has c1 = 0, and hence the above gives a constraint on the Betti numbers. One consequence is that the Betti number in the middle dimension, the Euler characteristic and the signature are necessarily even unless the dimension of M is divisible by 32. A. Beauville [J. Differential Geom. 18 (1983), no. 4, 755–782 (1984); [m] MR0730926] constructed two families K and Km of compact hyper-Kählermanifolds out of K3 surfaces. By examining these examples, the author shows the sharpness of the above result for the Euler characteristic. The Betti number constraint obtained from the vanishing of c1cn−1 may be interpreted as saying that either the Euler characteristic χ is zero, which can occur when the de Rham decomposition of M contains a flat factor, or a topological invariant φ2 is a certain constant multiple of the dimension. The invariant φ2 is constructed from the Poincarépolynomial of M and is defined for any compact even-dimensional manifold of nonzero Euler characteristic in such a way as to be additive with respect to Cartesian products. If S is a complex surface, write S[m] for the Hilbert scheme of 0-dimensional [m] subschemes of length m on S. Then it is shown that φ2(S ) = mφ2(S), whenever χ(S) 6= 0. Appropriate variations of these invariants are considered for (T 4)[m]. Andrew Swann c Copyright American Mathematical Society 1997, 2016

Citations From References: 61 From Reviews: 11

MR1288469 (95k:53059) 53C25 32L25 LeBrun, Claude (1-SUNYS); Salamon, Simon (4-OX) Strong rigidity of positive quaternion-K¨ahler manifolds. Invent. Math. 118 (1994), no. 1, 109–132.

The only known examples of quaternionic-Kählermanifolds with positive scalar curvature are symmetric, and in dimension four and eight it has been proved that nonsymmetric examples do not exist. In this paper it is proved that for any n, there are, up to isometries and rescalings, only finitely many quaternionic-Kähler4n-manifolds M of positive scalar curvature. Furthermore, it is proved that such geometries are simply connected and that the second homotopy group (a) vanishes iff M is the quaternionic projective plane, (b) equals Z iff M is the Grassmannian of complex 2-planes in C2n+2, (c) is finite with 2- torsion otherwise. The proof applies Mori theory to the twistor space which is a Fano contact manifold and therefore, via a result of Wi´sniewski,is very restricted. The Betti numbers are seen to satisfy some surprising relations, and a new proof that nonsymmetric examples do not exist in dimension 8 is given [see also Y. S. Poon and S. M. Salamon, J. Differential Geom. 33 (1991), no. 2, 363–378; MR1094461; C. R. LeBrun, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1205–1208; MR0955010]. Henrik Pedersen c Copyright American Mathematical Society 1995, 2016

Citations From References: 114 From Reviews: 31

MR1016448 (90i:53055) 53C25 53C57 Bryant, Robert L. (1-DUKE); Salamon, Simon M. (4-OX) On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58 (1989), no. 3, 829–850.

The article under review is a continuation of the earlier work of Bryant [Ann. of Math. (2) 126 (1987), no. 3, 525–576; MR0916718], where the existence of Riemannian metrics 7 8 on open sets of R and R with holonomy group equal to G2 and Spin(7) was established. In this paper the authors explicitly construct three distinct complete metrics with holonomy equal to G2, one complete metric with holonomy equal to Spin(7), and other incomplete examples. First, they demonstrate that there is a family of metrics on spin bundles over space forms (i.e., spaces of constant sectional curvature) whose holonomy 3 3 equals G2. Two cases are discussed in some detail: M = H and M = S . The latter metric is shown to be complete. 2 Next, the authors use similar techniques to study the Λ− bundle of anti-self-dual two- forms over a self-dual Einstein 4-manifold M. For M = S4 and M = CP2 they obtain complete metrics with G2 holonomy on the total space of the bundle. The last part of the paper is devoted to the study of the Spin(7) holonomy case. It is shown that the total space of the spin bundle of a self-dual Einstein 4-manifold M admits a Spin(7) holonomy metric. The only complete example is obtained when M = S4. However, the recent construction of 4-dimensional orbifolds with self-dual Einstein metrics of positive scalar curvature yields inﬁnitely many orbifold metrics with holonomy G2 or Spin(7). Krzysztof Galicki c Copyright American Mathematical Society 1990, 2016

Citations From References: 167 From Reviews: 13 MR1004008 (90g:53058) 53C25 32-02 53-02 53C35 53C55 Salamon, Simon (4-OX) FRiemannian geometry and holonomy groups. Pitman Research Notes in Mathematics Series, 201. Longman Scientiﬁc &Technical, Harlow; copublished in the United States with John Wiley &Sons, Inc., New York, 1989. viii+201 pp. $47.95. ISBN 0-582-01767-X

The book provides a very elegantly modern exposition of special Riemannian geometries distinguished by reduction of the holonomy group. All symmetric spaces were classified by E.´ Cartan and his classification supplies a long list of holonomy groups. On the other hand, it is a result of Berger that, if M is an oriented simply-connected n-dimensional Riemannian manifold which is neither locally a product nor symmetric, then its holo- n n n n nomy group is equal to one of SO(n), U( 2 ), SU( 2 ), Sp( 4 ) · Sp(1), Sp( 4 ), G2, Spin7. All these groups correspond to special geometries: the general Riemannian geometry, the Kählergeometry, the Kähler Ricci-flat geometry, the quaternionic Kählergeometry, the hyper-Kählergeometry, and G2 and Spin7 exceptional geometries, respectively. After a short introduction of the notation and definitions in the first two chapters, each geometry is studied separately. Chapters 3 and 4 take us into the vast realm of Kähler geometry. The theory of Kählermanifolds is in a sense an intersection of Riemannian geometry with complex geometry, just as SO(n, R) and GL(n, C) intersect in the unitary group U(n). In particular, the decomposition of the Riemann and the Kählercurvature tensors into their irreducible components is determined by representation theory and the relation between complex symplectic and Kählerstructures is presented. Chapter 5 is devoted to the study of the symmetric case in which the classification can be easily carried out. Some standard techniques in the representation theory are introduced in the following chapter and applied later. Chapter 7 brings us to the 4-dimensional case where both the bundle Λ2T ∗M and the Weyl curvature tensor decompose into two halves. This decomposition leads to self- dual metrics and the twistor correspondence. There have been many new results in this particular area and the author gives a very complete and up-to-date account of the recent advances. Chapters 8 and 9 are concerned with special Kählermanifolds and quaternionic manifolds. Many standard results on the existence of the Ricci-flat metrics on various Kählermanifolds are reviewed. We also learn here about some new constructions of hyper-Kählerand quaternionic Kählermetrics using generalized Marsden-Weinstein quotients. Finally, the last three chapters introduce us to some very recent constructions by the author and Bryant of metrics with exceptional holonomy groups. After proving Berger’s classification theorem a detailed exposition of G2 and Spin7 geometries is given. In 2 4 2 2 particular, we learn that the total spaces of Λ−S and Λ−CP admit complete Ricci- 4 flat metrics with G2-holonomy and that the total space of the spin bundle over S has a complete Ricci-flat metric with Spin7-holonomy. The book under review is an expanded version of a lecture course the author gave at Oxford in 1986. Even though it does not attempt to give an exhaustive survey of the subject, it does cover a range of topics in Riemannian geometry in which Lie groups play a fundamental role. The selection of topics and examples is excellent and makes the reading very enjoyable. Krzysztof Galicki c Copyright American Mathematical Society 1990, 2016 Citations From References: 51 From Reviews: 12

MR967469 (89k:58064) 58E15 53C55 Capria, M. Mamone [Mamone Capria, Marco] (I-FRNZ); Salamon, S. M. (4-OX) Yang-Mills ﬁelds on quaternionic spaces. Nonlinearity 1 (1988), no. 4, 517–530.

A quaternionic Kählermanifold is by definition a 4n-dimensional Riemannian manifold whose linear holonomy group is contained in the subgroup Sp(n) · Sp(1) of SO(4n). In this paper, the authors study the solutions of the Yang-Mills equations over a quaternionic Kählermanifold. The corresponding notion of self-duality is interpreted in terms of holomorphic geometry on a twistor space. In particular, self-dual connections are constructed on various vector bundles over quaternionic projective spaces. Yang Lian Pan c Copyright American Mathematical Society 1989, 2016