Citations

From References: 0From 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-Kahler 8-manifolds. (Englishsummary)

J. Geom. Phys. 91 (2015), 146–162.

A quaternion-Kahler manifold is a 4n-dimensional smooth manifold (n ≥ 2) endowedwith 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 ofdifferential forms on a quaternion-Kahler manifold admitting an isometric circle actionand establish the existence of direct links between quaternion-Kahler geometry indimension eight, half-flat geometry in dimension six, and G2 geometry in dimensionseven. Gabriel Eduard Vılcu

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Soc., 2001, pp. 70–89. MR187100142. D. Conti, S. Salamon, Generalized Killing spinors in dimension 5, Trans. Amer.

Math. Soc. 359 (2007) 5319–5343. MR232703243. V. Apostolov, S. Salamon, Kahler reduction of metrics with holonomy G2, Comm.

Math. Phys. 246 (2004) 43–61. MR204489044. S.M. Salamon, Almost parallel structures, in: Global Differential Geometry: The

Mathematical Legacy of Alfred Gray, in: Contemp. Math., vol. 288, AmericanMath. Soc., 2001, pp. 162–181. MR1871007

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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: 5From 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 manifoldM is defined by a pair consistingof a 3-form and a 4-form which satisfy some compatibility conditions. From a half-flatSU(3)-structure one can reconstruct a metric g with holonomy group G2 via Hitchinflow. In the simplest case when M is a nearly-Kahler space, the metric g is the conicalmetric 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-Kahler structure.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 afinite-dimensional symplectic quotient.

The matrix algebra is used also to simplify and interpret the Hitchin flow equationsfor the associated cohomogeneity one Ricci-flat metrics with holonomy G2. In the finalpart of the paper, the authors present results of a numerical study of Hitchin’s evolutionequations for S3×S3. They recover metrics that behave asymptotically locally conically.

Dmitriı Vladimir Alekseevsky

References

1. Apostolov, V., Salamon, S.: Kahler reduction 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

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4. Bedulli, L., Vezzoni, L,: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57(4),1125–1146 (2007) MR2287296

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6. Brandhuber, A., Gomis, J., Gubser, S., Gukov, S.: Gauge theory at large N and newG2 holonomy metrics. Nuclear Phys. B 611(1–3), 179–204 (2001) MR1857379

7. Bryant, R.: Non-embedding and non-extension results in special holonomy. Themany facets of geometry. Oxford Cniversity Press, Oxford (2010) MR2681703

8. Bryant, R., Salamon, S.: On the construction of some complete metrics with excep-tional holonomy. Duke Math. J. 58(3), 829–850(1989) MR1016448

9. Butruille, J.-P.: Espacede twisteurs d’une variete presque hermitienne de dimension6. Ann. Inst. Fourier (Grenoble) 57(5), 1451–485 (2007) MR2364136

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11. Calabi, R: Construction and properties of some 6-dimensional almost complexmanifolds. 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. Differentialgeometry, Valencia,: 115–133. World Sci. Publ. River Edge, NJ (2001). (2002)

14. Chong, Z., Cvetic, M., Gibbons, G.: H. L., C. Pope, P. Wagner, General metricsof G2 Coholonomy and contraction limits. Nuclear Phys. B 638(3), 459–482 (2002)MR1922498

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16. Cvetic, M., Gibbons, G., Lu, H., Pope. C.: Supersymmetric M3-branes and G2

manifolds. Nuclear Phys. B 620(1–2), 3–28 (2002) MR187314417. Cvetic, M., Gibbons, G., Lu, H., Pope, C.: A G2 unification of the deformed and

resolved conifolds. Phys. Lett. B 534(1–4), 172–180 (2002) MR191311818. Cvetic, M., Gibbons, G.. Lu, H., Pope, C.: Cohomogeneity one manifolds of Spin(7)

and G2 holonomy. Phys. Rev. D 65(3), 29 (2002) (no. 10, 106004) MR191903519. Cvetic, M., Gibbons, G., Lu, H., Pope, C.: Orientifolds and slumps in G2 and

Spin(7) metrics. Ann. Physics 310(2), 265–301 (2004) MR204474020. Dancer, A.: McKenzie Wang, Painleve expansions, cohomogeneity one metrics and

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equations. Comm. Math. Phys. 260(1), 75–115 (2005) MR217599022. Eschenburg, J.: McKenzie Wang, The initial value problem for cohomogeneity one

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Pura Appl. (4) 132, 19–45 (1982) MR069603725. 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. Symplecticgeometry, 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 Veranderlichen.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

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possible with no attempt to correct errors.

c© Copyright American Mathematical Society 2016

Citations

From References: 1From 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 conformallyEinstein nor half conformally flat.” Michael Roch Jablonski

References

1. V. Apostolov, D.M.J. Calderbank, P. Gauduchon, Ambitoric geometry I: Einsteinmetrics and extremal ambikahler structures, arXiv:1302.6975.

2. R. Bach, Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung desKrummungstensorbegriffs, 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. Berard Bergery, Les espaces homogenes riemanniens 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 gradientsteady Ricci solitons, arXiv:1107.4591.

7. A. Derzinski, Self-dual Kahler manifolds 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

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12. T.B. Madsen, A. Swann, Invariant strong KT geometry on four-dimensional solvableLie 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. Plebanski, Non-vacuum twisting type-N metrics, Class. QuantumGravity 18 (2001) 341–351. MR1807623

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16. G. Tian, J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent.Math. 160 (2005) 357–415. MR2138071

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possible with no attempt to correct errors.

c© Copyright American Mathematical Society 2016

Citations

From References: 5From 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-flatmanifolds (Mn, 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 notionof generalised Killing spinor [C. Bar, P. Gauduchon and A. Moroianu, Math. Z. 249

(2005), no. 3, 545–580; MR2121740]. This means that the derivative of the spinor isgiven by Clifford multiplication with the second fundamental form A of the submanifoldN : ∇Xψ = 1

2A(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 geom-etry 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 thereal 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: 20From Reviews: 3

MR2044890 (2005e:53070) 53C29 53C55 53D20

Apostolov, Vestislav (3-QU) ; Salamon, Simon (I-TRNP)

Kahler reduction 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) withholonomy equal to G2 are based on the inclusions of the subgroups SU(2) and SU(3) inthe exceptional group. They can be broadly divided into two types: the first is associatedto SU(2) geometry, the father of all constructions being the one that led to the completemetrics 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 theproperties of special Hermitian geometry in dimension six. A neat formalisation can befound in [N. J. Hitchin, in Global differential geometry: the mathematical legacy of AlfredGray (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 toa 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 classicalcomplete examples mentioned above are studied in great detail.

This paper more generally investigates holonomyG2 manifolds Y with a Killing vectorfield acting freely. When the quotient N6 = Y/S1 is Kahler, a second local isometryappears 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 endowedwith a one-parameter family of Kahler forms essentially depending on the fibres’ size inthe original fibration Y →N , an important feature for string theory purposes [see, e.g.,M. Cvetic et 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 ahyper-Kahler four-manifold M with an additional Ricci-flat structure, one can recoverexplicitly irreducible metrics with exceptional holonomy. The method generalises aconstruction first studied by physicists [G. W. Gibbons et al., Nuclear Phys. B 623

(2002), no. 1-2, 3–46; MR1883449] when M is hyper-Kahler. Indeed, this case too isdiscussed by the authors and linked to the complex Monge–Ampere operator.

Uniform and comprehensive presentation is given of some crucial examples, i.e. torusbundles corresponding to compact quotients of nilpotent Lie groups. Interestingly, thereare reasons for arguing that these manifolds, seen as hypersurfaces inside Y , are in

some sense dual to the quotients N obtained by Kahler reduction, as hinted in [S.Chiossi and S. M. Salamon, in Differential geometry, Valencia, 2001, 115–133, WorldSci. Publishing, River Edge, NJ, 2002; MR1922042]. Simon G. Chiossi

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3. Acharya, B., Witten, E.: Chiral fermions from manifolds of G2 holonomy. Availableat arXiv: hep-th/0109152

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12. Brandhuber, A., Gomis, J., Gubser, S.S., Gukov, S.: Gauge theory at large N andnew G2 holonomy metrics. Nucl. Phys. B 611, 179–204 (2001) MR1857379

13. Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)MR0916718

14. Bryant, R.L., Salamon, S.: On the construction of some complete metrics withexceptional holonomy. Duke Math. J. 58, 829–850 (1989) MR1016448

15. Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G2 structures. In:Differential Geometry, Valencia 2001. River Edge, NJ: World Sci. Publishing, 2002,pp. 115–133 MR1922042

16. Cvetic, M., Gibbons, G.W., Lu, H., Pope, C.N.: Almost special holonomy in typeIIA and M-theory. Nucl. Phys. B 638, 186–206 (2002) MR1924090

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20. Gibbons, G.W., Lu, H., Pope, C.N., Stelle, K.S.: Supersymmetric domain walls frommetrics of special holonomy. Nucl. Phys. B 623, 3–46 (2002) MR1883449

21. Hitchin, N.J.: Stable forms and special metrics. In: Global Differential Geometry:The Mathematical Legacy of Alfred Gray, Contemp. Math. 288, Providence RI: Am.Math. Soc., 2001, pp. 70–89 MR1871001

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28. Nurowski, P., Przanowski, M.: A four-dimensional example of Ricci flat metricadmitting almost Kahler non-Kahler structure. Classical Quant. Grav. 16, L9–L13(1999) MR1682582

29. Salamon, S.: Riemannian geometry and holonomy groups. Pitman Research Notesin Mathematics Vol. 201, London-New York: Longman, 1989 MR1004008

30. Santillan, O.P.: A construction of G2 holonomy spaces with torus symmetry. Nucl.Phys. B 660, 169–193 (2003) MR1983318

31. Sekigawa, K.: On some compact Einstein almost-Kahler manifolds. J. Math. Soc.Japan 36, 677–684 (1987) MR0905633

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 2005, 2016

Citations

From References: 9From 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 groupsthat 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 viaLie groups, but there is also some material on compact examples arising from theresolution 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)metric on R8. The article is based on a lecture of the author, and retains much of thatstyle, with the emphasis on indicating methods and main ideas. Andrew Swann

References

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2. V. Apostolov and S. Salamon, Kahler reduction of metrics with holonomy G2. Toappear 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

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8. M. Berger, Sur les groupes d’holonomie homogene des varietes a connexion affine etdes varietes riemanniennes. Bull.Soc.Math. France 83 (1955), 279–330. MR0079806

9. M. Berger, Les varietes riemanniennes homogene normales simplement conexesa courbure strictemente positive. Ann. Sc. Norm. Sup. Pisa 15 (1961), 179–246.MR0133083

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11. R. Bryant, Metrics with exceptional holonomy. Annals of Math. 126 (1987), 525–576.MR0916718

12. R. Bryant and R. Harvey, Submanifolds in hyper-Kahler geometry. J. Amer. Math.Soc. 2 (1989), 1–31. MR0953169

13. R. Bryant and S. Salamon, On the construction of some complete metrics withexceptional 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, Metriques kahleriennes et fibres holomorphes. Ann. Ec. Norm. Sup. 12

(1979), 269–294. MR054321817. G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis, and G. Zoupanos,

Non-Kahler string backgrounds and their five torsion classes. Nuclear Phys. B 652

(2003), 5–34. MR195932418. 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. MR192204220. M. Cvetic, G. W. Gibbons, H. Lu, and C. N. Pope. New complete noncompact Spin

(7) manifolds. Nuclear Phys. B 620 (2002), 29–54. MR187314521. I. G. Dotti and A. Fino, Abelian hypercomplex 8-dimensional nilmanifolds. Ann.

Global Anal. Geom. 18 (2000), 47–59. MR173952422. B. Fantechi and L. Gottsche, Orbifold cohomology for global quotients.

math.AG/0104207.23. M. Fernandez and A. Gray, Riemannian manifolds with structure group G2. Ann.

Mat. Pura Appl. 32 (1982), 19–45. MR069603724. M. Fernandez and L. Ugarte, Dolbeault cohomology for G2-structures. Geom. Dedi-

cata 70 (1998), 57–86. MR161234625. 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 Kahler V-manifolds of dimension four.

In Classification of algebraic and analytic manifolds (Katata, 1982), volume 39 ofProgr. Math.. Birkhauser, 1983, 71–250. MR0728609

28. G. W. Gibbons, H. Lu, C. N. Pope, and K. S. Stelle, Supersymmetric domain wallsfrom 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. MR104089330. E. Goldstein and S. Prokushkin, Geometric model for complex non-Kahler manifolds

with SU(3) structure. hep-th/0212307.31. A. Gray, Weak holonomy groups. Math. Z. 123 (1971), 290–300. MR029353732. A. Gray, The structure of nearly Kahler manifolds. Math. Ann. 223 (1976), 233–248.

MR041796533. 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.

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(2000), 547–576. MR186373336. N. Hitchin, Stable forms and special metrics. In Global Differential Geometry: The

Mathematical Legacy of Alfred Gray, volume 288 of Contemp. Math.. AmericanMath. 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 acomplex 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 anda 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. MR081231247. M. Parton, Private communication.48. R. Reyes-Carrion, A generalization of the notion of instanton. Differ. Geom. Appl.

8 (1998), 1–20. MR160154649. S. Salamon, Riemannian geometry and holonomy groups. Pitman Research Notes in

Mathematics 201. Longman, 1989. MR100400850. S. Salamon, Quaternion-Kahler geometry. 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. InGlobal 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: 51From 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 theintrinsic torsion τ1 of an SU(3)-structure on a 6-manifold and those of the intrinsictorsion τ2 of a G2-structure on a 7-manifold. Thus they begin with a quite detaileddescription of the torsion of an SU(3)-structure. This is done by refining the theory forU(3)-structures. Then they relate, in a purely algebraic sense, the torsion τ1 with thetorsion τ2. The relations obtained in this way are enough to study situations in whichthe inclusion SU(3) ⊆ G2 is fixed. Such a case, called “static”, is given by a product ofan 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 afamily of SU(3)-structures parametrized by t varying in some real interval I and theG2-structure is defined on a warped product manifold M × I fibring over I. This lastsituation is called “dynamic”, because the inclusion SU(3) ⊆ G2 varies from point topoint. They introduce the notion of half-flat SU(3) structure to interpret the corre-sponding 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 inG2 on M × I, for some interval I. The authors also provide additional examples of in-complete 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 prod-uct 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-manifoldsendowed with an appropriate structure. Thus they provide an explicit description of τ2as a function of τ1 and a curvature 2-form. In particular, they study the case of thecanonical circle bundle over a Kahler 3-fold. When the holonomy of the bundle spacereduces to G2, the base is a symplectic manifold with a type of generalized Calabi-Yaugeometry that is described in terms of τ1.{For the collection containing this paper see MR1919551}

Francisco Martın Cabrera

c© Copyright American Mathematical Society 2003, 2016

Citations

From References: 2From 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 Poincare polynomial P (t).When the Euler characteristic χ 6= 0, the authors introduce an invariant

φ2 :=P ′′(−1)

2χ− 1

2n2,

which is additive with respect to products of manifolds. Its value is computed explicitlyfor homogeneous spaces G/H with compact and simple G and nonzero Euler charac-teristic. In particular, it is shown that φ2 is non-negative and is 0 precisely when theuniversal cover of G/H is a product of 2-spheres.

The authors then specialize to the case when G/H is a 2n-dimensional irreducibleRiemannian symmetric space. In this case, if it is either Hermitian symmetric or if allroots of G have the same length, then φ2 = 1

6(h− 2)n, where h is the Coxeter number of

G. In the case of quaternionic symmetric spaces of real dimension 4m, then φ2 = 13m

2.The motivation for considering the invariant φ2 comes from studying the topology of

closed hyper-Kahler and quaternionic Kahler manifolds 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 basisof comparison between topological invariants of the model spaces and those of generalpositive quaternionic Kahler manifolds.{For the collection containing this paper see MR1423152}

McKenzie Y. Wang

c© Copyright American Mathematical Society 1998, 2016

Citations

From References: 20From 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 haverecently 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 giverise to infinitely many homotopy types of compact Einstein manifolds in each dimensioncongruent to 3 (mod 4).

A 3-Sasakian manifold S admits a triple of orthonormal Killing fields defining a localaction of SU(2). The quotient M is a quaternionic Kahler orbifold and S is said to beregular 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 paperunder review examines how these are manifested on S.

The main results are as follows. For any compact 3-Sasakian manifold of dimension4n+ 3, the odd Betti numbers bi vanish for i < 2n+1. This corresponds to the vanishingof 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 areonly finitely many such S in any given dimension; (2) if b2(S) > 0 then S is determineduniquely; (3)

∑nk=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 the3-Sasakian case. The Poincare polynomials of the homogeneous 3-Sasakian manifoldsare computed to illustrate these relations.

The authors also show that a compact regular 3-Sasakian manifold in dimension 15or 19 with b4 = 0 is a sphere or a real projective space. This follows from the strongresult that a compact quaternionic Kahler manifold in dimension 12 or 16 with b4 =0 is isometric to quaternionic projective space, which was announced by LeBrun andSalamon, but first proved in this paper.

If S is 3-Sasakian, then S1× S is a hypercomplex manifold which is locally conformalto hyper-Kahler without being hyper-Kahler, and any such manifold N which satisfiesa regularity condition is of this form. The authors close the paper by indicating how theabove results on the topology of S apply to N . Andrew Swann

c© Copyright American Mathematical Society 1998, 2016

Citations

From References: 26From Reviews: 6

MR1367278 (97f:32042) 32J27 19L10 53C25

Salamon, S. M. (4-OX)

On the cohomology of Kahler and hyper-Kahler manifolds.

Topology 35 (1996), no. 1, 137–155.

The main results of this paper are linear relations for the Hodge numbers of a compactKahler manifold with zero first Chern class and, most interestingly, for the Betti numbersof a compact hyper-Kahler manifold.

Riemann-Roch techniques are used to calculate the Chern number c1cn−1 of a com-pact Kahler manifold of real dimension 2n in terms of the Euler characteristics χp =∑

(−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 satisfythe mirror symmetry relation hp,q = hn−p,q, as is the case for a hyper-Kahler manifold,the expression for c1cn−1 may be written in terms of the Betti numbers.

For a hyper-Kahler manifold, one has c1 = 0, and hence the above gives a constraint onthe 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 ofM is divisible by 32. A. Beauville [J. Differential Geom. 18 (1983), no. 4, 755–782 (1984);MR0730926] constructed two families K [m] and Km of compact hyper-Kahler manifoldsout of K3 surfaces. By examining these examples, the author shows the sharpness ofthe above result for the Euler characteristic.

The Betti number constraint obtained from the vanishing of c1cn−1 may be interpretedas saying that either the Euler characteristic χ is zero, which can occur when the deRham decomposition of M contains a flat factor, or a topological invariant φ2 is acertain constant multiple of the dimension. The invariant φ2 is constructed from thePoincare polynomial of M and is defined for any compact even-dimensional manifold ofnonzero Euler characteristic in such a way as to be additive with respect to Cartesianproducts. If S is a complex surface, write S[m] for the Hilbert scheme of 0-dimensionalsubschemes of length m on S. Then it is shown that φ2(S[m]) = mφ2(S), wheneverχ(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: 61From Reviews: 11

MR1288469 (95k:53059) 53C25 32L25

LeBrun, Claude (1-SUNYS) ; Salamon, Simon (4-OX)

Strong rigidity of positive quaternion-Kahler manifolds.

Invent. Math. 118 (1994), no. 1, 109–132.

The only known examples of quaternionic-Kahler manifolds with positive scalar curva-ture are symmetric, and in dimension four and eight it has been proved that nonsym-metric 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-Kahler 4n-manifolds M of positive scalar curvature.Furthermore, it is proved that such geometries are simply connected and that thesecond 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 Fanocontact manifold and therefore, via a result of Wisniewski, is very restricted.

The Betti numbers are seen to satisfy some surprising relations, and a new proof thatnonsymmetric 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: 114From 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 metricson open sets of R7 and R8 with holonomy group equal toG2 and Spin(7) was established.

In this paper the authors explicitly construct three distinct complete metrics withholonomy equal to G2, one complete metric with holonomy equal to Spin(7), and otherincomplete examples. First, they demonstrate that there is a family of metrics on spinbundles over space forms (i.e., spaces of constant sectional curvature) whose holonomyequals G2. Two cases are discussed in some detail: M = H3 and M = S3. The lattermetric is shown to be complete.

Next, the authors use similar techniques to study the Λ2− bundle of anti-self-dual two-

forms over a self-dual Einstein 4-manifold M . For M = S4 and M = CP2 they obtaincomplete 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-manifoldM admits a Spin(7) holonomy metric. The only complete example is obtained whenM = S4. However, the recent construction of 4-dimensional orbifolds with self-dualEinstein metrics of positive scalar curvature yields infinitely many orbifold metrics withholonomy G2 or Spin(7). Krzysztof Galicki

c© Copyright American Mathematical Society 1990, 2016

Citations

From References: 167From 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 JohnWiley &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 geometriesdistinguished by reduction of the holonomy group. All symmetric spaces were classifiedby E. Cartan and his classification supplies a long list of holonomy groups. On the otherhand, it is a result of Berger that, if M is an oriented simply-connected n-dimensionalRiemannian manifold which is neither locally a product nor symmetric, then its holo-nomy group is equal to one of SO(n), U(n2 ), SU(n2 ), Sp(n4 ) ·Sp(1), Sp(n4 ), G2, Spin7. Allthese groups correspond to special geometries: the general Riemannian geometry, theKahler geometry, the Kahler Ricci-flat geometry, the quaternionic Kahler geometry, thehyper-Kahler geometry, and G2 and Spin7 exceptional geometries, respectively.

After a short introduction of the notation and definitions in the first two chapters, eachgeometry is studied separately. Chapters 3 and 4 take us into the vast realm of Kahlergeometry. The theory of Kahler manifolds is in a sense an intersection of Riemanniangeometry with complex geometry, just as SO(n,R) and GL(n,C) intersect in the unitarygroup U(n). In particular, the decomposition of the Riemann and the Kahler curvaturetensors into their irreducible components is determined by representation theory andthe relation between complex symplectic and Kahler structures is presented.

Chapter 5 is devoted to the study of the symmetric case in which the classificationcan be easily carried out. Some standard techniques in the representation theory areintroduced in the following chapter and applied later.

Chapter 7 brings us to the 4-dimensional case where both the bundle Λ2T ∗M andthe 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 thisparticular area and the author gives a very complete and up-to-date account of therecent advances.

Chapters 8 and 9 are concerned with special Kahler manifolds and quaternionicmanifolds. Many standard results on the existence of the Ricci-flat metrics on variousKahler manifolds are reviewed. We also learn here about some new constructions ofhyper-Kahler and quaternionic Kahler metrics using generalized Marsden-Weinsteinquotients.

Finally, the last three chapters introduce us to some very recent constructions by theauthor and Bryant of metrics with exceptional holonomy groups. After proving Berger’sclassification theorem a detailed exposition of G2 and Spin7 geometries is given. Inparticular, we learn that the total spaces of Λ2

−S4 and Λ2

−CP2 admit complete Ricci-flat metrics with G2-holonomy and that the total space of the spin bundle over S4 hasa complete Ricci-flat metric with Spin7-holonomy.

The book under review is an expanded version of a lecture course the author gave atOxford in 1986. Even though it does not attempt to give an exhaustive survey of thesubject, it does cover a range of topics in Riemannian geometry in which Lie groupsplay a fundamental role. The selection of topics and examples is excellent and makesthe reading very enjoyable. Krzysztof Galicki

c© Copyright American Mathematical Society 1990, 2016

Citations

From References: 51From 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 Kahler manifold is by definition a 4n-dimensional Riemannian manifoldwhose 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 aquaternionic Kahler manifold. The corresponding notion of self-duality is interpreted interms of holomorphic geometry on a twistor space. In particular, self-dual connectionsare constructed on various vector bundles over quaternionic projective spaces.

Yang Lian Pan

c© Copyright American Mathematical Society 1989, 2016