Bulletin of the American Mathematical Society

Author(s): M. Joachim; D. J. Wraith
Journal: Bull. Amer. Math. Soc. 45 (2008), 595-616.
MSC (2000): Primary 53C20
Posted: July 1, 2008
Retrieve article in: PDF

Abstract: Since their discovery by Milnor in 1956, exotic spheres have provided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres.

[Au]

T. Aubin, Métriques riemanniennes et courbure, J. Differential Geometry 4 (1970), 383-424. MR 0279731 (43:5452)

[Ad]

J.F. Adams, On the groups . IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)

[B]

A.L. Besse, Einstein Manifolds, Springer-Verlag, Berlin (2002). MR 867684 (88f:53087)

[BGK]

C.P. Boyer, K. Galicki, J. Kollár, Einstein metrics on spheres, Ann. of Math. (2) 162 no. 1 (2005), 557-580. MR 2178969 (2006j:53058)

[BGKT]

C.P. Boyer, K. Galicki, J. Kollár, E. Thomas, Einstein metrics on exotic spheres in dimensions , and , Experiment. Math. 14 no. 1 (2005), 59-64. MR 2146519 (2006a:53042)

[BGN1]

C.P. Boyer, K. Galicki, M. Nakamaye, On positive Sasakian geometry, Geom. Dedicata 101 (2003), 93-102. MR 2017897 (2005a:53072)

[BGN2]

C.P. Boyer, K. Galicki, M. Nakamaye, Sasakian geometry, homotopy spheres and positive Ricci curvature, Topology 42 (2003), 981-1002. MR 1978045 (2004c:53055)

[BH]

A. Back, W.Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987), 207-227. MR 906813 (88m:53073)

[Bk]

E. Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1-14. MR 0206972 (34:6788)

[Bl]

D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1874240 (2002m:53120)

[Br]

W. Browder, Surgery on simply-connected manifolds, Springer, Berlin, (1972). MR 0358813 (50:11272)

[Bu]

D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. MR 1835418 (2002e:53053)

[C]

J. Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geometry, 8 (1973), 623-628. MR 0341334 (49:6085)

[Ca]

E. Calabi, On Kähler manifolds with vanishing canonical class, Algebraic Geometry and Topology. A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, N.J., 1957, pp. 78-89. MR 0085583 (19:62b)

[Ce]

J. Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma_4=0)$ , Lecture Notes in Mathematics, 53, Springer-Verlag, Berlin-New York, 1968. MR 0229250 (37:4824)

[Ch]

I. Chavel, Riemannian geometry. A modern introduction, Cambridge Tracts in Mathematics, 108, Cambridge Univ. Press, 1993. MR 1271141 (95j:53001)

[D]

A. Dessai, On the topology of scalar-flat manifolds, Bull. London Math. Soc. 33 (2001), 203-209. MR 1815425 (2002b:53063)

[doC]

M. do Carmo, Riemannian geometry, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1138207 (92i:53001)

[DPR]

C. Durán, T Püttmann, A. Rigas, An infinite family of Gromoll-Meyer spheres, arXiv:math.DG/0610349.

[Du]

C. Durán, Pointed Wiedersehen metrics on exotic spheres and diffeomorphisms of , Geom. Dedicata 88 (2001), 199-210. MR 1877216 (2002i:57044)

[E1]

J.-H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982) no. 3, 469-480. MR 662603 (83i:53061)

[E2]

J.-H. Eschenburg, Freie isometrischen Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenreihe Math. Inst. Univ. Münster 32 (1984). MR 758252 (86a:53045)

[E3]

J.-H. Eschenburg, Local convexity and nonnegative curvature--Gromov's proof of the sphere theorem, Invent. Math. 84 (1986), 507-522. MR 837525 (87j:53080)

[E4]

J.-H. Eschenburg, Cohomology of biquotients, Manuscripta Math. 75 (1992) no. 2, 151-166. MR 1160094 (93e:57070)

[E5]

J.-H. Eschenburg, Almost positive curvature on the Gromoll-Meyer -sphere, Proc. Amer. Math. Soc. 130 no. 4, (2002), 1165-1167. MR 1873792 (2002i:53045)

[Eh]

P.E. Ehrlich, Metric deformations of curvature, I. Local convex deformations, Geom. Dedicata 5 (1976), 1-23. MR 0487886 (58:7482a)

[EK]

J.-H. Eschenburg, M. Kerin, Almost positive curvature on the Gromoll-Meyer -sphere, Proc. Amer. Math. Soc. posted on April 23, 2008, PII S0002-9939(08)09429-X (to appear in print).

[EKA]

A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Composito Math. 73 (1990), no. 1, 57-106. MR 1042454 (91f:58089)

[Et]

J. Etnyre, Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003), 81-107, Amer. Math. Soc., Providence, RI, 2003. MR 2024631 (2005b:53139)

[F]

A. Futaki, Scalar-flat closed manifolds not admitting positive scalar curvature metrics, Invent. Math. 112 (1993), 23-29. MR 1207476 (94f:53072)

[Fr]

M. Freedman, The topology of four dimensional manifolds, J. Differential Geom. 17 (1982) no. 3, 357-453. MR 679066 (84b:57006)

[FQ]

M. Freedman and F. Quinn, Topology of -manifolds, Princeton Mathematical Series 39, Princeton University Press, Princeton, N.J., 1990. MR 1201584 (94b:57021)

[G]

H. Geiges, Contact geometry, Chapter 5 of the Handbook of Differential Geometry, Volume 2, Elsevier/North-Holland, Amsterdam, 2006. MR 2194671 (2007c:53123)

[GL]

M. Gromov, H.B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), 423-434. MR 577131 (81h:53036)

[GM]

D. Gromoll, W.T. Meyer, An exotic sphere with nonnegative sectional curvature, Ann. of Math. (2) 100 (1974), 401-406. MR 0375151 (51:11347)

[GP]

K. Grove, P. Petersen, A pinching theorem for homotopy spheres, J. Amer. Math. Soc. 3 no.3, (1990), 671-677. MR 1049696 (91e:53040)

[GVWZ]

K. Grove, L. Verdiani, B. Wilking, W. Ziller, Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 159-170. MR 2244696 (2007h:53051)

[GW]

K. Grove, F. Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine Angew. Math. 487 (1997) 201-217. MR 1454266 (98d:53060)

[GZ1]

K. Grove, W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), 331-367. MR 1792298 (2001i:53047)

[GZ2]

K. Grove, W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), 619-646. MR 1923478 (2004b:53052)

[H]

F. Hirzebruch, Singularities and exotic spheres, Séminaire Bourbaki, 1966/67, Exp. 314, Textes des conférences, o.S., Paris: Institut Henri Poincaré 1967.

[HBJ]

F. Hirzebruch, T. Berger, R. Jung, Manifolds and modular forms, Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. MR 1189136 (94d:57001)

[HH]

W.-C. Hsiang, W.-Y. Hsiang, The degree of symmetry of homotopy spheres, Ann. of Math. (2) 89 (1969), 52-67. MR 0239621 (39:978)

[Hi]

N. Hitchin, Harmonic Spinors, Adv. in Math. 14 (1974), 1-55. MR 0358873 (50:11332)

[HM]

M. Hirsch, B. Mazur, Smoothings of piecewise linear manifolds, Annals of Mathematical Studies 80, Princeton Univ. Press, Princeton, NJ, 1974. MR 0415630 (54:3711)

[Hz]

H. Hernández-Andrade, A class of compact manifolds with positive Ricci curvature, Proc. Sympos. Pure Math. Vol. 27, 73-87, Amer. Math. Soc., Providence, RI, 1975.

MR 0380668 (52:1565)

[K]

J.L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Reg. Conf. Ser. Math., 57, Amer. Math. Soc., Providence, RI, 1985. MR 787227 (86h:53001)

[KM]

M. Kervaire, J. Milnor, Groups of homotopy spheres. I. Ann. of Math. (2) 77 (1963), 504-537. MR 0148075 (26:5584)

[KS]

R. Kirby, L. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematical Studies, No. 88, Princeton Univ. Press, Princeton, NJ, 1977. MR 0645390 (58:31082)

[KW1]

J.L. Kazdan, F.W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. 101 (1975), 317-331. MR 0375153 (51:11349)

[KW2]

J.L. Kazdan, F.W. Warner, A direct approach to the determination of Gaussian and scalar curvature functions, Invent. Math. 28 (1975), 227-230. MR 0375154 (51:11350)

[KW3]

J.L. Kazdan, F.W. Warner, Prescribing curvatures, Differential geometry, Proc. Sympos. Pure Math., Vol. XXVII Part 2, 309-319, Amer. Math. Soc., Providence, RI, 1975. MR 0394505 (52:15306)

[KZ]

V. Kapovitch, W. Ziller, Biquotients with singly generated rational cohomology, Geom. Dedicata 104 (2004), 149-160. MR 2043959 (2005e:22014)

[La]

T. Lance, Differentiable structures on manifolds, Surveys on Surgery Theory, Vol. 1, 73-104, Annals of Mathematical Studies, 145, Princeton Univ. Press, Princeton, NJ, 2000. MR 1747531 (2001d:57035)

[LM]

H.B. Lawson, M.-L. Michelsohn, Spin geometry, Princeton Math. Series, 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)

[Li]

A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9. MR 0156292 (27:6218)

[Lo]

J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), 655-683. MR 1307899 (95i:53042)

[M1]

J. Milnor, On manifolds homeomorphic to the -sphere, Ann. of Math. (2) 64 (1956), 399-405. MR 0082103 (18:498d)

[M2]

J. Milnor, Morse Theory, Annals of Mathematical Studies, 51, Princeton Univ. Press, Princeton, NJ, 1963. MR 0163331 (29:634)

[M3]

J. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology, 55-62, Princeton Univ. Press, Princeton, NJ, 1965. MR 0180978 (31:5208)

[M4]

J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematical Studies, 61, Princeton Univ. Press, Princeton, NJ, 1968. MR 0239612 (39:969)

[M5]

J. Milnor, Classification of -connected -dimensional manifolds and the discovery of exotic spheres, Surveys on Surgery Theory, Vol. 1, 25-30, Annals of Mathematical Studies 145, Princeton Univ. Press, Princeton, NJ, 2000. MR 1747528 (2001c:57002)

[MS]

J. Milnor, J. Stasheff, Characteristic Classes, Annals of Mathematical Studies, 76, Princeton Univ. Press, Princeton, NJ, 1974. MR 0440554 (55:13428)

[My]

S.B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401-404. MR 0004518 (3:18f)

[Na]

J. Nash, Positive Ricci curvature on fiber bundles, J. Differential Geom. 14 (1979), 241-254. MR 587552 (81k:53039)

[ON]

B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469. MR 0200865 (34:751)

[P]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer-Verlag, (1998). MR 1480173 (98m:53001)

[P1]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arxiv.org/math.DG/0211159

[P2]

G. Perelman, Ricci flow with surgery on three-manifolds, http://arxiv.org/ math.DG/0303109

[P3]

G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, http://arxiv.org/math.DG/0307245

[Po]

W.A. Poor, Some exotic spheres with positive Ricci curvature, Math. Ann. 216 (1975), 245-252. MR 0400110 (53:3945)

[PW]

P. Petersen, F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv: 0805.0812v1 [math.DG]

[R]

A. Ranicki, Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Oxford University Press, (2002). MR 2061749 (2005e:57075)

[Ri]

A. Rigas, Some bundles of nonnegative curvature, Math. Ann. 232 (1978), 187-193. MR 0464254 (57:4188)

[S]

K. Shiohama, Recent developments in sphere theorems, Differential geometry. Proc. Sympos. Pure Math., 551-576, Vol. 54, Part 3, Amer. Math. Soc., Providence, RI, 1993. MR 1216646 (94d:53071)

[Si]

W. Singhof, On the topology of double coset manifolds, Math. Ann. 297 (1993) no. 1, 133-146. MR 1238411 (94k:57054)

[ScY]

R. Schoen, S.T. Yau, On the structure of manifolds of positive scalar curvature, Manuscr. Math. 28 (1979), 159-183. MR 535700 (80k:53064)

[Sh]

N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59-69. MR 0096223 (20:2715)

[Sm1]

S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406. MR 0137124 (25:580)

[Sm2]

S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387-399. MR 0153022 (27:2991)

[St]

S. Stolz, Simply connected manifolds with positive scalar curvature, Ann. of Math. (2) 136 (1992), 511-540. MR 1189863 (93i:57033)

[SY]

J.P. Sha, D.G. Yang, Positive Ricci curvature on the connected sums of $S^n \times S^m$ , J. Differential Geometry 33 (1991), 127-138.

[Su]

Y. Suyama, A differentiable sphere theorem by curvature pinching. II, Tohoku Math. J. (2) 47 no. 1 (1995), 15-29. MR 1311439 (95k:53048)

[To]

B. Totaro, Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61 (2002), 397-452. MR 1979366 (2004b:53075)

[Vi]

J. Vilms, Totally geodesic maps, J. Differential Geom. 4 (1970), 73-79. MR 0262984 (41:7589)

[We]

M. Weiss, Pinching and concordance theory, J. Differential Geom. 38 (1993), 387-416. MR 1237489 (95a:53057)

[Wi1]

F. Wilhelm, Exotic spheres with lots of positive curvatures, J. Geom. Anal. 11 (2001), 163-188. MR 1829354 (2002c:53062)

[Wi2]

F. Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11 (2001), 519-560. MR 1857856 (2002f:53056)

[Wr1]

D. Wraith, Exotic spheres with positive Ricci curvature, J. Differential Geom. 45 (1997), 638-649. MR 1472892 (98i:53058)

[Wr2]

D. Wraith, Surgery on Ricci positive manifolds, J. Reine Angew. Math. 501 (1998), 99-113. MR 1637825 (99j:53050)

[Y]

S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation. I, Comm. Pure Appl. Math. 31 (1978), 339-411. MR 480350 (81d:53045)

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 53C20

M. Joachim
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: joachim@math.uni-muenster.de

D. J. Wraith
Affiliation: Department of Mathematics, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland
Email: David.Wraith@nuim.ie

DOI: 10.1090/S0273-0979-08-01213-5
PII: S 0273-0979(08)01213-5
Received by editor(s): May 27, 2008
Posted: July 1, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS