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Exotic spheres and curvature. (English) Zbl 1149.53020
Manifolds which are homeomorphic but not diffeomorphic to a standard sphere are called exotic spheres. In dimensions one, two, and three it is well known that every topological manifold admits a unique smooth structure. The lowest dimension in which exotic smooth structures exist is dimension four. Many four-dimensional manifolds are known to admit infinitely many distinct smooth structures. However, this phenomenon cannot occur in higher dimensions. In dimensions at least five, any manifold has at most a finite number of smooth structures, and in many cases this number is one.
In [Ann. Math. (2) 64, 399–405 (1956; Zbl 0072.18402)], J. Milnor showed that spheres, specifically $$S^ 7$$, were the first manifolds admitting exotic smooth structures. M. Kervaire and J. Milnor [Ann. Math. (2) 77, 504–537 (1963; Zbl 0115.40505)] were able to show how large the family of exotic spheres is in each dimension greater than four. Although in each case this number is finite, there are exotic spheres in infinitely many dimensions, and in dimensions $$4n + 3$$ the number of exotic spheres grows very rapidly with $$n$$. The question of whether there is an exotic $$S^ 4$$, and if so, how many there are is still open today. This is essentially the smooth Poincaré conjecture in dimension four.
A general approach to describing exotic spheres is provided by the twisted sphere construction. A twisted sphere is a manifold constructed by gluing two discs $$D^ n$$ along their boundaries using an orientation preserving diffeomorphism $$S^{n-1}\to S^{n-1}$$. Such a manifold is homeomorphic to $$S^ n$$. In dimensions other than four, every exotic sphere is a twisted sphere. In dimension four the only twisted sphere is the standard sphere, so if an exotic $$4$$-sphere exists, it cannot be a twisted sphere.
Another term used when discussing exotic spheres is a homotopy sphere that is a smooth manifold with the same homotopy type as $$S^ n$$. Thus every exotic sphere is a homotopy sphere. In dimensions at least five, any smooth manifold with the homotopy type of a sphere must be homeomorphic to a sphere. This is the Generalised Poincaré Conjecture, proved by S. Smale in [Ann. Math. (2) 74, 391–406 (1961; Zbl 0099.39202)]. Thus in these dimensions the set of diffeomorphism classes of homotopy spheres is precisely the union of the diffeomorphism class of the standard sphere with the diffeomorphism classes of the exotic spheres.
In dimensions one and two, homotopy spheres are always diffeomorphic to the standard spheres. In dimension three, the differentiable structure on $$S^ 3$$ is unique which means that there is no exotic $$S^ 3$$. Moreover, every $$3$$-manifold has a unique differentiable structure, and any two homeomorphic $$3$$-manifolds are necessarily diffeomorphic. On the other hand, it was not known if a homotopy $$S^ 3$$ was necessarily homeomorphic, and therefore diffeomorphic, to $$S^ 3$$. This is the famous Poincaré conjecture, which has recently been affirmatively resolved. Thus any homotopy $$3$$-sphere is diffeomorphic to $$S^ 3$$. In dimension four, any homotopy $$4$$-sphere is homeomorphic to $$S^ 4$$ but the question of whether such a manifold is necessarily diffeomorphic to $$S^ 4$$ still remains open.
In this paper, the authors survey what is known about the curvature of exotic spheres. Any smooth manifold can be equipped with a smooth Riemannian metric. A Riemannian metric endows the manifold with shape, so it makes sense to try and quantify the curvature of such an object. The most widely known measure of curvature is the Gaussian curvature in dimension two. If the Gaussian curvature at a point is positive, then the geometry in a neighborhood of the point resembles that of a sphere, if the Gaussian curvature is zero, the local geometry is that of a Euclidean plane, and if the Gaussian curvature is negative, the local geometry resembles that of a saddle.
Every closed compact surface admits a metric with constant Gaussian curvature. This, combined with the Gauss-Bonnet Theorem shows that the only closed compact surfaces admitting a metric of strictly positive Gaussian curvature are the $$2$$-sphere and the real projective plane. The only closed compact surfaces admitting a metric of identically zero Gaussian curvature are the torus and Klein bottle. The remaining closed compact surfaces admit a metric of strictly negative Gaussian curvature.
The simplicity of this topology-curvature relationship in dimension two is not mirrored in higher dimensions. The first natural generalization of the Gaussian curvature is the notion of sectional curvature. The next curvature to be introduced is the Ricci curvature, and finally the scalar curvature. The standard $$n$$-dimensional sphere of radius $$R$$ has constant sectional curvature $$1/R^ 2$$, Ricci curvature $$(n-1)/R^ 2$$, and scalar curvature $$n(n-1)/R^ 2$$. By studying the curvature of exotic spheres, the authors gain some insight into the extent to which the geometry of these objects resembles that of the standard sphere.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 57R60 Homotopy spheres, Poincaré conjecture 57R55 Differentiable structures in differential topology
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##### References:
  Thierry Aubin, Métriques riemanniennes et courbure, J. Differential Geometry 4 (1970), 383 – 424 (French). · Zbl 0212.54102  J. F. Adams, On the groups \?(\?). IV, Topology 5 (1966), 21 – 71. · Zbl 0145.19902  Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001  Charles P. Boyer, Krzysztof Galicki, and János Kollár, Einstein metrics on spheres, Ann. of Math. (2) 162 (2005), no. 1, 557 – 580. · Zbl 1093.53044  Charles P. Boyer, Krzysztof Galicki, János Kollár, and Evan Thomas, Einstein metrics on exotic spheres in dimensions 7, 11, and 15, Experiment. Math. 14 (2005), no. 1, 59 – 64. · Zbl 1112.53033  Charles P. Boyer, Krzysztof Galicki, and Michael Nakamaye, On positive Sasakian geometry, Geom. Dedicata 101 (2003), 93 – 102. · Zbl 1046.53029  Charles P. Boyer, Krzysztof Galicki, and Michael Nakamaye, Sasakian geometry, homotopy spheres and positive Ricci curvature, Topology 42 (2003), no. 5, 981 – 1002. · Zbl 1066.53089  Allen Back and Wu-Yi Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987), no. 1, 207 – 227. · Zbl 0632.53048  Egbert Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1 – 14 (German). · Zbl 0145.17804  David E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203, Birkhäuser Boston, Inc., Boston, MA, 2002. · Zbl 1011.53001  William Browder, Surgery on simply-connected manifolds, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65. · Zbl 0239.57016  Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. · Zbl 1232.53037  Jeff Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geometry 8 (1973), 623 – 628. · Zbl 0281.53040  Eugenio Calabi, On Kähler manifolds with vanishing canonical class, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 78 – 89. · Zbl 0080.15002  Jean Cerf, Sur les difféomorphismes de la sphère de dimension trois (\Gamma $$_{4}$$=0), Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin-New York, 1968 (French). · Zbl 0164.24502  Isaac Chavel, Riemannian geometry — a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. · Zbl 0810.53001  Anand Dessai, On the topology of scalar-flat manifolds, Bull. London Math. Soc. 33 (2001), no. 2, 203 – 209. · Zbl 1035.53057  Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. · Zbl 0752.53001  C. Durán, T Püttmann, A. Rigas, An infinite family of Gromoll-Meyer spheres, arXiv:math.DG/0610349. · Zbl 1258.53039  Carlos E. Durán, Pointed Wiedersehen metrics on exotic spheres and diffeomorphisms of \?$$^{6}$$, Geom. Dedicata 88 (2001), no. 1-3, 199 – 210. · Zbl 1002.53026  J.-H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982), no. 3, 469 – 480. · Zbl 0484.53031  Jost-Hinrich Eschenburg, Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 32, Universität Münster, Mathematisches Institut, Münster, 1984 (German). · Zbl 0551.53024  J.-H. Eschenburg, Local convexity and nonnegative curvature — Gromov’s proof of the sphere theorem, Invent. Math. 84 (1986), no. 3, 507 – 522. · Zbl 0594.53034  J.-H. Eschenburg, Cohomology of biquotients, Manuscripta Math. 75 (1992), no. 2, 151 – 166. · Zbl 0769.53029  J.-H. Eschenburg, Almost positive curvature on the Gromoll-Meyer 7-sphere, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1165 – 1167. · Zbl 1029.53043  Paul Ehrlich, Metric deformations of curvature. I. Local convex deformations, Geometriae Dedicata 5 (1976), no. 1, 1 – 23. , https://doi.org/10.1007/BF00148134 Paul Ehrlich, Metric deformations of curvature. II. Compact 3-manifolds, Geometriae Dedicata 5 (1976), no. 2, 147 – 161. , https://doi.org/10.1007/BF00145952 Paul Ehrlich, Deformations of scalar curvature, Geometriae Dedicata 5 (1976), no. 1, 25 – 26. , https://doi.org/10.1007/BF00148135 Paul Ehrlich, Local convex deformations, Hermitian metrics, and Hermitian connections, Geometriae Dedicata 5 (1976), no. 1, 27 – 29. · Zbl 0345.53024  J.-H. Eschenburg, M. Kerin, Almost positive curvature on the Gromoll-Meyer $$7$$-sphere, Proc. Amer. Math. Soc. posted on April 23, 2008, PII S0002-9939(08)09429-X (to appear in print). · Zbl 1153.53023  Aziz El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), no. 1, 57 – 106 (French, with English summary). · Zbl 0697.57014  John B. Etnyre, Introductory lectures on contact geometry, Topology and geometry of manifolds (Athens, GA, 2001) Proc. Sympos. Pure Math., vol. 71, Amer. Math. Soc., Providence, RI, 2003, pp. 81 – 107. · Zbl 1045.57012  Akito Futaki, Scalar-flat closed manifolds not admitting positive scalar curvature metrics, Invent. Math. 112 (1993), no. 1, 23 – 29. · Zbl 0792.53036  Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357 – 453. · Zbl 0528.57011  Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. · Zbl 0705.57001  Hansjörg Geiges, Contact geometry, Handbook of differential geometry. Vol. II, Elsevier/North-Holland, Amsterdam, 2006, pp. 315 – 382. · Zbl 1147.53068  Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423 – 434. · Zbl 0463.53025  Detlef Gromoll and Wolfgang Meyer, An exotic sphere with nonnegative sectional curvature, Ann. of Math. (2) 100 (1974), 401 – 406. · Zbl 0293.53015  Karsten Grove and Peter Petersen V, A pinching theorem for homotopy spheres, J. Amer. Math. Soc. 3 (1990), no. 3, 671 – 677. · Zbl 0717.53025  Karsten Grove, Luigi Verdiani, Burkhard Wilking, and Wolfgang 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. · Zbl 1170.53307  Karsten Grove and Frederick Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine Angew. Math. 487 (1997), 201 – 217. · Zbl 0882.53030  Karsten Grove and Wolfgang Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), no. 1, 331 – 367. · Zbl 0991.53016  Karsten Grove and Wolfgang Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), no. 3, 619 – 646. · Zbl 1038.53034  F. Hirzebruch, Singularities and exotic spheres, Séminaire Bourbaki, 1966/67, Exp. 314, Textes des conférences, o.S., Paris: Institut Henri Poincaré 1967.  Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. · Zbl 0767.57014  Wu-chung Hsiang and Wu-yi Hsiang, The degree of symmetry of homotopy spheres, Ann. of Math. (2) 89 (1969), 52 – 67. · Zbl 0198.56601  Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1 – 55. · Zbl 0284.58016  Morris W. Hirsch and Barry Mazur, Smoothings of piecewise linear manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 80. · Zbl 0298.57007  Horacio Hernández-Andrade, A class of compact manifolds with positive Ricci curvature, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 73 – 87.  Jerry L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conference Series in Mathematics, vol. 57, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. · Zbl 0561.53048  Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504 – 537. · Zbl 0115.40505  Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah; Annals of Mathematics Studies, No. 88. · Zbl 0361.57004  Jerry L. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317 – 331. · Zbl 0297.53020  Jerry L. Kazdan and F. W. Warner, A direct approach to the determination of Gaussian and scalar curvature functions, Invent. Math. 28 (1975), 227 – 230. · Zbl 0297.53021  Jerry L. Kazdan and F. W. Warner, Prescribing curvatures, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 309 – 319. · Zbl 0313.53017  Vitali Kapovitch and Wolfgang Ziller, Biquotients with singly generated rational cohomology, Geom. Dedicata 104 (2004), 149 – 160. · Zbl 1063.53055  Timothy Lance, Differentiable structures on manifolds, Surveys on surgery theory, Vol. 1, Ann. of Math. Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 73 – 104. · Zbl 0941.57027  H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. · Zbl 0688.57001  André Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7 – 9 (French). · Zbl 0136.18401  Joachim Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), no. 3, 655 – 683. · Zbl 0824.53033  John Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399 – 405. · Zbl 0072.18402  J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. · Zbl 0108.10401  John W. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 55 – 62. · Zbl 0132.19602  John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. · Zbl 0184.48405  John Milnor, Classification of (\?-1)-connected 2\?-dimensional manifolds and the discovery of exotic spheres, Surveys on surgery theory, Vol. 1, Ann. of Math. Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 25 – 30. · Zbl 0941.57001  John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. · Zbl 0298.57008  S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401 – 404. · Zbl 0025.22704  John C. Nash, Positive Ricci curvature on fibre bundles, J. Differential Geom. 14 (1979), no. 2, 241 – 254. · Zbl 0464.53035  Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459 – 469. · Zbl 0145.18602  Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. · Zbl 0914.53001  G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arxiv.org/math.DG/0211159 · Zbl 1130.53001  G. Perelman, Ricci flow with surgery on three-manifolds, http://arxiv.org/ math.DG/0303109 · Zbl 1130.53002  G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, http://arxiv.org/math.DG/0307245 · Zbl 1130.53003  W. A. Poor, Some exotic spheres with positive Ricci curvature, Math. Ann. 216 (1975), no. 3, 245 – 252. · Zbl 0293.53016  P. Petersen, F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv: 0805.0812v1 [math.DG]  Andrew Ranicki, Algebraic and geometric surgery, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2002. Oxford Science Publications. · Zbl 1003.57001  A. Rigas, Some bundles of non-negative curvature, Math. Ann. 232 (1978), no. 2, 187 – 193. · Zbl 0354.53039  Katsuhiro Shiohama, Recent developments in sphere theorems, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 551 – 576. · Zbl 0844.53035  W. Singhof, On the topology of double coset manifolds, Math. Ann. 297 (1993), no. 1, 133 – 146. · Zbl 0793.57019  R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159 – 183. · Zbl 0423.53032  Nobuo Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59 – 69. · Zbl 0145.20303  Stephen Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391 – 406. · Zbl 0099.39202  S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387 – 399. · Zbl 0109.41103  Stephan Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no. 3, 511 – 540. · Zbl 0784.53029  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. · Zbl 0728.53027  Yoshihiko Suyama, A differentiable sphere theorem by curvature pinching. II, Tohoku Math. J. (2) 47 (1995), no. 1, 15 – 29. · Zbl 0828.53032  Burt Totaro, Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61 (2002), no. 3, 397 – 451. · Zbl 1071.53529  Jaak Vilms, Totally geodesic maps, J. Differential Geometry 4 (1970), 73 – 79. · Zbl 0194.52901  Michael Weiss, Pinching and concordance theory, J. Differential Geom. 38 (1993), no. 2, 387 – 416. · Zbl 0783.53027  Frederick Wilhelm, Exotic spheres with lots of positive curvatures, J. Geom. Anal. 11 (2001), no. 1, 161 – 186. · Zbl 1023.53024  Frederick Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11 (2001), no. 3, 519 – 560. · Zbl 1039.53037  David Wraith, Exotic spheres with positive Ricci curvature, J. Differential Geom. 45 (1997), no. 3, 638 – 649. · Zbl 0910.53027  David Wraith, Surgery on Ricci positive manifolds, J. Reine Angew. Math. 501 (1998), 99 – 113. · Zbl 0915.53018  Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339 – 411. · Zbl 0369.53059
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