The Michigan Mathematical Journal
- Michigan Math. J.
- Volume 56, Issue 2 (2008), 419-451.
A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications
Carlos Durán and Thomas Püttmann
Full-text: Open access
Article information
Source
Michigan Math. J., Volume 56, Issue 2 (2008), 419-451.
Dates
First available in Project Euclid: 23 October 2008
Permanent link to this document
https://projecteuclid.org/euclid.mmj/1224783521
Digital Object Identifier
doi:10.1307/mmj/1224783521
Mathematical Reviews number (MathSciNet)
MR2492402
Zentralblatt MATH identifier
1158.53025
Subjects
Primary: 53C22: Geodesics [See also 58E10] 57S25: Groups acting on specific manifolds
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 57S15: Compact Lie groups of differentiable transformations
Citation
Durán, Carlos; Püttmann, Thomas. A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications. Michigan Math. J. 56 (2008), no. 2, 419--451. doi:10.1307/mmj/1224783521. https://projecteuclid.org/euclid.mmj/1224783521
References
- U. Abresch, C. E. Durán, T. Püttmann, and A. Rigas, Wiedersehen metrics and exotic involutions of Euclidean spheres, J. Reine Angew. Math. 605 (2007), 1--21.
- Ya. Bazaikan, A manifold with positive sectional curvature and fundamental group $\Bbb Z_3\oplus\Bbb Z_3,$ Sibersk. Mat. Zh. 40 (1999), 994--996.Mathematical Reviews (MathSciNet): MR1726845
- A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., 93, Springer-Verlag, Berlin, 1978.Mathematical Reviews (MathSciNet): MR496885
- G. E. Bredon, Exotic actions on spheres, Proceedings of conference on transformation groups (New Orleans, 1967), pp. 47--76, Springer-Verlag, New York, 1968.Mathematical Reviews (MathSciNet): MR266239
- ------, Introduction to compact transformation groups, Pure Appl. Math., 46, Academic Press, New York, 1972.Mathematical Reviews (MathSciNet): MR413144
- E. Brieskorn, Beispiele zur Differentialgeometrie von Singularitäten, Invent. Math. 2 (1966), 1--14.
- W. Browder, Free $Z_p$-actions on homotopy spheres, Topology of manifolds (Athens, 1969), pp. 217--226, Markham, Chicago, 1970.Mathematical Reviews (MathSciNet): MR276982
- J. Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geom. 8 (1972), 623--628.
- C. E. Durán, Pointed Wiedersehen metrics on exotic spheres and diffeomorphisms of $\Bbb S^6,$ Geom. Dedicata 88 (2001), 199--210.
- C. E. Durán, A. Mendoza, and A. Rigas, Blakers--Massey elements and exotic diffeomorphisms of $S^6$ and $S^14$ via geodesics, Trans. Amer. Math. Soc. 356 (2004), 5025--5043.Mathematical Reviews (MathSciNet): MR2084409
Digital Object Identifier: doi:10.1090/S0002-9947-04-03469-5 - J.-H. Eschenburg and M. Kerin, Almost positive curvature on the Gromoll--Meyer sphere, Proc. Amer. Math. Sci. 136 (2008), 3263--3270.Mathematical Reviews (MathSciNet): MR2407092
Digital Object Identifier: doi:10.1090/S0002-9939-08-09429-X - D. Gromoll and W. Meyer, An exotic sphere with nonnegative curvature, Ann. of Math. (2) 100 (1974), 401--406.Mathematical Reviews (MathSciNet): MR375151
Digital Object Identifier: doi:10.2307/1971078
JSTOR: links.jstor.org - K. Grove and K. Shankar, Rank two fundamental groups of positively curved manifolds, J. Geom. Anal. 10 (2000), 679--682.
- K. Grove, K. Shankar, and W. Ziller, Symmetries of Eschenburg spaces and the Chern problem, Asian J. Math. 10 (2006), 647--661.
- K. Grove, L. Verdiani, B. Wilking, and W. Ziller, Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5 (2006), 159--170.Mathematical Reviews (MathSciNet): MR2244696
- K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), 331--367.Mathematical Reviews (MathSciNet): MR1792298
Digital Object Identifier: doi:10.2307/2661385
JSTOR: links.jstor.org - M. Hirsch and J. Milnor, Some curious involutions of spheres, Bull. Amer. Math. Soc. 70 (1964), 372--377.Mathematical Reviews (MathSciNet): MR176479
Digital Object Identifier: doi:10.1090/S0002-9904-1964-11103-4
Project Euclid: euclid.bams/1183526016 - F. Hirzebruch and K. H. Mayer, $\text\rm O(n)$-Mannigfaltigkeiten, exotische Sphären und Singularitäten, Lecture Notes in Math., 57, Springer-Verlag, Berlin, 1968.Mathematical Reviews (MathSciNet): MR229251
- W.-Y. Hsiang and H. B. Lawson, Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1--38.
- V. Kapovitch and W. Ziller, Biquotients with singly generated rational cohomology, Geom. Dedicata 104 (2004), 149--160.Mathematical Reviews (MathSciNet): MR2043959
Digital Object Identifier: doi:10.1023/B:GEOM.0000022860.89824.2f - T. Lance, Differentiable structures on manifolds, Surveys on surgery theory, vol. 1, Ann. of Math. Stud., 145, pp. 73--104, Princeton Univ. Press, Princeton, NJ, 2000.Mathematical Reviews (MathSciNet): MR1747531
- W. S. Massey, Cross products of vectors in higher dimensional Euclidean spaces, Amer. Math. Monthly 90 (1983), 697--701.Mathematical Reviews (MathSciNet): MR723943
Digital Object Identifier: doi:10.2307/2323537
JSTOR: links.jstor.org - J. W. Milnor, On the $3$-dimensional Brieskorn manifolds $M(p,q,r),$ Knots, groups and 3-manifolds (L. P. Neuwirth, ed.), Ann. of Math. Stud., 84, pp. 175--225, Princeton Univ. Press, Princeton, NJ, 1975.Mathematical Reviews (MathSciNet): MR418127
- P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458--480.
- P. Orlik, Smooth homotopy lens spaces, Michigan Math. J. 16 (1969), 245--255.Mathematical Reviews (MathSciNet): MR248831
Digital Object Identifier: doi:10.1307/mmj/1029000268
Project Euclid: euclid.mmj/1029000268 - G. P. Paternain and R. J. Spatzier, New examples of manifolds with completely integrable geodesic flow, Adv. Math. 108 (1994), 346--366.
- M. G. Scharlemann and L. C. Siebenmann, The Hauptvermutung for smooth singular homeomorphisms, Topology of manifolds (Tokyo, 1973), pp. 85--91, Univ. of Tokyo Press, Tokyo, 1975.Mathematical Reviews (MathSciNet): MR372871
- L. Schwachhöfer and W. Tuschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), 59--91.Mathematical Reviews (MathSciNet): MR2091679
- K. Shankar, On the fundamental groups of positively curved manifolds, J. Differential Geom. 49 (1998), 179--182.
- ------, Isometry groups of homogeneous spaces with positive sectional cuvature, Differential Geom. Appl. 14 (2001), 57--78.Mathematical Reviews (MathSciNet): MR1812525
Digital Object Identifier: doi:10.1016/S0926-2245(00)00038-3 - E. Straume, Compact differentiable transformation groups on exotic spheres, Math. Ann. 299 (1994), 355--389.
- B. Totaro, Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61 (2002), 397--451.
- F. Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal. 11 (2001), 519--560.Mathematical Reviews (MathSciNet): MR1857856
- K. Yamato, The spectrum of the Milnor--Gromoll--Meyer sphere, J. Math. Soc. Japan 54 (2002), 409--445.Mathematical Reviews (MathSciNet): MR1883525
Digital Object Identifier: doi:10.2969/jmsj/05420409
Project Euclid: euclid.jmsj/1213024074 - C. T. Yang, On involutions of the five sphere, Topology 5 (1966), 17--19.Mathematical Reviews (MathSciNet): MR185605
Digital Object Identifier: doi:10.1016/0040-9383(66)90003-6

- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- The spectrum of the Milnor-Gromoll-Meyer sphere
YAMATO, Kazuo, Journal of the Mathematical Society of Japan, 2002 - The integer valued SU(3) Casson invariant for Brieskorn spheres
Boden, Hans U., Herald, Christopher M., and Kirk, Paul A., Journal of Differential Geometry, 2005 - The Minimal Cycles over Brieskorn Complete Intersection
Surface Singularities
Meng, Fanning, Yuan, Wenjun, and Wang, Zhigang, Taiwanese Journal of Mathematics, 2016
- The spectrum of the Milnor-Gromoll-Meyer sphere
YAMATO, Kazuo, Journal of the Mathematical Society of Japan, 2002 - The integer valued SU(3) Casson invariant for Brieskorn spheres
Boden, Hans U., Herald, Christopher M., and Kirk, Paul A., Journal of Differential Geometry, 2005 - The Minimal Cycles over Brieskorn Complete Intersection
Surface Singularities
Meng, Fanning, Yuan, Wenjun, and Wang, Zhigang, Taiwanese Journal of Mathematics, 2016 - Critical points of non-$C^{2}$ functionals
Duc, Duong Minh, Hung, Tran Vinh, and Khai, Nguyen Tien, Topological Methods in Nonlinear Analysis, 2007 - Classification of Willmore two-spheres in the 5-dimensional
sphere
Ma, Xiang, Wang, Changping, and Wang, Peng, Journal of Differential Geometry, 2017 - The Conley index and the critical groups via an extension of Gromoll-Meyer theory
Chang, K. C. and Ghoussoub, N., Topological Methods in Nonlinear Analysis, 1996 - Ruled Minimal Lagrangian Submanifolds of Complex Projective 3-Space
Bolton, J. and Vrancken, L., Asian Journal of Mathematics, 2005 - Local Floer homology and infinitely many simple Reeb orbits
McLean, Mark, Algebraic & Geometric Topology, 2012 - Hamiltonian Minimal Lagrangian Cones in ${\mathbb C}^{m}$
IRIYEH, Hiroshi, Tokyo Journal of Mathematics, 2005 - A Morse lemma for degenerate critical points with low differentiability
de Moura, Adriano A. and de Souza, Fausto M., Abstract and Applied Analysis, 2000
