Topology and Curvature of Isoparametric Families in Spheres

Chao Qian , Zizhou Tang , Wenjiao Yan

Communications in Mathematics and Statistics ›› 2023, Vol. 11 ›› Issue (2) : 439 -475.

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Communications in Mathematics and Statistics ›› 2023, Vol. 11 ›› Issue (2) : 439 -475. DOI: 10.1007/s40304-021-00259-2
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Topology and Curvature of Isoparametric Families in Spheres

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Abstract

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

Keywords

Isoparametric hypersurface / Focal submanifold / Homotopy equivalent / Homeomorphism / Diffeomorphism / Parallelizability / Lusternik–Schnirelmann category / Sectional curvature / Ricci curvature

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Chao Qian, Zizhou Tang, Wenjiao Yan. Topology and Curvature of Isoparametric Families in Spheres. Communications in Mathematics and Statistics, 2023, 11(2): 439-475 DOI:10.1007/s40304-021-00259-2

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References

[1]

Adams JF. On the non-existence of elements of Hopf invariant one. Ann. Math.. 1960, 72 1 20-104

[2]

Adams JF. Vector fields on spheres. Ann. Math.. 1962, 75 3 603-632

[3]

Adams JF. On the groups J(X)-IV. Topology. 1966, 5 21-71

[4]

Berstein I. On the Lusternik–Schnirelmann category of Grassmannians. Math. Proc. Camb. Philos. Soc.. 1976, 79 129-134

[5]

Cecil TE, Chi QS, Jensen GR. Isoparametric hypersurfaces with four principal curvatures. Ann. Math.. 2007, 166 1 1-76

[6]

Chi QS. Isoparametric hypersurfaces with four principal curvatures, II. Nagoya Math. J.. 2011, 204 1-18

[7]

Chi QS. Isoparametric hypersurfaces with four principal curvatures, III. J. Differ. Geom.. 2013, 94 469-504

[8]

Chi QS. Isoparametric hypersurfaces with four principal curvatures, IV. J. Differ. Geom.. 2020, 115 225-301

[9]

Cecil TE, Ryan PJ. Geometry of Hypersurfaces. Springer Monographs in Mathematics. 2015 New York: Springer

[10]

Cornea O, Lupton G, Oprea J, Tanré D. Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs. 2003 Providence: American Mathematical Society

[11]

Dorfmeister J, Neher E. Isoparametric hypersurfaces, case $g = 6, m =1$. Commun. Algebra. 1985, 13 2299-2368

[12]

Dranishnikov AN, Katz MG, Rudyak YB. Small values of the Lusternik–Schnirelmann category for manifolds. Geom. Topol.. 2008, 12 1711-1727

[13]

Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry-Methods and Applications, Part III: Introduction to Homology Theory, Graduate Texts in Mathematics, vol. 124. Translated by R. G. Burns, Springer, New York (1990)

[14]

Fang FQ. On the topology of isoparametric hypersurfaces with four distinct principal curvatures. Proc. Am. Math. Soc.. 1999, 127 259-264

[15]

Fang FQ. Dual submanifolds in rational homology spheres. Sci. China Math.. 2017, 60 1549-1560

[16]

Ferus D, Karcher H, Münzner HF. Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z.. 1981, 177 479-502

[17]

Ge JQ, Tang ZZ. Geometry of isoparametric hypersurfaces in Riemannian manifolds. Asian J. Math.. 2014, 18 117-126

[18]

Gómez-Larrañaga JC, González-Acuña F. Lusternik–Schnirelmann category of 3-manifolds. Topology. 1992, 31 791-800

[19]

Grove K, Halperin S. Dupin hypersurfaces, group actions and the double mapping cylinder. J. Differ. Geom.. 1987, 26 429-459

[20]

Hsiang WY, Lawson HB. Minimal submanifolds of low cohomogenity. J. Differ. Geom.. 1971, 5 1-38

[21]

Hu S-T. Homotopy Theory, Pure and Applied Mathematics. 1959 New York: Academic Press

[22]

Immervoll S. On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann. Math.. 2008, 168 1011-1024

[23]

Iwase N. Lusternik–Schnirelmann category of a sphere-bundle over a sphere. Topology. 2003, 42 701-713

[24]

James I. On category, in the sense of Lusternik–Schnirelmann. Topology. 1978, 17 331-348

[25]

James, I., Whitehead, J.H.C.: On the homotopy theory of sphere-bundles over spheres (I), (II). Proc. Lond. Math. Soc. 4, 196–218 (1954), 5, 148–166 (1955)

[26]

Kervaire MA. Some nonstable homotopy groups of Lie groups. Illinois J. Math.. 1960, 4 161-169

[27]

Li QC, Yan WJ. On Ricci tensor of focal submanifolds of isoparametric hypersurfaces. Sci. China Math.. 2015, 58 1723-1736

[28]

Miatello ID, Miatello RJ. On stable parallelizability of ${\widetilde{G}}_{k, n}$ and related manifolds. Math. Ann.. 1982, 259 343-350

[29]

Miyaoka R. The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces. Osaka J. Math.. 1993, 30 179-202

[30]

Miyaoka R. Geometry of $G_2$ orbits and isoparametric hypersurfaces. Nagoya Math. J.. 2011, 203 175-189

[31]

Miyaoka R. Isoparametric hypersurfaces with (g, m) = (6,2). Ann. Math.. 2013, 177 53-110

[32]

Miyaoka R. Errata of "isoparametric hypersurfaces with (g, m) = (6, 2) ". Ann. Math.. 2016, 183 1057-1071

[33]

Münzner HF. Isoparametrische hyperflächen in sphären, I. Math. Ann.. 1980, 251 57-71

[34]

Münzner HF. Isoparametrische hyperflächen in sphären, II. Math. Ann.. 1981, 256 215-232

[35]

Nishimoto T. On the Lusternik–Schnirelmann category of Stiefel manifilds. Topol. Appl.. 2007, 154 1956-1960

[36]

Nomizu, K.: Some results in E. Cartan’s theory of isoparametric families of hypersurfaces. Bull. Am. Math. Soc. 79, 1184–1189 (1973)

[37]

Ozeki H, Takeuchi M. On some types of isoparametric hypersurfaces in spheres, I. Tôhoku Math. J.. 1975, 27 515-559

[38]

Ozeki H, Takeuchi M. On some types of isoparametric hypersurfaces in spheres, II. Tôhoku Math. J.. 1976, 28 7-55

[39]

Qian C, Tang ZZ. Isoparametric foliations, a problem of Eells–Lemaire and conjectures of Leung. Proc. Lond. Math. Soc.. 2016, 112 979-1001

[40]

Qian C, Tang ZZ, Yan WJ. New examples of Willmore submanifolds in the unit sphere via isoparametric functions, II. Ann. Glob. Anal. Geom.. 2013, 43 47-62

[41]

Rudyak Y. On category weight and its applications. Topology. 1999, 38 37-55

[42]

Schweitzer PA. Secondary cohomology operations induced by the diagonal mapping. Topology. 1965, 3 337-355

[43]

Singhof W. On the Lusternik–Schnirelmann category of Lie groups. Math. Z.. 1975, 145 111-116

[44]

Solomon B. Quartic isoparametric hypersurfaces and quadratic forms. Math. Ann.. 1992, 293 387-398

[45]

Steenrod N. The Topology of Fiber Bundles. 1951 Princeton: Princeton Univ. Press

[46]

Strom, J.A.: Two special cases of Ganea’s conjecture. Trans. Am. Math. Soc. 352, 679–688 (1999)

[47]

Sutherland WA. A note on the parallelizability of sphere-bundles over spheres. J. Lond. Math. Soc.. 1964, 39 55-62

[48]

Takens F. The minimal number of critical points of a function on a compact manifold and the Lusternik–Schnirelman category. Invent. Math.. 1968, 6 197-244

[49]

Tang ZZ. Isoparametric hypersurfaces with four distinct principal curvatures. Chin. Sci. Bull.. 1991, 36 1237-1240

[50]

Tang ZZ. Codimension two immersions of oriented Grassmann manifolds. Manuscripta Math.. 1995, 88 165-170

[51]

Tang ZZ. Some existence and nonexistence results of isometric immersions of Riemannian manifolds. Commun. Contemp. Math.. 2004, 6 867-879

[52]

Takagi R, Takahashi T. On the Principal Curvatures of Homogeneous Hypersurfaces in a Sphere, Differential Geometry, in Honor of K. Yano. 1972 Tokyo: Kinokuniya

[53]

Tang ZZ, Xie YQ, Yan WJ. Schoen–Yau–Gromov–Lawson theory and isoparametric foliations. Commun. Anal. Geom.. 2012, 20 989-1018

[54]

Tang ZZ, Xie YQ, Yan WJ. Isoparametric foliation and Yau conjecture on the first eigenvalue, II. J. Funct. Anal.. 2014, 266 6174-6199

[55]

Tang ZZ, Yan WJ. Isoparametric foliation and Yau conjecture on the first eigenvalue. J. Differ. Geom.. 2013, 94 521-540

[56]

Tang ZZ, Yan WJ. Isoparametric foliation and a problem of Besse on generalizations of Einstein condition. Adv. Math.. 2015, 285 1970-2000

[57]

Tang, Z.Z., Yan, W.J.: On the Chern conjecture for isoparametric hypersurfaces. arXiv: 2001.10134 (submitted)

[58]

Wang QM. Isoparametric functions on Riemannian manifolds. I. Math. Ann.. 1987, 277 639-646

[59]

Wang QM. On the topology of Clifford isoparametric hypersurfaces. J. Differ. Geom.. 1988, 27 55-66

[60]

Wu BL. A finiteness theorem for isoparametric hypersurfaces. Geom. Dedicata. 1994, 50 247-250

[61]

Ziller, W.: Riemannian Manifolds with Positive Sectional Curvature, Geometry of Manifolds with Non-negative Sectional Curvature. Lecture Notes in Mathematics, vol. 2110, pp. 1–19. Springer, Cham (2014)

Funding

National Natural Science Foundation of China(11722101)

National Natural Science Foundation of China(11931007)

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