Lie sphere transformations and the focal sets of certain taut immersions

Buyske, Steven G.

doi:10.1090/S0002-9947-1989-0965743-X

Lie sphere transformations and the focal sets of certain taut immersions
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by Steven G. Buyske PDF

Trans. Amer. Math. Soc. 311 (1989), 117-133 Request permission

Abstract:

We study the images of certain taut or Dupin hypersurfaces, including their focal sets, under Lie sphere transformations (generalizations of conformal transformations of euclidean or spherical space). After the introduction, the method of studying hypersurfaces as Lie sphere objects is developed. In two recent papers, Cecil and Chern use submanifolds of the space of lines on the Lie quadric. Here we use submanifolds of the Lie quadric itself instead. The third section extends the concepts of tightness and tautness to semi-euclidean space. The final section shows that if a hypersurface is the Lie sphere image of certain standard constructions (tubes, cylinders, and rotations) over a taut immersion, the resulting family of curvature spheres is taut in the Lie quadric. The sheet of the focal set will be tight in euclidean space if it is compact. In particular, if a hypersurface in euclidean space is the Lie sphere image of an isoparametric hypersurface each compact sheet of the focal set will be tight.

References

Vorlesungen über Differentialgeometrie

Sheila Carter and Alan West, Tight and taut immersions, Proc. London Math. Soc. (3) 25 (1972), 701–720. MR 314071, DOI 10.1112/plms/s3-25.4.701
Thomas E. Cecil and Shiing Shen Chern, Tautness and Lie sphere geometry, Math. Ann. 278 (1987), no. 1-4, 381–399. MR 909233, DOI 10.1007/BF01458076

Dupin submanifolds in Lie sphere geometry

Thomas E. Cecil and Patrick J. Ryan, Focal sets, taut embeddings and the cyclides of Dupin, Math. Ann. 236 (1978), no. 2, 177–190. MR 503449, DOI 10.1007/BF01351390

Tight and taut immersions of manifolds

Nicolaas H. Kuiper, On convex maps, Nieuw Arch. Wisk. (3) 10 (1962), 147–164. MR 145545
Nicolaas H. Kuiper, Tight embeddings and maps. Submanifolds of geometrical class three in $E^{N}$, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 97–145. MR 609559

Geometrie der Berührungsformationen

Morse theory

Reiko Miyaoka, Compact Dupin hypersurfaces with three principal curvatures, Math. Z. 187 (1984), no. 4, 433–452. MR 760045, DOI 10.1007/BF01174180
Reiko Miyaoka, Taut embeddings and Dupin hypersurfaces, Differential geometry of submanifolds (Kyoto, 1984) Lecture Notes in Math., vol. 1090, Springer, Berlin, 1984, pp. 15–23. MR 775141, DOI 10.1007/BFb0101563

Isoparametrische hyperflächen in Sphären

251

256

Katsumi Nomizu, Élie Cartan’s work on isoparametric families of hypersurfaces, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 191–200. MR 0423260
Katsumi Nomizu and Lucio Rodríguez, Umbilical submanifolds and Morse functions, Nagoya Math. J. 48 (1972), 197–201. MR 315630
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Tetsuya Ozawa, On critical sets of distance functions to a taut submanifold, Math. Ann. 276 (1986), no. 1, 91–96. MR 863709, DOI 10.1007/BF01450927
Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957), iii+123. MR 121424
U. Pinkall, Dupin hypersurfaces, Math. Ann. 270 (1985), no. 3, 427–440. MR 774368, DOI 10.1007/BF01473438
Ulrich Pinkall, Curvature properties of taut submanifolds, Geom. Dedicata 20 (1986), no. 1, 79–83. MR 823161, DOI 10.1007/BF00149273
Ulrich Pinkall and Gudlaugur Thorbergsson, Deformations of Dupin hypersurfaces, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1037–1043. MR 975655, DOI 10.1090/S0002-9939-1989-0975655-9
Patrick J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J. (2) 21 (1969), 363–388. MR 253243, DOI 10.2748/tmj/1178242949
Shigeo Sasaki and Tsuneo Suguri, On the problems of equivalence of plane curves in the Lie’s higher circle geometry and of minimal curves in the conformal geometry, Tôhoku Math. J. 47 (1940), 77–86. MR 2256
Chuu-Lian Terng, Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), no. 1, 79–107. MR 806704
Gudlaugur Thorbergsson, Dupin hypersurfaces, Bull. London Math. Soc. 15 (1983), no. 5, 493–498. MR 705529, DOI 10.1112/blms/15.5.493