A general theory of canonical forms
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- by Richard S. Palais and Chuu-Lian Terng PDF
- Trans. Amer. Math. Soc. 300 (1987), 771-789 Request permission
Abstract:
If $G$ is a compact Lie group and $M$ a Riemannian $G$-manifold with principal orbits of codimension $k$ then a section or canonical form for $M$ is a closed, smooth $k$-dimensional submanifold of $M$ which meets all orbits of $M$ orthogonally. We discuss some of the remarkable properties of $G$-manifolds that admit sections, develop methods for constructing sections, and consider several applications.References
- Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029. MR 105694, DOI 10.2307/2372843
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 72877, DOI 10.2307/2372597
- Lawrence Conlon, Variational completeness and $K$-transversal domains, J. Differential Geometry 5 (1971), 135–147. MR 295252
- Lawrence Conlon, A class of variationally complete representations, J. Differential Geometry 7 (1972), 149–160. MR 377954
- Jiri Dadok, On the $C^{\infty }$ Chevalley’s theorem, Adv. in Math. 44 (1982), no. 2, 121–131. MR 658537, DOI 10.1016/0001-8708(82)90002-0
- Jiri Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125–137. MR 773051, DOI 10.1090/S0002-9947-1985-0773051-1
- Michael Davis, Smooth $G$-manifolds as collections of fiber bundles, Pacific J. Math. 77 (1978), no. 2, 315–363. MR 510928
- Dirk Ferus, Hermann Karcher, and Hans Friedrich Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), no. 4, 479–502 (German). MR 624227, DOI 10.1007/BF01219082
- Robert Hermann, Variational completeness for compact symmetric spaces, Proc. Amer. Math. Soc. 11 (1960), 544–546. MR 124436, DOI 10.1090/S0002-9939-1960-0124436-6
- Robert Hermann, Totally geodesic orbits of groups of isometries, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 291–298. MR 0139115 W. T. Hsiang, W. Y. Hsiang and I. Sterling, On the construction of codimension two minimal immersions of exotic spheres into Euclidean spheres, preprint.
- Wu-yi Hsiang, On the compact homogeneous minimal submanifolds, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 5–6. MR 205203, DOI 10.1073/pnas.56.1.5
- Wu-Yi Hsiang, Minimal cones and the spherical Bernstein problem. I, Ann. of Math. (2) 118 (1983), no. 1, 61–73. MR 707161, DOI 10.2307/2006954
- Wu-yi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1–38. MR 298593
- W.-Y. Hsiang, R. S. Palais, and C. L. Terng, Geometry and topology of isoparametric submanifolds in Euclidean spaces, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), no. 15, 4863–4865. MR 799109, DOI 10.1073/pnas.82.15.4863 S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience, New York, 1969.
- Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57–71 (German). MR 583825, DOI 10.1007/BF01420281
- Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865 H. Ozeki and M. Takeuchi, On some types of isoparametric hypersurfaces in spheres. 1, 2, Tôhoku Math. J. 27 (1975), 515-559; 28 (1976), 7-55.
- Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335
- Richard S. Palais and Chuu-Lian Terng, Reduction of variables for minimal submanifolds, Proc. Amer. Math. Soc. 98 (1986), no. 3, 480–484. MR 857946, DOI 10.1090/S0002-9939-1986-0857946-2
- Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68. MR 370643, DOI 10.1016/0040-9383(75)90036-1
- Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37–135. MR 573821
- J. Szenthe, A generalization of the Weyl group, Acta Math. Hungar. 41 (1983), no. 3-4, 347–357. MR 703746, DOI 10.1007/BF01961321
- J. Szenthe, Orthogonally transversal submanifolds and the generalizations of the Weyl group, Period. Math. Hungar. 15 (1984), no. 4, 281–299. MR 782429, DOI 10.1007/BF02454161
- Chuu-Lian Terng, Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), no. 1, 79–107. MR 806704
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 300 (1987), 771-789
- MSC: Primary 57S15; Secondary 53C20, 58E30
- DOI: https://doi.org/10.1090/S0002-9947-1987-0876478-4
- MathSciNet review: 876478