R. S. Palais and C. L. Terng :
“Natural bundles have finite order Topology
16 : 3
(1977 ),
pp. 271–277 .
MR
467787 Zbl
0359.58004 article
People
BibTeX
@article {key467787m,
AUTHOR = {Palais, Richard S. and Terng, Chuu Lian},
TITLE = {Natural bundles have finite order},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {16},
NUMBER = {3},
YEAR = {1977},
PAGES = {271--277},
DOI = {10.1016/0040-9383(77)90008-8},
NOTE = {MR:467787. Zbl:0359.58004.},
ISSN = {0040-9383},
}
R. S. Palais, A. Derdzinski, and C.-L. Terng :
“Warped products and Einstein manifolds and Hessian PDE ,”
pp. 44–47
in
Proceedings of differential geometry meeting, Münster
(Münster, Germany, 1982 ).
1982 .
incollection
People
BibTeX
@incollection {key31433097,
AUTHOR = {Palais, Richard S. and Derdzinski, A.
and Terng, C.-L.},
TITLE = {Warped products and {E}instein manifolds
and {H}essian {PDE}},
BOOKTITLE = {Proceedings of differential geometry
meeting, {M}\"unster},
YEAR = {1982},
PAGES = {44--47},
NOTE = {(M\"unster, Germany, 1982).},
}
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 : 15
(August 1985 ),
pp. 4863–4865 .
MR
799109 Zbl
0573.53033 article
Abstract
People
BibTeX
@article {key799109m,
AUTHOR = {Hsiang, W.-Y. and Palais, R. S. and
Terng, C. L.},
TITLE = {Geometry and topology of isoparametric
submanifolds in {E}uclidean spaces},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {82},
NUMBER = {15},
MONTH = {August},
YEAR = {1985},
PAGES = {4863--4865},
DOI = {10.1073/pnas.82.15.4863},
NOTE = {MR:799109. Zbl:0573.53033.},
ISSN = {0027-8424},
}
R. S. Palais and C.-L. Terng :
“Reduction of variables for minimal submanifolds Proc. Am. Math. Soc.
98 : 3
(November 1986 ),
pp. 480–484 .
MR
857946 Zbl
0627.53050 article
Abstract
People
BibTeX
If \( G \) is a compact Lie group and \( M \) a Riemannian \( G \) manifold, then the orbit map
\[ \Pi : M \to M/G \]
is a stratified Riemannian submersion and the well-known “cohomogeneity method” pioneered by Hsiang and Lawson [1971] reduces the problem of finding codimension \( k \) minimal submanifolds of \( M \) to a related problem in \( M/G \) . We show that this reduction of variables technique depends only on a certain natural Riemannian geometric property of the map \( \Pi \) which we call \( h \) -projectability and which is shared by certain other naturally occurring and important classes of Riemannian submersions.
@article {key857946m,
AUTHOR = {Palais, Richard S. and Terng, Chuu-Lian},
TITLE = {Reduction of variables for minimal submanifolds},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {98},
NUMBER = {3},
MONTH = {November},
YEAR = {1986},
PAGES = {480--484},
DOI = {10.2307/2046207},
NOTE = {MR:857946. Zbl:0627.53050.},
ISSN = {0002-9939},
}
R. S. Palais and C.-L. Terng :
“A general theory of canonical forms Trans. Am. Math. Soc.
300 : 2
(1987 ),
pp. 771–789 .
MR
876478 Zbl
0652.57023 article
Abstract
People
BibTeX
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.
@article {key876478m,
AUTHOR = {Palais, Richard S. and Terng, Chuu-Lian},
TITLE = {A general theory of canonical forms},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {300},
NUMBER = {2},
YEAR = {1987},
PAGES = {771--789},
DOI = {10.2307/2000369},
NOTE = {MR:876478. Zbl:0652.57023.},
ISSN = {0002-9947},
}
R. S. Palais and C.-L. Terng :
“Geometry of canonical forms 133–151
in
The legacy of Sonya Kovalevskaya
(Cambridge, MA and Amherst, MA, 25–28 October 1985 ).
Contemporary Mathematics 64 .
American Mathematical Society (Providence, RI ),
1987 .
MR
881460 Zbl
0607.57023 incollection
Abstract
People
BibTeX
If \( G \) is a compact Lie group and \( X \) a Riemannian \( G \) -manifold with principal orbits of codimension \( k \) then a section or canonical form for \( X \) is a closed, smooth \( k \) -dimensional submanifold of \( X \) which meets all orbits of \( M \) orthogonally. We discuss some of the remarkable properties of \( G \) -manifolds that admit sections, develop methods for construction sections, and consider several applications.
@incollection {key881460m,
AUTHOR = {Palais, Richard S. and Terng, Chuu-Lian},
TITLE = {Geometry of canonical forms},
BOOKTITLE = {The legacy of {S}onya {K}ovalevskaya},
SERIES = {Contemporary Mathematics},
NUMBER = {64},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1987},
PAGES = {133--151},
DOI = {10.1090/conm/064/881460},
NOTE = {(Cambridge, MA and Amherst, MA, 25--28
October 1985). MR:881460. Zbl:0607.57023.},
ISSN = {0271-4132},
ISBN = {9780821850671},
}
W.-Y. Hsiang, R. S. Palais, and C.-L. Terng :
“The topology of isoparametric submanifolds J. Diff. Geom.
27 : 3
(1988 ),
pp. 423–460 .
MR
940113 Zbl
0618.57018 article
Abstract
People
BibTeX
It has been known since a famous paper of Bott and Samelson that, using Morse theory, the homology and cohomology of certain homogeneous spaces can be computed algorithmically from Dynkin diagram and multiplicity data. L. Conlon and J. Dadok noted that these spaces are the orbits of the isotropy representations of symmetric spaces. Recently the theory of isoparametric hypersurfaces has been generalized to a theory of isoparametric submanifolds of arbitrary codimension in Euclidean space, and these same orbits turn out to be exactly the homogeneous examples. Even the nonhomogeneous examples have associated to them Weyl groups with Dynkin diagrams marked with multiplicities. We extend and simplify the Bott-Samelson method to compute the homology and cohomology of isoparametric submanifolds from their marked Dynkin diagrams.
@article {key940113m,
AUTHOR = {Hsiang, Wu-Yi and Palais, Richard S.
and Terng, Chuu-Lian},
TITLE = {The topology of isoparametric submanifolds},
JOURNAL = {J. Diff. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {27},
NUMBER = {3},
YEAR = {1988},
PAGES = {423--460},
DOI = {10.4310/jdg/1214442003},
URL = {http://projecteuclid.org/euclid.jdg/1214442003},
NOTE = {MR:940113. Zbl:0618.57018.},
ISSN = {0022-040X},
}
R. S. Palais and C.-L. Terng :
Critical point theory and submanifold geometry Lecture Notes in Mathematics 1353 .
Springer (Berlin ),
1988 .
MR
972503 Zbl
0658.49001 book
People
BibTeX
@book {key972503m,
AUTHOR = {Palais, Richard S. and Terng, Chuu-Lian},
TITLE = {Critical point theory and submanifold
geometry},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1353},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1988},
PAGES = {x+272},
DOI = {10.1007/BFb0087442},
NOTE = {MR:972503. Zbl:0658.49001.},
ISSN = {0075-8434},
ISBN = {9783540503996},
}
incollection
R. S. Palais and C.-L. Terng :
“The life and mathematics of Shiing Shen Chern ,”
pp. 18–62
in
S. S. Chern: A great geometer of the twentieth century .
Edited by S.-T. Yau .
Monographs in geometry and topology .
International Press (Hong Kong ),
1992 .
MR
1201340
People
BibTeX
@incollection {key1201340m,
AUTHOR = {Palais, Richard S. and Terng, Chuu-Lian},
TITLE = {The life and mathematics of {S}hiing
{S}hen {C}hern},
BOOKTITLE = {S.~{S}. {C}hern: {A} great geometer
of the twentieth century},
EDITOR = {Yau, Shing-Tung},
SERIES = {Monographs in geometry and topology},
PUBLISHER = {International Press},
ADDRESS = {Hong Kong},
YEAR = {1992},
PAGES = {18--62},
NOTE = {MR:1201340.},
ISBN = {9781571460981},
}
E. Heintze, R. Palais, C.-L. Terng, and G. Thorbergsson :
“Hyperpolar actions and \( k \) -flat homogeneous spaces J. Reine Angew. Math.
1994 : 454
(1994 ),
pp. 163–179 .
MR
1288683 Zbl
0804.53074 article
People
BibTeX
@article {key1288683m,
AUTHOR = {Heintze, E. and Palais, R. and Terng,
C.-L. and Thorbergsson, G.},
TITLE = {Hyperpolar actions and \$k\$-flat homogeneous
spaces},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal f\"ur die Reine und Angewandte
Mathematik. [Crelle's Journal]},
VOLUME = {1994},
NUMBER = {454},
YEAR = {1994},
PAGES = {163--179},
DOI = {10.1515/crll.1994.454.163},
NOTE = {MR:1288683. Zbl:0804.53074.},
ISSN = {0075-4102},
}
E. Heintze, R. S. Palais, C.-L. Terng, and G. Thorbergsson :
“Hyperpolar actions on symmetric spaces ,”
pp. 214–245
in
Geometry, topology, & physics: For Raoul Bott
(Cambridge, MA, 23–25 April 1993 ).
Edited by S.-T. Yau .
Conference Proceedings and Lecture Notes in Geometry and Topology 4 .
International Press (Cambridge, MA ),
1995 .
Lectures of a conference in honor of Raoul Bott’s 70th birthday.
MR
1358619 Zbl
0871.57035 incollection
Abstract
People
BibTeX
An isometric action of a compact Lie group \( G \) on a Riemannian manifold \( M \) is called polar if there exists a closed, connected submanifold of \( M \) that meets all \( G \) -orbits and meets orthogonally. Such a \( \Sigma \) is called a section . A section is automatically totally geodesic in \( M \) , and if it is also flat in the induced metric then the action is called hyperpolar . In this paper we study hyperpolar actions on compact symmetric spaces, prove some structure and classification theorems for them, and study their relation to polar actions on Hilbert space and to involutions of affine Kac–Moody algebras.
@incollection {key1358619m,
AUTHOR = {Heintze, Ernst and Palais, Richard S.
and Terng, Chuu-Lian and Thorbergsson,
Gudlaugur},
TITLE = {Hyperpolar actions on symmetric spaces},
BOOKTITLE = {Geometry, topology, \& physics: {F}or
{R}aoul {B}ott},
EDITOR = {Yau, Shing-Tung},
SERIES = {Conference Proceedings and Lecture Notes
in Geometry and Topology},
NUMBER = {4},
PUBLISHER = {International Press},
ADDRESS = {Cambridge, MA},
YEAR = {1995},
PAGES = {214--245},
NOTE = {(Cambridge, MA, 23--25 April 1993).
Lectures of a conference in honor of
Raoul Bott's 70th birthday. MR:1358619.
Zbl:0871.57035.},
ISBN = {9781571460240},
}
