Isoparametric Foliations and their Buildings

Thorbergsson, Gudlaugur

doi:10.2307/2944343

Cited by 79 publications

(73 citation statements)

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“…The Algebraicity Theorem reduces to Hilbert's Theorem in the homogeneous case, and to [Mün80, Satz 2] and [Ter85,Theorem D] in the isoparametric case (see also [Tho91]). …”

Section: Preliminariesmentioning

confidence: 99%

A slice theorem for singular Riemannian foliations, with applications

Mendes¹,

Radeschi

2018

Trans. Amer. Math. Soc.

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“…The Algebraicity Theorem reduces to Hilbert's Theorem in the homogeneous case, and to [Mün80, Satz 2] and [Ter85,Theorem D] in the isoparametric case (see also [Tho91]). …”

Section: Preliminariesmentioning

confidence: 99%

A slice theorem for singular Riemannian foliations, with applications

Mendes¹,

Radeschi

2018

Trans. Amer. Math. Soc.

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“…Classification of isoparametric submanifolds. Another application of the topological spherical Tits buildings to differential geometry concerns isoparametric submanifolds in Euclidean spaces [Tho2] [Ter1].…”

Section: This Is Proved In [Mosmentioning

confidence: 99%

“…In fact, it is the major part of the paper [Tho2], which depends on a local approach of Tits on constructing buildings.…”

Section: Theorem 2105 Any Compact Irreducible Full Isoparametric mentioning

confidence: 99%

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Buildings and their Applications in Geometry and Topology

Ji¹

2006

Asian Journal of Mathematics

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Abstract. Buildings were first introduced by J. Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic K-theories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and S-arithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications:1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces and also appear as limiting objects under degeneration or scaling of metrics. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups. 3. Buildings of higher rank are rigid and hence objects which contain or induce higher rank buildings tend to be rigid. 4. Additional structures on buildings, for example, topological buildings, are important in applications for infinite groups.

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“…[14]. By contrast, (irreducible) isoparametric foliations in spheres either have cohomogeneity one, or arise from a polar representation [17]. (3) Any singular Riemannian foliation F on KP n , (K = R, C, H) lifts to a foliation F * on a sphere via the Hopf map S m → KP n .…”

Section: Introductionmentioning

confidence: 99%

Mean Curvature Flow of Singular Riemannian Foliations

Alexandrino

Radeschi

2015

J Geom Anal

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Abstract. Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. These results generalize previous results of Liu and Terng, Pacini and Koike. In particular our results can be applied to partitions of Riemannian manifolds into orbits of actions of compact groups of isometries.

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Isoparametric Foliations and their Buildings

Cited by 79 publications

References 0 publications

A slice theorem for singular Riemannian foliations, with applications

A slice theorem for singular Riemannian foliations, with applications

Buildings and their Applications in Geometry and Topology

Mean Curvature Flow of Singular Riemannian Foliations

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