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• W155227778 abstract "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. The origin of the theory of hyperpolar actions can be traced back at least to the 1950’s, with Bott [B] and Bott and Samelson’s [BS] introduction of the closely related concept of a “variationally complete” action. We will give a brief review below of what we know of the history of this subject, but first we recall some basic notation and terminology from the theory of symmetric spaces that will be needed throughout the paper. Let G be a compact, semi-simple, connected Lie group, and G = TG e its Lie algebra. A subgroup K of G is called a symmetric subgroup of G (and the pair (G, K) is called a symmetric pair) if there is an involution σ of G such that (Kσ)0 ⊂ K ⊂ Kσ, where Kσ is the fixed point set of σ and (Kσ)0 is the connected component of Kσ. In this case G/K equipped with a G-invariant metric induced from any AdG invariant inner product for G is a symmetric space. The corresponding orthogonal decomposition G = K⊕ P is called the Cartan decomposition associated to σ. It is completely determined by σ (independent of the inner product on G); namely K and P are respectively the +1 and −1 eigenspaces of dσe : G→G . It follows that K and P satisfy the characteristic bracket relations [K, K] ⊆K , [K, P] ⊆P , and [P, P] ⊆K . 1.1 Definition. If (G, K) is a symmetric pair and G = K⊕ P the correspond- “refsrep ing Cartan decomposition, then the Adjoint representation of K on P is called an s-representation. Of course, up to equivalence the s-representation can be identified with the isotropy representation of the symmetric space G/K at eK (from which it gets its name)." @default.
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• W155227778 date "2001-01-01" @default.
• W155227778 modified "2023-09-26" @default.
• W155227778 title "Hyperpolar Actions on Symmetric Spaces" @default.
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