Carnot spaces and the k-stein condition
Copyright policy )Carnot spaces and the k-stein condition
Abstract A riemannian manifold M with associated Jacobi operators R X (R X Y = R(Y, X) X), X in TM, is said to be k-stein, k ≥ 1, if there exists a function μk on M such that tr(R k X ) = μk |X|2k for all X in TM. We study the k-stein condition on Lie groups of Iwasawa type and in particular in those which are Carnot spaces. We show that a Carnot space which is k-stein for some k > 1 is necessarily a Damek–Ricci space; Damek–Ricci spaces are Einstein and 2-stein and they are not k-stein for any k ≥ 3, unless they are symmetric. Moreover, we show that a harmonic Lie group of Iwasawa type which is 3-stein is a symmetric space of noncompact type and rank one.
- National University of Córdoba Argentina
Damek-Ricci spaces, Differential geometry of homogeneous manifolds, harmonic spaces, Carnot spaces, \(k\)-stein condition, rank one symmetric spaces, Global differential geometry of Hermitian and Kählerian manifolds, Lie group of Iwasawa type
Damek-Ricci spaces, Differential geometry of homogeneous manifolds, harmonic spaces, Carnot spaces, \(k\)-stein condition, rank one symmetric spaces, Global differential geometry of Hermitian and Kählerian manifolds, Lie group of Iwasawa type
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