Carnot spaces and the k-stein condition

DRUETTA MARIA J.

ADVANCES IN GEOMETRY

de Gruyter

Lugar: Berlin; Año: 2006 vol. 6 p. 439 - 465

1615-715X

A riemannian manifold M with associated Jacobi operators R(X)=R(Y,X)X,  X a unitary vector in TM, is said to be k-stein (k>0), if there exists a function f on M such that tr ((R(X) with exponent  k ) equals to f ; that is it does not depend on unitary tangent vectors 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>2 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.