The metric at infinity on Damek-Ricci spaces

Abstract

Let S = N A be a Damek-Ricci space, identified with the unit ball B in s via the Cayley transform. Let S p+q = @B be the unit sphere in s, p = dim v, q = dim z. The metric in the ball model was computed in [1] both in Euclidean (or geodesic) polar coordinates and in Cartesian coordinates on B. The induced metric on the Euclidean sphere S(R) of radius R is the sum of a constant curvature term, plus a correction term proportional to h1, where h1 is a suitable differential expression which is smooth on S(R) for R < 1, but becomes (possibly) singular on the unit sphere at the pole (0, 0, 1). It has a simple geometric interpretation, namely h1 = |2|2, where 2 is, up to a conformal factor, the pull-back of the canonical 1-form on the group N (defining the horizontal distribution on N ) by the generalized stereographic projection. In the symmetric case h1, as well as the transported distribution on S p+q \ {(0, 0, 1)}, have a smooth extension to the whole sphere. This can be interpreted by the Hopf fibration of S p+q . In the general case no such structure is allowed on the unit sphere, and the question was left open in [1] whether or not h1 extends smoothly at the pole. In this paper we prove that h1 does not extend, except in the symmetric case. More precisely, writing h1 in the coordinates (V, Z) on S p+q as h1 = P h i j dzi dz j + P h i j dvi dv j + P h i j dzi dv j , we prove that, in the non-symmetric case, the coefficients h i j do not have a limit at the pole, but remain bounded there, whereas the coefficients h i j and h (zv) i j extend smoothly at the pole. In order to do this, we obtain an explicit formula for the 1-form 2 valid for any Damek-Ricci space. From this formula we deduce that2 does not extend to the pole, except for q = 1 (Hermitian symmetric case). The square of 2 and the distribution ker2 do not extend, unless S is symmetric. Indeed, we prove that the singular part of h1 vanishes identically if and only if the J2-condition holds. Mathematics Subject Classification (2010): 22E25 (primary); 43A85, 53C30 (secondary).