Characterisation of the sub-Riemannian isometry groups of theH-type groups

Abstract

For a H-type group G, we first give explicit equations for its shortest sub-Riemannian geodesies. We use properties of sub-Riemannian geodesies in G to characterise the isometry group ISO(G) with respect to the Carnot-Caratheodory metric. It turns out that ISO(G) coincides with the isometry group with respect to the standard Riemannian metric of G.

Lipschitz-Free Spaces Over Ultrametric Spaces

“…Our task is now to show that t´1, 1u ˆR`ˆI sopH n q and t´1, 1u ˆR`ˆI sopH m q are not isomorphic. By [26,Theorem 3.5], IsopH n q is the same group regardless of H n being equipped with the left-invariant Riemannian distance or the Carnot-Carathéodory metric. Furthermore, by [21,Corollary 2.3.5] it is also the same group if H n is equipped with the Korányi metric.…”

“…We have dimph i q " 2i `1, so we need to compute dimpa i q. Since by [26,Corollary 3.6 (1)] the connected component of the unit of A i can be identified with the unitary group U piq, we get that a i is isomorphic to the Lie algebra of U piq which is known to have dimension i 2 (see e.g. [18,Example 5.1.6 (v)]).…”

Gradient Estimates for the Heat Semigroup on H-Type Groups

“…The inverse function of μ is denoted by μ −1 . Then the Carnot-Carathéodory distance from o = (0, 0) to g = (x, t) is given by (see [27] or [28] pp. 90-91):…”

A Classification of Sub-Riemannian Structures on the Heisenberg Groups

“…The Heisenberg groups are the simplest, non-Euclidean Carnot groups. Structures on the Heisenberg groups (and their generalizations) have been extensively studied in the last few decades (see, e.g., [4,9,14,15,19]).…”