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In this paper, we consider homogeneous groups of Heisenberg type, which are generally reductive, generated with a heat kernels. We derive an equation to the generating heat kernel, which takes the form

where  and  are elements of the Heisenberg group and  and are constants. The generating kernel is then used to construct the Lie algebra associated with the Heisenberg group. Using the Lie algebra, we define a homogeneous space associated with the Heisenberg group, which is a Riemannian manifold of Euclidian type. We provide a detailed analysis of the geometry of this homogeneous space and its relationship to the Lie algebra. In particular, we show how the Heisenberg group can be used to construct an invariant metric and study its properties.

homogeneous groups, Riemannian manifold, Heisenberg-type, Lie algebra, heat kernels.