Rank 5 Trivializable Subriemannian Structure on $$\mathbb {S}^7$$ and Subelliptic Heat Kernel

Abstract

Abstract We present an explicit form of the subelliptic heat kernel of the intrinsic sublaplacian $$\Delta _{\textrm{sub}}^5$$ Δ sub 5 induced by a rank 5 trivializable subriemannian structure on the Euclidean seven dimensional sphere $$\mathbb {S}^7$$ S 7 . This completes the heat kernel analysis of trivializable subriemannian structures on $$\mathbb {S}^7$$ S 7 induced by a Clifford module action on $$\mathbb {R}^8$$ R 8 . As an application we derive the spectrum of $$\Delta _{\textrm{sub}}^5$$ Δ sub 5 and the Green function of the conformal sublaplacian in an explicit form.