TY - JOUR

T1 - Trivializable and Quaternionic Subriemannian Structures on S7 and Subelliptic Heat Kernel

AU - Bauer, Wolfram

AU - Laaroussi, Abdellah

N1 - Funding Information: Both authors have been supported by the priority program SPP 2026 geometry at infinity of Deutsche Forschungsgemeinschaft (Project Number BA 3793/6-1)

PY - 2022/8

Y1 - 2022/8

N2 - On the seven dimensional Euclidean sphere S7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution HT is induced by a Clifford module structure of R8. More precisely, HT is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution HQ of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration S3↪ S7→ S4. Answering a question in: Markina and Godoy Molina (Rev Mat Iberoam 27(3), 997–1022, 2011) we first show that HQ does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures [20], the intrinsic sublaplacians ΔsubT and ΔsubQ and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

AB - On the seven dimensional Euclidean sphere S7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution HT is induced by a Clifford module structure of R8. More precisely, HT is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution HQ of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration S3↪ S7→ S4. Answering a question in: Markina and Godoy Molina (Rev Mat Iberoam 27(3), 997–1022, 2011) we first show that HQ does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures [20], the intrinsic sublaplacians ΔsubT and ΔsubQ and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

KW - Heat kernel

KW - Spectrum of geometric operators

KW - Sublaplacian

KW - Subriemannian geometry

UR - http://www.scopus.com/inward/record.url?scp=85131759953&partnerID=8YFLogxK

U2 - 10.1007/s12220-022-00954-8

DO - 10.1007/s12220-022-00954-8

M3 - Article

AN - SCOPUS:85131759953

VL - 32

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 8

M1 - 219

ER -