Spectral analysis and geometry of a sub-Riemannian structure on S3S3 and S7S7

The purpose of this paper is to study the spectral properties of a sub-Laplacian on S3S3, i.e., we discuss the analytic continuation of its spectral zeta function, give explicit expressions of the residues and especially, we provide an expression of the zeta-regularized determinant of the sub-Laplacian   on S3S3. Also, we describe sub-Riemannian curves on S3S3 based on the Hopf bundle structure, together with a proof of Chow’s theorem for this case in a strong sense (= connecting property by globally smooth curves). A characterization of sub-Riemannian geodesics on S3S3 via an isoperimetric problem through the Hopf bundle is explained. Incidentally, we introduce a hypo-elliptic operator on P1CP1C descended from the sub-Laplacian on S3S3, which we call a spherical Grushin operator  . We determine the subspace where it degenerates and give an expression of the trace of its heat kernel by making use of the trace of the heat kernel of the sub-Laplacian. In case of S7S7, we limit ourselves to present the spectral zeta function of a sub-Laplacian.