TY - JOUR

T1 - Spectral analysis and geometry of a sub-Riemannian structure on S3 and S7

AU - Bauer, Wolfram

AU - Furutani, Kenro

N1 - Copyright: Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008/8/5

Y1 - 2008/8/5

N2 - The purpose of this paper is to study the spectral properties of a sub-Laplacian on S3, 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 S3. Also, we describe sub-Riemannian curves on S3 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 S3 via an isoperimetric problem through the Hopf bundle is explained. Incidentally, we introduce a hypo-elliptic operator on P1 C descended from the sub-Laplacian on S3, 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 S7, we limit ourselves to present the spectral zeta function of a sub-Laplacian.

AB - The purpose of this paper is to study the spectral properties of a sub-Laplacian on S3, 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 S3. Also, we describe sub-Riemannian curves on S3 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 S3 via an isoperimetric problem through the Hopf bundle is explained. Incidentally, we introduce a hypo-elliptic operator on P1 C descended from the sub-Laplacian on S3, 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 S7, we limit ourselves to present the spectral zeta function of a sub-Laplacian.

KW - Chow condition

KW - Heat kernel

KW - Hörmander condition

KW - Quaternion and Cayley number fields

KW - Spectral zeta function

KW - Sub-Laplacian

KW - Sub-Riemannian manifold

UR - http://www.scopus.com/inward/record.url?scp=55549100932&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2008.07.011

DO - 10.1016/j.geomphys.2008.07.011

M3 - Article

AN - SCOPUS:55549100932

VL - 58

SP - 1693

EP - 1738

JO - Journal of geometry and physics

JF - Journal of geometry and physics

SN - 0393-0440

IS - 12

ER -