Details
| Original language | English |
|---|---|
| Pages (from-to) | 1693-1738 |
| Number of pages | 46 |
| Journal | Journal of geometry and physics |
| Volume | 58 |
| Issue number | 12 |
| Publication status | Published - 5 Aug 2008 |
| Externally published | Yes |
Abstract
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.
Keywords
- Chow condition, Heat kernel, Hörmander condition, Quaternion and Cayley number fields, Spectral zeta function, Sub-Laplacian, Sub-Riemannian manifold
ASJC Scopus subject areas
- Mathematics(all)
- Mathematical Physics
- Physics and Astronomy(all)
- Mathematics(all)
- Geometry and Topology
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In: Journal of geometry and physics, Vol. 58, No. 12, 05.08.2008, p. 1693-1738.
Research output: Contribution to journal › Article › Research › peer review
}
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 -