Abstract
We study the spectral geometry of the Laplacian with Dirichlet and Neumann boundary conditions and the spectral geometry of the conformal Laplacian with Dirichlet and Robin boundary conditions. We show in §1 geometric properties of the boundary such as totally geodesic boundary, constant mean curvature, and totally umbillic are spectrally determined. In §2, we expand the invariants of the heat equation on a small geodesic ball in a power series in the radius. We characterize Einstein, conformally flat, and constant sectional curvature manifolds by the spectral geometry of their geodesic balls. Also, some characterizations are obtained for the rank 1 symmetric spaces S n, CP n, QP n, CaP 2 and their noncompact duals. MOS subject classification: 58G25
This paper is in final form and no version of it will be submited for publication elsewhere.
Research of P. Gilkey partially supported by the NSF and NSA.
Research of N. Blažić and N. Bokan partially supported by the conference “Global Differential Geometry and Global Analysis”, TU Berlin, June 15–20, 1990.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. V. Alekseevskij, “On holonomy groups of Riemannian manifolds,” Ukrain. Math. Z. 19 (1967), 100–104.
P. Amsterdamski, A. Berkin, and D. O'Connor, “b 8'Hamidew’ coefficient for a scalar field”, Class. Quantum Grav. 6 (1989), 1981–1991.
I. Avramidi, “The covariant technique for calculation of one-loop effective action,” to appear in Nucl. Phys. B.
T. Branson and P. Gilkey, “The asymptotics of the Laplacian on a manifold with boundary,” Communications on PDE, 15 (1990), 245–272.
T. Branson and B. Orsted, “Conformal indices of Riemannian manifolds,” Compositio Math. 60(1986), 261–293.
R. B. Brown and A. Gray, “Manifolds whose holonomy group is subgroup of Spin(9),” Differential geometry (in honor of K. Yano), Tokyo, 1972, 41–59.
B.Y. Chen and L. Vanhecke, “Differential geometry of geodesic spheres,” Journal für die reine und angewandte Mathematik, 325(1981), 28–67.
M. Ferraratti and L. Vanhecke, “Curvature invariants and symmetric spaces,” in preparation.
P. Gilkey, “The spectral geometry of a Riemannian manifold,” J. Diff. Geom. 10(1975), 601–618
A. Gray, “The volume of a small geodesic ball in a Riemannian manifold,” Michigan Math. J. 20(1973), 329–344.
A. Gray and L. Vanhecke, “Riemannian geometry as determined by the volumes of small geodesic balls,” Acta Math. 142(1979), 157–198.
L. Karp and M. Pinsky, “The first eigenvalue of a small geodesic ball in a Riemannian manifold,” Bull. Sc. Math 111 (1987), 229–239.
G. Kennedy, R. Critchley, and J.S. Dowker, “Finite temperature field theory with boundaries: stress tensor and surface action renormalization,” Annals of Physics 125(1980), 346–400.
O. Kowalski, F. Tricerri, and L. Vanhecke, “Curvature homogenous Riemannian manifolds,” J. Math. Pures Appl., to appear.
F. Lastaria, “Homogeneous metrics with the same curvature,” Simon Stevin, to appear.
A. Levy-Bruhl, “Invariants infinitesimaux,” C. R. Acad. Sc. Paris. T 279 (1974), 197–200.
I. Moss and J.S. Dowker, “The correct B 4coefficient,” (preprint).
H. McKean and I. Singer, “Curvature and the eigenvalues of the Laplacian,” J. Diff. Geom. 1(1967), 43–69.
T. Sakai, “On eigenvalues of Laplacian and curvature of Riemannian manifolds,” Tohoku Math J 23(1971), 589–603.
F. Tricerri and L. Vanhecke, “Curvature homogenous Riemannian spaces,” Ann. Sc. Ecole Norm. Sup. 22(1989), 535–554.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Blažić, N., Bokan, N., Gilkey, P. (1991). The spectral geometry of the laplacian and the conformal laplacian for manifolds with boundary. In: Ferus, D., Pinkall, U., Simon, U., Wegner, B. (eds) Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol 1481. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083623
Download citation
DOI: https://doi.org/10.1007/BFb0083623
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54728-0
Online ISBN: 978-3-540-46445-7
eBook Packages: Springer Book Archive