Abstract
Following Mark Kac, it is said that a geometric property of a compact Riemannian manifold can be heard if it can be determined from the eigenvalue spectrum of the associated Laplace operator on functions. On the contrary, D’Atri spaces, manifolds of type \({\mathcal{A}}\), probabilistic commutative spaces, \({\mathfrak{C}}\)-spaces, \({\mathfrak{TC}}\)-spaces, and \({\mathfrak{GC}}\)-spaces have been studied by many authors as symmetric-like Riemannian manifolds. In this article, we prove that for closed Riemannian manifolds, none of the properties just mentioned can be heard. Another class of interest is the class of weakly symmetric manifolds. We consider the local version of this property and show that weak local symmetry is another inaudible property of Riemannian manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arias-Marco, T.: Study of homogeneous D’Atri spaces of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica ©. Dissertation, Universitat de València, Valencia, Spain (2007) ISBN: 978-84-370-6838-1, http://www.tdx.cat/TDX-0911108-110640
Arias-Marco T.: The classification of 4-dimensional homogeneous D’Atri spaces revisited. Differ. Geom. Appl. 25, 29–34 (2007)
Arias-Marco T., Kowalski O.: The classification of 4-dimensional homogeneous D’Atri spaces. Czechoslovak Math. J. 58(133), 203–239 (2008)
Berndt J., Vanhecke L.: Two natural generalizations of locally symmetric spaces. Differ. Geom. Appl. 2, 57–80 (1992)
Berndt J., Vanhecke L.: Geometry of weakly symmetric spaces. J. Math. Soc. Japan 48, 745–760 (1996)
Berndt J., Prüfer F., Vanhecke L.: Symmetric-like Riemannian manifolds and geodesic symmetries. Proc. Roy. Soc. Edinburgh Sect. A 125(2), 265–282 (1995)
Berndt J., Tricerri F., Vanhecke L.: Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, Lecture Notes in Mathematics 1598. Springer-Verlag, Berlin/Heidelberg/New York (1995)
Cho J.T., Sekiwaga K., Vanhecke L.: Volume-preserving geodesicsymmetries on four-dimensional Hermitian Einstein spaces. Nagoya Math. J. 146, 13–29 (1997)
D’Atri J.E., Nickerson H.K.: Divergence preserving geodesicsymmetries. J. Diff. Geom. 3, 467–476 (1969)
Eberlein P.: Geometry of two-step nilpotent groups with a leftinvariant metric. Ann. Sci. Ecole Norm. Sup. 27(4), 611–660 (1994)
Gordon C.S., Gornet R., Schueth D., Webb D., Wilson E.N.: Isospectral deformations of closed Riemannian manifolds with different scalar curvature. Ann. Inst. Fourier 48(2), 593–607 (1998)
Gray A.: Einstein-like manifolds which are not Einstein. Geometriae Dedicata 7, 259–280 (1978)
Kobayashi S., Nomizu K.: Foundations of Differential Geometry. Vol. 1, Interscience, New York/London (1963)
Kowalski O.: Some curvature identities for commutative spaces. Czechoslovak Math. J. 32, 389–396 (1982)
Kowalski, O.: Spaces with volume-preserving symmetries and related classes of Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, pp. 131–158 (1983)
Kowalski O., Prüfer F.: On probabilistic commutative spaces. Monasth. Math. 107, 57–68 (1989)
Kowalski O., Prüfer F., Vanhecke L.: D’Atri spaces. Prog. Nonlinear Differ. Equ. Appl 20, 241–284 (1996)
Lauret J.: Commutative spaces which are not weakly symmetric. Bull London Math. Soc. 30, 29–36 (1998)
Pedersen H., Tod P.: The Ledger curvature conditions and D’Atri geometry. Differ. Geom. Appl. 11, 155–162 (1999)
Roberts P.H., Ursell H.D.: Random walk on a sphere and on a Riemannian manifold. Philos. Trans. Roy. Soc. London, Ser. A 252, 317–356 (1960)
Schueth D.: Isospectral manifolds with different local geometries. J. Reine Angew. Math. 534, 41–94 (2001)
Sekiwaga K., Vanhecke L.: Volume-preserving geodesic symmetries on four-dimensional 2-stein spaces. Kodai Math. J. 9, 215–224 (1986)
Sekiwaga K., Vanhecke L.: Volume-preserving geodesic symmetries on four-dimensional Kähler manifolds. Lect. Notes Math. 1209, 275–291 (1986)
Selberg A.: Harmonic analysis and discontinuous groups inweakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956)
Szabó Z.I.: Spectral theory for operator families on Riemannian manifolds. Proc. Symp. Pure Math. 54(3), 615–665 (1993)
Szabó Z.I.: Locally non-isometric yet super isospectral spaces. Geom. Funct. Anal. 9(1), 185–214 (1999)
Vanhecke L., Willmore T.J.: Interaction of tubes and spheres. Math. Ann. 263, 31–42 (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
Teresa Arias-Marco would like to dedicate this article to Prof. Angel Montesinos Amilibia, on the occasion of his retirement.
The authors were partially supported by DFG Sonderforschungsbereich 647. T. Arias-Marco has also been supported by D.G.I. (Spain) and FEDER Project MTM 2007-65852 and the network MTM2008-01013-E.
Rights and permissions
About this article
Cite this article
Arias-Marco, T., Schueth, D. On inaudible curvature properties of closed Riemannian manifolds. Ann Glob Anal Geom 37, 339–349 (2010). https://doi.org/10.1007/s10455-009-9189-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-009-9189-1
Keywords
- Laplace operator
- Isospectral manifolds
- Curvature tensor
- Weak symmetry
- D’Atri spaces
- Type \({\mathcal{A}}\) spaces
- Probabilistic commutative spaces
- \({\mathfrak{C}}\)-spaces
- \({\mathfrak{TC}}\)-spaces
- \({\mathfrak{GC}}\)-spaces