Tohoku Mathematical Journal

The first eigenvalues of finite Riemannian covers

Katsuhiro Yoshiji

Full-text: Open access


There exists a Riemannian metric on the real projective space such that the first eigenvalue coincides with that of its Riemannian universal cover, if the dimension is bigger than 2. For the proof, we deform the canonical metric on the real projective space. A similar result is obtained for lens spaces, as well as for closed Riemannian manifolds with Riemannian double covers. As a result, on a non-orientable closed manifold other than the real projective plane, there exists a Riemannian metric such that the first eigenvalue coincides with that of its Riemannian double cover.

Article information

Tohoku Math. J. (2), Volume 52, Number 2 (2000), 261-270.

First available in Project Euclid: 3 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]


Yoshiji, Katsuhiro. The first eigenvalues of finite Riemannian covers. Tohoku Math. J. (2) 52 (2000), no. 2, 261--270. doi:10.2748/tmj/1178224610.

Export citation


  • [1] C ANNE, Spectre du laplacien et ecrasement d'anses, Ann Sci Ecole Norm Sup 20 (1987), 271-280
  • [2] C ANNE ETB COLBOIS, Spectre du laplacien agissant sur les /7-formes differentielle et ecrasement d'anses, Math Ann 303 (1995), 545-573.
  • [3] M BERGER, P GAUDUCHONETE MAZET, Le spectre d'une variete riemannienne, Lecture Notes in Mat 194, Springer, 1974
  • [4] J CHEEGER, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis: A Sympo sium in Honor of Salomon Bochoner, Princeton, 1970, pp. 195-199
  • [5] A IKEDA, On lens spaces which are isospectral but not isometric, Ann Sci Ecole Norm Sup 13 (1980), 303-315
  • [6] T SAKAI, On the spectrum of lens spaces, Kodai Math Sem Rep 29 (1975), 249-25
  • [7] T SUNADA, The Fundamental Group and the Laplacian, Kinokuni-ya, 198
  • [8] S TANNO, The first eigenvalue of the Laplacian on spheres, Thoku Math J 31 (1979), 179-18
  • [9] H URAKAWA, On the positive eigenvalue of the Laplacian for compact group manifolds, J Math Soc Japa 31 (1979), 209-226
  • [10] J WOLF, Spaces of Constant Curvature, McGraw-Hill, New York-London-Sydney, 1967
  • [11] K YOSHIJI, On the first eigenvalue of non-orientable closed surfaces, Tsukuba J Math 22 (1998), 741-74