Tohoku Mathematical Journal
2000
June
SECOND SERIES VOL. 52, NO. 2

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

Title THE FIRST EIGENVALUES OF FINITE RIEMANNIAN COVERS

Author Katsuhiro Yoshiji

(Received October 5, 1998, revised January 12, 2000)
Abstract. 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.

1991 Mathematics Subject Classification. Primary 58G25; Secondary 53C20.

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