We study the first positive eigenvalue λ(p)1 of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality λ(1)1 ≤ λ(0)1 holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ(1)1 < λ(0)1 holds. We show that any oriented closed manifold M with the first Betti number b1(M) = 0 whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.
|Number of pages||14|
|Journal||Journal of the Mathematical Society of Japan|
|Publication status||Published - 2001 Apr 1|
- Einstein manifold
- Laplacian on forms
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