We study the first positive eigenvalue of the Laplacian on -forms for oriented closed Riemannian manifolds. It is known that the inequality holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality holds. We show that any oriented closed manifold with the first Betti number whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.
Junya TAKAHASHI. "On the gap between the first eigenvalues of the Laplacian on functions and -forms." J. Math. Soc. Japan 53 (2) 307 - 320, April, 2001. https://doi.org/10.2969/jmsj/05320307