Trace formulae forS 1 invariant green’s operators onS 2

Summary We study the spectrum of the Laplacian acting on 1-forms for a surface of revolution diffeomorphic toS ² and obtain, for theS ¹ invariant spectrum, a trace formula for the eigenvalues of its Green’s operator. The trace formula is used to “hear” negative curvature for some metrics and to prove the existence of metrics onS ² with rather unusual spectral characteristics. This article was processed by the author using the L^AT_EX style file from Springer-Verlag.