Abstract.We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, "can one hear the shape of a drum?" In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold. Kac's questionLet (M, g) be a compact Riemannian manifold with boundary. Then M has a Laplace operator A, defined by A(f ) = -div(grad / ), that acts on smooth functions on M. The spectrum of M is the sequence of eigenvalues of A. Two Riemannian manifolds are isospectral if their spectra coincide (counting multiplicities). A natural question concerning the interplay of analysis and geometry is: must two isospectral Riemannian manifolds actually be isometric? (When M has nonempty boundary, one can consider the Dirichlet spectrum, i.e., the spectrum of A acting on smooth functions that vanish on the boundary, or the Neumann spectrum, that of A acting on functions with vanishing normal derivative at the boundary.) If M is a domain in the Euclidean plane then the Dirichlet eigenvalues of A are essentially the frequencies produced by a drumhead shaped like M, so the question has been phrased by Bers and Kac [16] (the latter attributes the problem to Bochner) as "can one hear the shape of a drum?" We answer this question negatively by constructing a pair of nonisometric simply connected plane domains that have both the same Dirichlet spectra and the same Neumann spectra. The domains are depicted in Figure 1.The simple idea exploited here also permits us to construct the following: ( 1 ) a pair of isospectral flat surfaces (with boundary) one of which has a unitlength closed geodesic while the other has only a unit-length closed billiard trajectory; (2) a pair of isospectral potentials for the Schrödinger operator onReceived by the editors July 11, 1991 and, in revised form, November 5, 1991 1991 Mathematics Subject Classification. Primary 58G25, 35P05, 53C20.The authors gratefully acknowledge partial support from NSF grants. (4) As will be clear from the discussion of Sunada's theorem below, most known pairs of isospectral manifolds have a common Riemannian cover. Thus it is also of interest to exhibit simply connected isospectral manifolds. Sunada's TheoremAlthough the early examples of isospectral manifolds seemed rather ad hoc, a coherent explanation for most of them has since been provided. Sunada [19] introduced a general method for constructing pairs of isospectral manifolds with a common finite covering:Theorem (Sunada). Let M be a Riemannian manifold upon which a finite group G acts by isometries; let H and K be subgroups of G that act freely. Suppose that H and K are almost conjugate, i.e., there is a bijection f: H -> K carrying every element h of H to an element f(h) of K that is conjugate in G to h . Then the quotient manifolds Mx = H\M and M2 = K\M are isospectral.Choosing conjug...

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Two vector bundles associated to the moduli space of compact Riemann surfaces have a Hermitian metric derived from the hyperbolic geometry of Riemann surfaces. Briefly our purpose is to determine the connection and curvature forms for these metrics.The first bundle is the holomorphic tangent bundle of the Teichmfiller space of genus g, g > 2, Riemann surfaces; the metric is the Weil-Petersson metric. Weil introduced a K/ihler metric for the Teichmfiller space T o based on Petersson's Hermitian pairing for automorphic forms. Ahlfors considered the differential geometry of this metric; in particular he obtained integral formulas for the associated Riemann curvature tensor, [1,2]. As an application he found that the Ricci, holomorphic sectional, and scalar curvatures are all negative. Royden latter showed that the holomorphic sectional curvature is bounded away from zero, [16]. More recently Tromba found that the sectional curvature is also negative, [32]. After this result Royden and then the author also found proofs of the negative sectional curvature, [17]. In the present work we develop a formalism for computing second variations of a hyperbolic structure and consider as the first application a formula for the Riemann tensor. Before presenting the formula recall that the holomorphic tangent space of Teichmfiller space at the marked Riemann surface (S) is naturally isomorphic to ~ (S), the space of harmonic Beltrami differentials ((-1, 1) tensors) on S. Now denoting by dA the area element of the hyperbolic metric on S and by D the Laplacian of the hyperbolic metric then the Riemann tensor is given as

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