Spaces and moduli spaces of Riemannian metrics

These notes present and survey results about spaces and moduli spaces of complete Riemannian metrics with curvature bounds on open and closed manifolds, here focussing mainly on connectedness and disconnectedness properties. They also discuss several open problems and questions in the field.     

Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold M ⁿ suitable uniform structures, obtain completed spaces ^b,m M or M ^r (I, B _k ), respectively and calculate for each component of M ^r (I, B _k ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.     

Degenerate Riemannian metrics

We give a criterion for the existence of a degenerate Riemannian metric on a paracompact C^∞ manifold. We give a criterion for the existence of a corresponding connection without torsion on manifolds with degenerate Riemannian metrics.     

Isometric immersions of Riemannian Spaces in Euclidean Spaces

Questions of the theory of isometric immersions of Riemannian spaces in Euclidean spaces beginning with the very first results onthis topic and also results on immersions of pseudo-Riemannian spaces in pseudo-Euclidean spaces and applications of the theory of immersions in the general theory of relativity are considered.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 15, pp. 173–211, 1977.     

Group-Theoretic Description of Riemannian Spaces

We show that the geometric structure of an arbitrarily curved Riemannian space is locally determined by a deformed group of its diffeomorphisms.     

Stable projective planes with Riemannian metrics

We consider the question whether the system of lines of a two-dimensional stable plane can be described as the system of geodesics of a Riemannian metric and vice versa; we present two results: A complete two-dimensional Riemannian manifold with the property that every two points are joined by a unique geodesic and its family of geodesics form a stable plane. On the other hand every stable projective plane whose lines are geodesics of a Riemannian metric is isometric to the real projective plane. Combining both results it follows that it is impossible to realize the lines of a non-desarguesian projective plane using the geodesics of a complete Riemannian manifold.     

Wave Equations in Riemannian Spaces

With regard to applications in quantum theory, we consider the classical wave equation involving the scalar curvature with an arbitrary coefficient . General properties of this equation and its solutions are studied based on modern results in group analysis with the aim to fix a physically justified value of . These properties depend essentially not only on the values of and the mass parameter but also on the type and dimension of the space. Form invariance and conformal invariance must be distinguished in general. A class of Lorentz spaces in which the massless equation satisfies the Huygens principle and its Green's function is free of a logarithmic singularity exists only for the conformal value of . The same value of follows from other arguments and the relation to the known WKB transformation method that we establish.     

Conformal Mappings of Riemannian Spaces onto Ricci Symmetric Spaces

Mathematical Notes -     

Finsler metrics with special Riemannian curvature properties

The Weyl curvature is one of the fundamental quantities in Finsler geometry because it is a projective invariant. By determining the Weyl curvature of a class of Finsler metrics, we find a lot of Finsler metrics of quadratic Weyl curvature which are non-trivial in the sense that they are not of quadratic Riemann curvature.     

Riemannian metrics on Teichmüller space

On each compact Riemann surface Σ of genusp≥1, we have the Bergman metric obtained by pulling back the flat metric on its Jacobian via the Albanese map. Taking theL ²-product of holomorphic quadratic differentials w.r.t. this metric induces a Riemannian metric on the Teichmüller spaceT _p that is invariant under the action of the modular group. We investigate geometric properties of this metric as an alternative to the usually employed Weil-Petersson metric. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.     

Smoothing Riemannian metrics with Ricci curvature bounds

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds. The authors were supported in part by NSF Grant. The first author was also supported in part by Alfred P. Sloan Fellowship This article was processed by the author using the LATEX style filecljourl from Springer-Verlag.     

Natural connections on the bundle of Riemannian metrics

Let be the bundles of linear frames and Riemannian metrics of a manifold M, respectively. The existence of a unique Diff M-invariant connection form on , which is Riemannian with respect to the universal metric on , is proved. Applications to the construction of universal Pontryagin and Euler forms, are given. Authors’ addresses: R. Ferreiro Pérez, Departamento de Economía Financiera y Contabilidad I, UCM, Campus de Somosaguas, 28223 Madrid, Spain; J. Mu?oz Masqué, Insituto de Física Aplicada, CSIC, C/Serrano 144, 28006 Madrid, Spain