Most real analytic Cauchy-Riemann manifolds are nonalgebraizable

We give a simple argument to the effect that most germs of generic real analytic Cauchy-Riemann manifolds of positive CR dimension are not holomorphically embeddable into a generic real algebraic CR manifold of the same real codimension in a finite dimensional space. In particular, most such germs are not holomorphically equivalent to a germ of a generic real algebraic CR manifold.Mathematics Subject Classification (2000): Primary 32V20, 32V30Supported in part by Research Program P1-0291, Republic of SloveniaAcknowledgement I wish to thank Peter Ebenfelt and Alexander Sukhov for their invaluable advice concerning the state of knowledge on the question considered in the paper.     

Comparison results for the volume of geodesic celestial spheres in Lorentzian manifolds

Volume comparison results are obtained for the volume of geodesic celestial spheres in Lorentzian manifolds and the corresponding objects in Lorentzian space forms. Also, as a rigidity result it is shown that the volume of geodesic celestial spheres is independent of the instantaneous observer if and only if the spacetime has constant curvature.     

Pompeiu sets in compact Heisenberg manifolds

A necessary and sufficient condition is presented for a set to be a Pompeiu subset of any compact homogeneous space with a finite invariant measure. The condition, which is expressed in terms of the intertwining operators of each primary summand of the quasi-regular representation, is then interpreted in the case of the compact Heisenberg manifolds. Examples are presented demonstrating that the condition to be Pompeiu in these manifolds is quite different from the corresponding condition for a torus of the same dimension. This provides a contrast with the existing comparison between the Heisenberg group itself and Euclidean space in terms of Pompeiu sets. In addition, the closed linear span of all translates of any square integrable function on any compact homogeneous space is determined.     

On the dynamics of certain actions of free groups on closed real analytic manifolds

Let M be a closed connected real analytic manifold; let be a free group on two generators. The set of analytic actions of on M endowed with Taken‚s topology contains a nonempty open subset whose corresponding actions share three properties: (a) they have every orbit dense, (b) they leave invariant no geometric structure on M, (c) any homeomorphism conjugating two of them is analytic. Received: October 3, 2001     

Dual Radon transforms on affine Grassmann manifolds

Fix , and let and denote the affine Grassmann manifolds of - and -planes in . We investigate the Radon transform associated with the inclusion incidence relation. For the generic case and n$">, we will show that the range of this transform is given by smooth functions on annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case .

     

Totally geodesic hypersurfaces in manifolds of nonpositive curvature

In this paper we determine the structure of an embedded totally geodesic hypersurfaceF or, more generally, of a totally geodesic hypersurfaceF without selfintersections under arbitrarily small angles in a compact manifoldM of nonpositive sectional curvature. Roughly speaking, in the case of locally irreducibleM the result says thatF has only finitely many ends, and each end splits isometrically asK×(0, ∞), whereK is compact. This article was processed by the author using the Springer-Verlag T_EX PJour1g macro package 1991.     

The geodesic hypothesis and non-topological solitons on pseudo-Riemannian manifolds

A class of solitary wave solutions to a semi-linear wave equation on a pseudo-Riemannian manifold is studied. A construction of solutions which concentrate on geodesics is given.     

On Finsler manifolds with hyperbolic geodesic flows

Archiv der Mathematik - Let (M, F) be a closed $$C^\infty $$ Finsler manifold and $$\varphi $$ its geodesic flow. In the case that $$\varphi $$ is Anosov, we extend to the Finsler setting...     

Spiraling spectra of geodesic lines in negatively curved manifolds

Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of elements of ${\mathbb{R},\mathbb{C}}$ or the Heisenberg group by quadratic irrational ones.     

Harmonic transforms of complete Riemannian manifolds

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.     

Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds

Summary We show that a closed embedded totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface. Namely, we construct a tubular neighborhood function and show that an embedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface (and hence not on the geometry of the ambient manifold). The implications of this result for volumes of hyperbolic manifolds is discussed. In particular, we show that ifM is a hyperbolic 3-manifold containingn rank two cusps andk disjoint totally geodesic embedded closed surfaces, then the volume ofM is bigger than . We also derive a (hyperbolic) quantitative version of the Klein-Maskit combination theorem (in all dimensions) for free products of fuchsian groups. Using this last result, we construct examples to illustrate the qualitative sharpness of our tubular neighborhood function in dimension three. As an application of our results we give an eigenvalue estimate.Oblatum IX-1992 & 18-VIII-1993Research supported in part by NSF Grant DMS-9207019