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.     

Hurwitz and Tasoev Continued Fractions

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e^1/a (a 1), ae^1/a and (1/a)e^1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.     

Notes on the peripheral volume of hyperbolic 3‐manifolds

We consider orientable hyperbolic 3‐manifolds with either non‐empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion can only be maximal in some combinatorially extremal configurations. The techniques we employ are very elementary but the result is in our opinion of some interest.     

Counting Horoballs and Rational Geodesics

Let M be a geometrically finite pinched negatively curved Riemannianmanifold with at least one cusp. The asymptotics of the numberof geodesics in M starting from and returning to a given cusp,and of the number of horoballs at parabolic fixed points inthe universal cover of M, are studied in this paper. The caseof SL(2, Z), and of Bianchi groups, is developed. 2000 MathematicsSubject Classification 53C22, 11J06, 30F40, 11J70.     

The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of Symplectic Paths

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic. Project 10071040 supported by NNSF, 200014 supported by Excellent. Ph.D. Funds of ME of China, and PMC Key Lab. of ME of China     

Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space

Let M be an n-dimensional complete non-compact submanifold in a hyperbolic space with the norm of its mean curvature vector bounded by a constant . We prove in this paper that . In particular when M is minimal we have and this is sharp because equality holds when M is totally geodesic. Received September 14, 1999; in final form November 12, 1999 / Published online December 8, 2000     

Classes of submersions of Riemannian manifolds with compact fibers

In Riemannian geometry and its applications, the most popular is the class of Riemannian submersions (and foliations) [1–4] which are characterized by simplest mutual disposition of fibers. The purpose of the present article is to introduce other, more general, classes of submersions of Riemannian manifolds which, as well as the class of Riemannian submersions, are described by simple local properties of configuration tensors and to begin their study.Given a submersion :MM of differentiable manifolds with compact connected fibers and any metric onM, we define a metric on the base with the help of theL ²-norm of horizontal fields. In this caseT¯ M becomes a subbundle of some larger bundleM. The main class of totally geodesic submersions introduced in the article (Definition 1) corresponds to the metrics onM with simplest disposition ofT¯ M inM. In the article we obtain a criterion for such submersions (Corollary 1); existence is proved by means of the product with a metric varying along fibers (Example 2). To study totally geodesic submersions, we use ideas from the theory of Riemannian submersions and submanifolds with degenerate second form (Theorems 1 and 2 and Corollary 4).Foliations modeled by totally geodesic submersions (see equality (13)) are of interest too, but we leave them beyond the scope of the article.This work was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00271).Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1154–1164, September–October, 1994.     

On The diophantine equation Fn+Fm=2a

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F_k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.     

Teichmüller curves,Galois actions and $ \widehat {GT} $‐relations

Teichmüller curves are geodesic discs in Teichmüller space that project to algebraic curves C in the moduli space M_g. Some Teichmüller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group G_? acts faithfully on the set of these embedded curves. We also compare the action of G_? on π₁(C) with the one on π₁(M_g) and obtain a relation in the Grothendieck–Teichmüller group, seemingly independent of the known ones. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limiting angle of Brownian motion on certain manifolds

Summary. Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire S ^m−1 . This improves a result of Hsu and March. Received: 7 December 1994/In revised form: 2 September 1995     

Unclouding the sky of negatively curved manifolds

Let M be a complete simply connected Riemannian manifold, with sectional curvature K ≤ −1. Under certain assumptions on the geometry of ∂M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x₀ outside these horoballs, it is possible to shrink uniformly, by a finite amount depending only on M, these horoballs so that some geodesic ray starting from x₀ avoids the shrunk horoballs. As an application, we give a uniform upper bound on the infimum of the heights of the closed geodesics in the finite volume quotients of M.Received: January 2004 Accepted: August 2004     

Expansive dynamics and hyperbolic geometry

LetM be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. Then we show that the universal Riemannian covering ofM is a hyperbolic geodesic space according to the definition of M. Gromov. This allows us to extend a series of relevant geometric and topological properties of negatively curved manifolds toM and in particular, geometric group theory applies to the fundamental group ofM.     

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.     

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C³ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C¹ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.     

Minimal helix surfaces in <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℝ

An immersed surface M in N ⁿ ×ℝ is a helix if its tangent planes make constant angle with ∂ _t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N ⁿ ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.     

On totally real surfaces of the nearly Kaehler 6-sphere

Let M be a minimal totally real surface of the nearly Kaehler 6-sphere. We prove that if M is homeomorphic to a sphere, then M is totally geodesic. Consequently, if M is compact and has non-negative Gaussian curvature K, then eithe K=0 or K=1. Finally, we derive from these results that if M has constant Gaussian curvature K, then either K=0 or K=1.Aspirant Navorser N.F.W.O. (Belgium).     

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).     

Goldie Extending Modules

In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.     

Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces

We present a theory of harmonic maps where the target is a complete geodesic space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov and the domain is a measure space with a symmetric Markov kernel p on it. Our theory is a nonlinear generalization of the theory of symmetric Markov kernels and reversible Markov chains on M. It can also be regarded as a particular case of the theory of generalized (= nonlinear) Dirichlet forms and energy minimizing maps between singular spaces, initiated by Jost (1994) and Korevaar, Schoen (1993) and developed further by Jost (1997a), (1998) and Sturm (1997). We investigate the discrete and continuous time heat flow generated by p and show that various properties of the linear heat flow carry over to this nonlinear heat flow. In particular, we study harmonic maps, i.e. maps which are invariant under the heat flow. These maps are identified with the minimizers of the energy. Received April 2, 2000 / Accepted May 9, 2000 /Published online November 9, 2000     

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let