Orders of accumulation of entropy on manifolds

For a continuous self-map T of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold M and any countable ordinal α, we construct a continuous, surjective self-map ofM having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism.     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

Index Theorem for Equivariant Dirac Operators on Noncompact Manifolds

Let D be a (generalized) Dirac operator on a noncompact complete Riemannian manifold M acted on by a compact Lie group G. Let v: M g = Lie G be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the transversal cotangent bundle to M. Hence, by embedding of M into a compact manifold, one can define a topological index of the pair (D,v) as an element of the completed ring of characters of G. We define an analytic index of (D,v) as an index space of certain deformation of D and we prove that the analytic and topological indexes coincide. As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of noncompact cobordisms. As an application, we extend the Atiyah–Segal–Singer equivariant index theorem to our noncompact setting. In particular, we obtain a new proof of this theorem for compact manifolds.     

Ergodic Group Rotations, Hartman Setsand Kronecker Sequences

 Let be a homomorphism with dense image in the compact group C. If is a continuity set, i.e. its topological boundary has Haar measure 0, then is called a Hartman set. If M is aperiodic then S contains the essential information about (C, ι) or, equivalently, about the dynamical system (C, T) where T is the ergodic group rotation . Using Pontryagin’s duality the paper presents a new method to get this information from S: The set S induces a filter on which is an isomorphism invariant for (C, T) and turns out to be a complete invariant for ergodic group rotations. If one takes , , , , one gets the interesting special case of Kronecker sequences (nα) which are classical objects in number theory and diophantine analysis.     

Ergodic Group Rotations,Hartman Setsand Kronecker Sequences

 Let be a homomorphism with dense image in the compact group C. If is a continuity set, i.e. its topological boundary has Haar measure 0, then is called a Hartman set. If M is aperiodic then S contains the essential information about (C, ι) or, equivalently, about the dynamical system (C, T) where T is the ergodic group rotation . Using Pontryagin’s duality the paper presents a new method to get this information from S: The set S induces a filter on which is an isomorphism invariant for (C, T) and turns out to be a complete invariant for ergodic group rotations. If one takes , , , , one gets the interesting special case of Kronecker sequences (nα) which are classical objects in number theory and diophantine analysis. Received 3 November 2000; in final form 25 January 2002     

On Birkhoff Theorems for Lagrangian invariant tori with closed orbits

We show that a C¹ torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T², and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T², shows that the second Birkhoff Theorem holds for C¹ Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T² and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection. Partially supported by CNPq, FAPERJ, TWAS     

On a Theorem Due to Birkhoff

The manifold M being closed and connected, we prove that every submanifold of T*M that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.     

Rigidity of the critical point equation

On a compact n ‐dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3‐dimensional manifold M is isometric to the round 3‐sphere S³ if ker s^′*_g ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3‐dimensional compact manifold under certain topological conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

Compact symplectic solvmanifolds not admitting complex structures

In this paper the authors exhibit a family of 4-dimensional compact solvemanifolds. Each member M ³(k) of the family possesses all of the topological properties of a compact Kähler manifold, yet M ³(k) can have no complex structure. The proof uses Kodaira's classification of compact surfaces.     

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

We investigate the finiteness structure of a complete non-compact n-dimensional Riemannian manifold M whose radial curvature at a base point of M is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann functions on M are exhaustions, and that there exists a compact subset of M such that the compact set contains all critical points for any Busemann function on M. As corollaries by the main theorem, M has finite topological type, and the isometry group of M is compact.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

Connectivity graphs on two-dimensional surfaces and their application in robotics

The problem of robot motion planning in an environment with obstacles can often be reduced to the study of connectivity of the robot's free configuration space. In turn, space connectivity can be analysed via the corresponding connectivity graph. For two-degree-of-freedom robots, the free configuration space presents a two-dimensional (2D) surface—a compact subspace of a 2D orientable compact manifold. This paper addresses the following abstract problem: given a compact 2D surface bounded by simple closed curves and lying in an orientable 2D manifold (a sphere, a torus, etc.) and given two points in the subspace, suggest a systematic way of defining the connectivity graph in the subspace, based on its topological properties. The use of space topology results in powerful, from the robotics standpoint, provable algorithms capable of on-line motion planning in an environment with unknown obstacles of arbitrary shapes. This makes the method distinct from other techniques, which require complete information, algebraic representation of space geometry, and off-line computation.     

Partial linearization of differentiable systems

For any given differentiable dynamical system with discrete time on a compact Riemannian manifold of finite dimension d, along its orbits, in this paper, we introduce a type of global linearization under natural moving orthonormal q-frames of the tangent space of the base manifold, 1 ≤ q ≤ d. As an application, we give a new proof of the Oseledec-Million??ikov Multiplicative Ergodic Theorem (MET) for a ergodic smooth systems (M, ν; φ).     

Wave Invariants of the Spectrum of the G-Invariant Laplacian and the Basic Spectrum of a Riemannian Foliation

Given a compact boundaryless Riemannian manifold upon which a compact Lie group G acts by isometries, recall that the G-invariant Laplacian is the restriction of the ordinary Laplacian on functions to the space of functions which are constant along the orbits of the action. By considering the wave trace of the invariant Laplacian and the connection between G manifolds and Riemannian foliations, invariants of the spectrum of the G-invariant Laplacian can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the principal orbits and contained in the open dense set of principal orbits are associated to the singularities of the wave trace of the G-invariant spectrum. If the action admits finite orbits, then the invariants also include the lengths of certain geodesics arcs connecting the finite orbit to itself. Under additional hypotheses, we obtain partial wave traces. As an application, a partial trace formula for Riemannian foliations with bundle-like metrics is also presented, as well as several special cases where better results are available.