Invariant Metrics on Cohomogeneity One Manifolds

Let G be one of the connected subgroups of the orthogonal group of ⁿ which acts transitively on the unit sphere S ^n–1. We get the necessary and sufficient condition for G-invariant metrics g on ⁿ \{0} to be extendend to the origin. For n=2 this is a classical result of Berard–Bergery. The curvature tensor and the sectional curvature of any such Riemannian G-manifold ( ⁿ , g) are described in terms of the length of the Killing vector fields, as well as the second fundamental form of the regular orbits G(P)=S ^n–1. As an application we describe all G-invariant metrics which are Kähler, hyperKähler or have constant principal curvatures. Some of these results are generalized to the case of any cohomogeneity one G-manifold which, in a neighbourhood of a singular orbit, can be identified with a twisted product.     

Cohomogeneity Two Actions on Flat Riemannian Manifolds

In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.     

Computing Degree of Maps Between Manifolds

We determine the degrees of maps between two given closed （n-1）-connected 2n-mamifolds when n ≡ 1 （mod 8）.     

One Nonparabolic End Theorem on Kahler Manifolds

In this paper, the complete noncompact Kahler manifolds satisfying the weighted Poincare inequality are considered and one nonparabolic end theorem which generalizes Munteanu＇s result is obtained.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2 × V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

On the Degree of Approximation by Manifolds of Finite Pseudo-Dimension

The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A class of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width = which measures the worst-case approximation error over all functions by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρ _n (W ^r,d _p , L _q ) , both being a constant factor of n ^-r/d , for a Sobolev class W ^r,d _p , . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of W ^r,d _p by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators. March 12, 1997. Dates revised: August 26, 1997, October 24, 1997, March 16, 1998, June 15, 1998. Date accepted: June 25, 1998.     

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).     

Proper Holomorphic Self-Maps of Quasi-balanced Domains

We show that proper holomorphic self-maps of smoothly bounded pseudoconvex quasi-balanced domains of finite type are automorphisms. This generalizes the classical Alexander’s theorem on proper holomorphic self-maps of the unit ball.     

NMS Flows on Three-Dimensional Manifolds with One Saddle Periodic Orbit

The simplest NMS flow is a polar flow formed by an attractive periodic orbit and a repulsive periodic orbit as limit sets.In this paper we show that the only orientable,simple,compact,3-dimensional manifolds without boundary that admit an NMS flow with none or one saddle periodic orbit are lens spaces. We also see that when a fattened round handle is a connected sum of tori, the corresponding flow is also a trivial connected sum of flows.     

Cohomogeneity and positive curvature in low dimensions

Mathematische Zeitschrift -     

Cohomogeneity one manifolds and hypersurfaces of the euclidean space

This paper is aimed at studying compact hypersurfaces of the euclidean space which are supposed to be Riemannian manifolds of cohomogeneity one, namely acted on by a Lie group of intrinsic isometries with principal orbits of codimension one. We give necessary and sufficient conditions on the structure of the Riemannian metric in order that a hypersurface of such kind turns out to be a revolution hypersurface.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Invariant Measures for Univalent Self-Maps of the Unit Disc

We develop a translation-type model for univalent self-maps φ of the unit disc having an interior fixed-point and use the model to classify the φ-invariant measures on . We are particularly interested in maps which can be embedded in continuous semigroups of holomorphic self-maps of . Received: February 2, 2007. Revised: June 18, 2007. Accepted: July 4, 2007.     

利用Dirac理论进行Poisson流形及Jacobi流形的约化(英文)

通过将可约的Dirac以及Jacobi-Dirac结构分别分为两种类型,给出对应于Poisson流形和Jacobi流形的约化定理.这些约化定理的证明只需要进行一些直接的计算,而不需要借助于矩映射或者相容函数等复杂概念的引入.另外,给出了一些相应的例子和应用.     

Dilatation and Order of Contact for Holomorphic Self-Maps of Strongly Convex Domains

Let D be a bounded strongly convex domain and let f be a holomorphicself-map of D. In this paper we introduce and study the dilatation(f) of f defined, if f has no fixed points in D, as the usualboundary dilatation coefficient of f at its Wolff point, or,if f has some fixed points in D, as the ratio of shrinking ofthe Kobayashi balls around a fixed point of f. In particular,we show that the map , defined as : f (f) [0,1], is lowersemicontinuous. Among other things, this allows us to studythe limits of a family of holomorphic self-maps of D. In thecase of an inner fixed point, the dilatation is an intrinsicmeasure of the order of contact of f(D) to D. Finally, using complex geodesics, we define and study a directionaldilatation, which is a measure of the shrinking of the domainalong a given direction. Again, results of semicontinuity aregiven and applied to a family of holomorphic self-maps. 2000Mathematical Subject Classification: primary 32H99; secondary30F99, 32H15.     

Nontrivial Alexander polynomials of knots and links

In this paper we present a sequence of link invariants, definedfrom twisted Alexander polynomials, and discuss their effectivenessin distinguishing knots. In particular, we recast and extendby geometric means a recent result of Silver and Williams onthe nontriviality of twisted Alexander polynomials for nontrivialknots. Furthermore we prove that these invariants decide ifa genus one knot is fibered. Finally we also show that theseinvariants distinguish all mutants with up to 12 crossings.     

Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case

This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with 1-degree of freedom in the degenerate case H = q⁴ + H₅ + H₆ +.... Our main results complement the study initiated by Markeev in [9].     

On a Submersion Between Reinhart Lightlike Manifolds and Semi-Riemannian Manifolds

In this paper, we introduce a new submersion, namely, screen lightlike submersion between a lightlike manifold and a semi-Riemannian manifold. We give an example and obtain a characterization for lightlike manifold to be Reinhart under such submersion. Then, we investigate the geometry of a screen lightlike submersion when the total manifold is a Reinhart lightlike manifold. Received: March 6, 2007 Revised: October 4, 2007 and November 2, 2007 Accepted: December 6, 2007