Metric ends,fibers and automorphisms of graphs

Several results on the action of graph automorphisms on ends and fibers are generalized for the case of metric ends. This includes results on the action of the automorphisms on the end space, directions of automorphisms, double rays which are invariant under a power of an automorphism and metrically almost transitive automorphism groups. It is proved that the bounded automorphisms of a metrically almost transitive graph with more than one end are precisely the kernel of the action on the space of metric ends. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geodetic rays and fibers in periodic graphs

Using the notion of fibers, where two rays belong to the same fiber if and only if they lie within bounded Hausdorff‐distance of one another, we study how many fibers of a graph contain a geodetic ray and how many essentially distinct geodetic rays such “geodetic fibers” must contain. A complete answer is provided in the case of locally finite graphs that admit an almost transitive action by some infinite finitely generated, abelian group. Such graphs turn out to have either finitely many or uncountably many geodetic fibers. Furthermore, with finitely many possible exceptions, each of these fibers contains uncountably many geodetic rays. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 67–88, 2000     

Homotopy ends and Thomason model categories

No Abstract.       

Graph-like compacta: Characterizations and Eulerian loops

A compact graph-like space is a triple $\left(X,V,E\right)$, where $X$ is a compact, metrizable space, $V\subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X⧹V\cong \text{◂+▸}E\times \left(0,1\right)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.     

Normal Vietoris Implies Compactness: A Short Proof

One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.     

Adapted Metrics and Webster Curvature in Finslerian 2-Dimensional Geometry

The Webster scalar curvature is computed for the sphere bundle $T_1S$ of a Finsler surface $(S, F)$ subject to the Chern-Hamilton notion of adapted metrics. As an application, it is derived that in this setting $(T_1S, g_{\rm Sasaki})$ is a Sasakian manifold homothetic with a generalized Berger sphere, and that a natural Cartan structure is arising from the horizontal $1$-forms and the author associates a non-Einstein pseudo-Hermitian structure. Also, one studies when the Sasaki type metric of $T_1S$ is generally adapted to the natural co-frame provided by the Finsler structure.     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

Generalized hermann theorems and conformal holomorphic submersions

It is shown that if a Kähler manifold admits a holomorphic Riemann submersion, then this manifold is locally reducible. Hermann's well-known theorems are generalized to conformal and holomorphic submersions. A method for constructing Kähler fiber spaces with holomorphic conformal (non-Riemannian) projection and totally geodesic isomorphic fibers is suggested. The method allows us to construct complete, including compact, Kähler fiber spaces of the specified type.     

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer , let be the space of complete embedded singly periodic minimal surfaces in , which in the quotient have genus zero and Scherk-type ends. Surfaces in can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that consists of the -parameter family of singly periodic Scherk minimal surfaces. We prove that for each , there exists a natural one-to-one correspondence between and the space of convex unitary nonspecial polygons through the map which assigns to each the polygon whose edges are the flux vectors at the ends of (a special polygon is a parallelogram with two sides of length and two sides of length ). As consequence, reduces to the saddle towers constructed by Karcher.

     

Atsuji completions vis-à-vis hyperspaces

A metric space (X, d) is called an Atsuji space if every real-valued continuous function on (X, d) is uniformly continuous. It is well-known that an Atsuji space must be complete. A metric space (X, d) is said to have an Atsuji completion if its completion (, d) is an Atsuji space. In this paper, we study twelve equivalent (external) characterizations for a metric space to have an Atsuji completion in terms of hyperspace topologies. We also characterize topologically those metrizable spaces whose completions are Atsuji spaces. The first author was supported by the SPM fellowship awarded by the Council of Scientific and Industrial Research, India, during the work of this paper.     

Analysis of the physical Laplacian and the heat flow on a locally finite graph

We study the physical Laplacian and the corresponding heat flow on an infinite, locally finite graph with possibly unbounded valence.     

局部对称共形平坦黎曼流形中的紧致子流形

本文讨论局部对称共形平坦黎曼流形中紧子流形问题．改进了［１］的结果并将［２］中关于球面子流形的一个结果推广到局部对称共形平坦黎曼流形子流形．     

Nayatani's metric and conformal transformations of a Kleinian manifold

According to Schoen and Yau (1988), an extensive class of conformally flat manifolds is realized as Kleinian manifolds. Nayatani (1997) constructed a metric on a Kleinian manifold which is compatible with the canonical flat conformal structure. He showed that this metric has a large symmetry if is a complete metric. Under certain assumptions including the completeness of , the isometry group of coincides with the conformal transformation group of . In this paper, we show that may have a large symmetry even if is not complete. In particular, every conformal transformation is an isometry when corresponds to a geometrically finite Kleinian group.

     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

Morse theory, Milnor fibers and minimality of hyperplane arrangements

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of -cells equals the -th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are Eilenberg-MacLane spaces.

     

关于切丛和单位切球丛的度量的一个注记

本文在黎曼流形$(M,g)$的切丛$TM$ 上研究与参考文献[10]中平行的一类度量$G$以及相容的近复结构$J$.证明了切丛$TM$关于这些度量和相应的近复结构是局部共形近K\"{a}hler流形,并且把这些结构限制在单位切球丛上得到了切触度量结构的新例子.     

Mappings of Domains in Rn and Their Metric Tensors

We consider quasi-isometric mappings of domains in multidimensional Euclidean spaces. We establish that a mapping depends continuously in the sense of the topology of Sobolev classes on its metric tensor to within isometry of the space. In the space of metric tensors we take the topology determined by means of almost everywhere convergence. We show that if the metric tensor of a mapping is continuous then the length of the image of a rectifiable curve is determined by the same formula as in the case of mappings with continuous derivatives. (Continuity of the metric tensor of a mapping does not imply continuity of its derivatives.)     

On Connection Between the Nonholonomic Metric on the Heisenberg Group and the Grushin Metric

The purpose of this paper is to compute geodesics on the Grushin plane and examine an assertion on connection between spheres of the Grushin plane and spheres of the Heisenberg group. The assertion turns out to require correction that the spheres of the Heisenberg group are directly obtained by rotation of the Grushin spheres. We find a modified Grushin metric for which the last assertion holds. Also, we prove several theorems about connections between the Grushin plane and Heisenberg group.