A spherical decomposition for Riemannian open 3-manifolds

以偶维球面空间形为纤维的丛及纤维向四分之一夹挤黎曼度量

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

本文证明：若光滑丛的每个纤维（实射影空间）带有连续变化的正曲率逐点强1/4夹挤度量，则丛的结构群可约化为标准圆度量的等距群。

Geometric decompositions of surfaces with spherical metric and conical singularities

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical analog of the geometric triangulations for flat surfaces with conical singularities. The irreducible components include not only spherical triangles but also other interesting spherical polygons. In particular, we present the class of \textit{half-spherical concave polygons} that are spherical polygons without diagonals and that can be arbitrarily complicated.…

Thin-thick decomposition for real definable isolated singularities

Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thin-thick decomposition of an isolated singularity germ - which happens to be a natural blow-spherical invariant. This decomposition is a generalization of the thin-thick decomposition of normal complex surface singularity germs introduced in [7]

Analysis on Riemannian foliations of bounded geometry

A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.

Similarity structures and de Rham decomposition

A similarity structure on a connected manifold M is a Riemannian metric on its universal cover such that the fundamental group of M acts by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham decomposition with at most two factors, one of which is Euclidean. Very recently, after Belgun and Moroianu conjectured that the number of factors was at most one, Matveev and Nikolayevsky found an example with two factors. When the non-flat factor has dimension 2, we give a complete classification of the examples with two factors. In greater dimensions, we make t…

Short loop decompositions of surfaces and the geometry of Jacobians

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any $λ\in (0,1)$ there exists a constant $C_λ$ such that every closed Riemannian surface of genus $g$ whose area is normalized at $4π(g-1)$ has at least $[λg]$ homologically independent loops of length at most $C_λ\log(g)$. This result extends Gromov's asy…