Let X be a separable locally compact metrizable space and G a separable completely metrizable ANR topological group without isolated points. By C(X, G), we denote the topological group of all continuous maps f : X → G endowed with the Whitney (graph) topology and by Cc(X, G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the spac...

We prove the Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature. Given a smooth closed manifold M, it is a long-standing question to determine whether or notM admits a Riemannian metri...

Let (Mn, g), n ≥ 4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0 < l ≤ L, we prove that there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g satisfies l ≤ s ≤ L and the Einstein tensor satisfies |Ric − s n g| ≤ ε then M is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result...

We study the magic manifold N which is a hyperbolic and fibered 3-manifold. We give an explicit construction of a fiber Fa and its monodromy : Fa → Fa of the fibration associated to each fibered class a of N . Let δg (resp. δ + g ) be the minimal dilatation of pseudo-Anosovs (resp. pseudo-Anosovs with orientable invariant foliations) defined on an orientable closed surface of genus g. As a cons...

Let (M, g) be a spacetime, where M is a smooth, connected, Hausdorff four-dimensional manifold and g is smooth Lorentzian metric of signature (+ -) defined on M . The manifold M and the metric g are assumed smooth (C). A smooth vector field ξ is said to preserve a matter symmetry [1] on M if, for each smooth local diffeomorphism φt associated with ξ, the tensors T and φ∗tT are equal on the doma...

In the context of paracontact geometry, η-Ricci solitons are considered on manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0, S · R(ξ,X) = 0, W2(ξ,X) · S = 0 and S · W2(ξ,X) = 0. We prove that on a para-Kenmotsu manifold (M,φ, ξ, η, g), the existence of an η-Ricci soliton implies that (M, g) is quasi-Einstein and if the Ricci curvature satisfies R(ξ,X) · S = 0, then (M, g) is Ei...

G. Leibon and D. Letscher showed that for general and sufficiently dense point set its Delaunay triangulation and Voronoi diagram in Riemannian manifold exist. They also proposed an algorithm to construct them for a given set. In this paper we estimate the necessary number of points for computing the Voronoi diagram in the manifold by using sectional curvature of the manifold. Moreover, we show...

We investigate the relations between the cut number, c(M), and the first Betti number, b1(M), of 3-manifolds M. We prove that the cut number of a “generic” 3-manifold M is at most 2. This is a rather unexpected result since specific examples of 3-manifolds with large b1(M) and c(M) ≤ 2 are hard to construct. We also prove that for any complex semisimple Lie algebra g there exists a 3-manifold M...

There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by ...

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. We first give a geometric construction of a non-degenerate triangular dynamical rmatrix from a fat reductive decomposition g = h⊕m using symplectic fibrations. We also prove that a triangular dynamical r-matrix r : h∗ −→ ∧g corresponds to a Poisson manifold h∗ ×G. A special type of quantizations o...