Exponential iterated integrals and the relative solvable completion of the fundamental group of a manifold

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K.T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of π₁(M,x) can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called the relative solvable completions of π₁(M,x). We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.     

A Wilson group of non-uniformly exponential growth

This Note constructs a finitely generated group W whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 – in other words, of non-uniformly exponential growth.This answers a question by Mikhael Gromov (Structures métriques pour les variétés riemanniennes, in: J. Lafontaine, P. Pansu (Eds.), CEDIC, Paris, 1981).The construction also yields a group of intermediate growth V that locally resembles W in that (by changing the generating set of W) there are isomorphic balls of arbitrarily large radius in V and W's Cayley graphs. To cite this article: L. Bartholdi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Almost sure exponential behaviour for a parabolic SPDE on a manifold

We derive an upper bound on the large-time exponential behavior of the solution to a stochastic partial differential equation on a compact manifold with multiplicative noise potential. The potential is a random field that is white-noise in time, and Hölder-continuous in space. The stochastic PDE is interpreted in its evolution (semigroup) sense. A Feynman–Kac formula is derived for the solution, which is an expectation of an exponential functional of Brownian paths on the manifold. The main analytic technique is to discretize the Brownian paths, replacing them by piecewise-constant paths. The error committed by this replacement is controlled using Gaussian regularity estimates; these are also invoked to calculate the exponential rate of increase for the discretized Feynman–Kac formula. The error is proved to be negligible if the diffusion coefficient in the stochastic PDE is small enough. The main result extends a bound of Carmona and Viens (Stochast. Stochast. Rep. 62 (3–4) (1998) 251) beyond flat space to the case of a manifold.     

共形平坦黎曼流形中具有平行第二基本形式的超曲面

In this paper, we prove the followingTheorem. Let Cⁿ⁺¹ ( n >5 ) be a conformally flat Riemannian anifola of dimension n + 1 . If Mn is a hypersurface immersed isometrically in Cⁿ⁺¹ over which the second fundamental form is covariant constant, then there are three posible cases only:I . locally Mⁿ= S^p×S^q×S^r, p+q+r=n;Ⅱ . locally Mⁿ=S^p×S^q, p + q = n , where S^k is k- dimensional Riemannian space of constant curvature;III. Mⁿ is umbilical and conformally flat. Moreover, if Mⁿ is connected and complete, then the result holds globally.     

The fundamental group of a unitary orbit

The unitary orbit of a complex n × n matrix A is simply connected if and only if the portion of the commutant {A} which resides in the special unitary group is path connected.     

On the affine group of a normal homogeneous manifold

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this study, we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous space. The naturally reductive case is also treated. This completes the geometric calculation of the isometry group of naturally reductive spaces. In addition, we prove that for normal homogeneous spaces the set of fixed points of the full isotropy is a torus. As an application of our results it follows that the holonomy group of a homogeneous fibration is contained in the group of (canonically) affine transformations of the fibers; in particular, this holonomy group is a Lie group (this is a result of Guijarro and Walschap).     

On the group of homotopy equivalences of a manifold

We consider the group of homotopy equivalences of a simply connected manifold which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group for special manifolds , for example if is a connected sum of products of spheres. In particular the group is determined completely. Also the connection of with the group of isotopy classes of diffeomorphisms of is studied.

     

Relations between a manifold and its focal set

Summary Throughout this paper, smooth meansC . All manifolds and embeddings will be smooth. By aclosed m-manifold we mean a compact connected manifold of dimensionm, without boundary.LetM be a closedm-manifold (m>0), andf: ME ⁿ an embedding in Euclideann-space. The focal points off are the centres of principal curvature (with respect to some normal direction) of the embedded manifoldf(M). These points form thefocal set C(f) off.The starting point for our investigation is the following problem. Is there any relation between the topological structure ofM and the relative positions ofC(f) andf(M) inE ⁿ ? In particular, canf be so chosen thatC(f) andf(M) are disjoint? We say that such an embedding isnonfocal.We find that there are manifolds for which no such embedding exists.     

The fundamental group scheme of a non-reduced scheme

We extend the definition of fundamental group scheme to non-reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.     

The fundamental group of a compact metric space

We give a forcing free proof of a conjecture of Mycielski that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the continuum.