On the canonical extension of a differentiable manifold

In this paper, the differential geometry of second canonical extension² M of a differentiable manifoldM is studied. Some vector fields tangent to² M inTTM are determined. In addition we obtain that the second canonical extensions ofM and a totally geodesic submanifold inM are totally geodesic submanifolds inTTM and² M respectively.     

On the Hantzsche-Wendt manifold

We explicitly compute the outer automorphism group Out ₁ M of the fundamental group of the Hantzsche — Wendt manifoldM. It is an extension 1(₂)³Out₁ MS ₃₂1, but not the semidirect product (₂)³(S ₃₂) as claimed in [3] (see also [4]). As a consequence, we get a quick algebraic computation of the symmetry groups of the Borromean rings and the figure-8-knot.     

On the recovery of a manifold with boundary in Rn

If the Riemann curvature tensor associated with a smooth field $\text{C}$ of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset $\text{Ω?}\text{R}{}^{\mathrm{n}}$, then $\text{C}$ is the metric tensor field of a manifold isometrically immersed in $\text{R}{}^{\mathrm{n}}$.In this Note, we first show how, under a mild smoothness assumption on the boundary of $\text{Ω}$, this classical result can be extended “up to the boundary”. When $\text{Ω}$ is bounded, we also establish the continuity of the manifold with boundary obtained in this fashion as a function of its metric tensor field, the topologies being those of the Banach spaces $\text{C}{}^{\mathrm{?}}\text{(}\text{Ω}\text{)}$. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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.

     

On the periodic points of functions on a manifold

In a conference on fixed point theory, B. Halpern of Indiana University considered the problem of reducing the number of periodic points of a map by homotopy. He also asked whether the number of periodic points of a function could be increased by a homotopy. In this paper, we will show that for any map on a closed manifold, an arbitrarily small perturbation can always create infinitely many periodic points of arbitrarily high periods.

     

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 meromorphic convexity of normality domains in a Stein manifold

Let X be a Stein manifold. Then we prove that for any family ℱ⊂?(X) the normality domain Dℱ) is a meromorphically ?(X)-convex open set of X. Received: 4 November 1999     

On the depth of a planar graph

In this paper, we have defined the concept of the depth of a planar graph. We show that, if G is a simple finite planar graph with p vertices and q edges and q > 3(p ? 1) ? p/2^s?1, then the depth of G is at least equal to s.     

On the quantum flag manifold

Rostov State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 3, pp. 90–92, July–September, 1992.