We provide a simple characterization of codimension two submanifolds of ${\mathbb{P}}^n\left(ℝ\right)$ that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when $n\ge 6$. If the codimension two submanifold is a nonsingular algebraic subset of ${\mathbb{P}}^n\left(ℝ\right)$ whose Zariski closure in ${\mathbb{P}}^n\left(ℂ\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^n\left(ℝ\right)$.

It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation ${ℱ}_G$. A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing $C^r$ diffeomorphism group G is simple iff the foliation ${ℱ}_{[G,G]}$ defined by [G,G] admits no proper minimal sets....

Dans ce texte, on définit, pour les immersions lagrangiennes de variétés fermées dans ${ℂ}^n$, une notion d’aire symplectique enlacée. Puis on construit, dans le cas $n=2$, un certain nombre de surfaces lagrangiennes enlaçant une aire infinie. Dans le cas des surfaces exactes, elles ont le minimum de points doubles possible permis par la théorie (sauf la sphère), c’est-à-dire moins que prévu par quelques conjectures.

For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in ${ℝ}^{n+2}$ (or ${}^{n+2}$), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the $n$-dimensional torus, its identity component is a simple group. For $U\left(1\right)$ fibered manifolds, for manifolds admitting special semi-free $U\left(1\right)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U\left(1\right)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

We consider smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in ${ℝ}^{n+2}$ (or $S^{n+2}$), for the cases n=2 or n=3. In a previous paper we have generalized the notion of the Reidemeister moves of classical knot theory. In this paper we examine in more detail the above mentioned dimensions. Examples are given; in particular we examine projections of twist-spun knots. Knot moves are given which demonstrate the triviality of the 1-twist spun trefoil. Another application is a smooth...

Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree...