We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus $T^n$, $n\ge 2$. We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are $\pm \mathrm{Id}$ and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and...

Let $A\subset {ℝ}^n$ be a set-germ at $0\in {ℝ}^n$ such that $0\in \overline{A}$. We say that $r\in S^{n-1}$ is a direction of $A$ at $0\in {ℝ}^n$ if there is a sequence of points $\left\{x_i\right\}\subset A\setminus \left\{0\right\}$ tending to $0\in {ℝ}^n$ such that $\frac{x_i}{\parallel x_i\parallel }\to r$ as $i\to \infty$. Let $D\left(A\right)$ denote the set of all directions of $A$ at $0\in {ℝ}^n$.Let $A,B\subset {ℝ}^n$ be subanalytic set-germs at $0\in {ℝ}^n$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\right(A)\cap D(B\left)\right)$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular if two subanalytic...

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on ${ℝ}^{2n}\times S^1$ and $ℝP^{2n+1}$. On the other hand we also show by elementary arguments that the...

We prove the “End Curve Theorem,” which states that a normal surface singularity $(X,o)$ with rational homology sphere link $\Sigma$ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function $(X,o)\to (ℂ,0)$ whose zero set intersects $\Sigma$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” $(X,o)$ is described by giving an explicit set of equations describing...

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A-1 is also invertible and that the equation [B,(A-1)(A,B)] = 0 called the fundamental equation is satisfied. Then this defines a representation of the algebra ℱ = A, B | [B,(A-1)(A,B)] = 0. An invariant R-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when R is the quaternions.