This is a detailed study of the infinitesimal variation of the varieties of lines and osculating lines through a point of a low degree hypersurface in projective space. The motion is governed by a system of partial differential equations which we describe explicitly.

For most curves of genus 4 and characteristic > 3 the second osculating cone of the theta divisor is the cone over the canonical curve. Let G be a nonhyperelliptic complete smooth curve of genus 4. The canonical morphism G —> P3 is an embedding and the image is defined as the intersection of a quadric and a cubic. In this paper we will discuss a geometric way of constructing such equations from...

2 Under the microscope 9 2.1 Fundamental forms of projective varieties . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Osculating spaces of homogeneous varieties . . . . . . . . . . . . . . . . . . . . . 11 2.3 Decomposing the tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Second fundamental forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2...

In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3. Namely, we show i) that the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them...

This paper aims to analyze trajectories behavior and attractor structure of chaotic dynamical systems with the Differential Geometry and Mechanics formalism. Applied to slow-fast autonomous dynamical systems (S-FADS), this approach provides: on the one hand a kinematics interpretation of the trajectories motion, and on the other hand, a direct determination of the slow manifold equation. The at...

We show how some geometric elements of the path of a particle moving in a plane – the osculating circle and its radius of curvature – can be used to construct the parabolic trajectory of projectiles in motion under gravity.

For integral geometric and local topological properties of two sets of positive reach one xed and the other moving or translating certain exceptional relative positions have to be excluded. We give a measure geometric justiication in form of a Sarddtype theorem which extends to more general singular sets. A slightly stronger result has been proved by Schneider for the special case of convex bod...