On a Problem of Elementary Differential Geometry and the Number of its Solutions

J. Wallner

Journal for geometry and graphics, Tome 10 (2006) no. 2, pp. 155-16.

Résumé

If $M$ and $N$ are submanifolds of ${\mathbb R}^k$, and $a$, $b$ are points in ${\mathbb R}^k$, we may ask for points $x\in M$ and $y\in N$ such that the vector $\vec{ax}$ is orthogonal to $y$'s tangent space, and vice versa for $\vec{by}$ and $x$'s tangent space. If $M,N$ are compact, critical point theory is employed to give lower bounds for the number of such related pairs of points. Interestingly, we also employ the curvature theory of hypersurfaces in a pseudo-Euclidean space, where curvatures are not considered as real numbers, but as linear forms in the normal space of a point.

Détail

Classification : 53A05, 53A30, 57D70
Mots-clés : Curves and surfaces, critical points, pseudo-euclidean distance