
Journal for Geometry and Graphics 10 (2006), No. 2, 155160 Copyright Heldermann Verlag 2006 On a Problem of Elementary Differential Geometry and the Number of its Solutions Johannes Wallner Institute of Geometry, Technical University, Kopernikusgasse 24, 8010 Graz, Austria j.wallner@tugraz.at [Abstractpdf] 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 pseudoEuclidean space, where curvatures are not considered as real numbers, but as linear forms in the normal space of a point. Keywords: Curves and surfaces, critical points, pseudoeuclidean distance. MSC: 53A05; 53A30, 57D70 [ Fulltextpdf (124 KB)] for subscribers only. 