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Journal for Geometry and Graphics 10 (2006), No. 2, 155--160
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


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.

Keywords: Curves and surfaces, critical points, pseudo-euclidean distance.

MSC: 53A05; 53A30, 57D70

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