Eulidean hypersurfaces with reflection properties
According to the author a smooth connected hypersurface S in Euclidean n- space is said to have the reflection property with respect to the points p and q, if the following conditions are satisfied: i) For each point x of S the directions from x to q and p and the unit normal N of S at x are coplanar. ii) The line of direction N through x bisects one of the pairs of opposite angles at x formed by the lines spanned by x with p and q. The points p and q are not allowed to belong to S, but they can coincide or they can be points at infinity (in the projective sense). If both of them belong to infinity, then they have to be different from each other. The author's main result is the following theorem: A smooth connected hypersurface of Euclidean n-space has the reflection property with respect to certain foci p and q, if and only if it is a connected open subset of one of the following: a hyperplane, an (n-1)-sphere, or a paraboloid, ellipsoid, or hyperboloid that is rotationally symmetric about the axis spanned by the two foci. The proof is only sketched in the different steps \(n=2\), \(n=3\), and for general n. [Reviewer's remark: The proof must contain a gap, because cones with general bases satisfy the conditions required for the reflection property, if the vertex is removed. Furthermore the proof of the non- trivial `only if' part can be reduced to a simple argument by using the uniqueness of integral manifolds for integrable distributions.]
scientific article; zbMATH DE number 4133029