2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65286Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra does. We also describe a connection between our result and the Hurwitz--Radon theorem about sums of squares.12 pages; to appear in Funct. Anal. and its Appl.; v4: minor correctionsMetric GeometryDifferential Geometry15A66; 51M04text