Paolo Pizzetti: the forgotten originator of triangle comparison geometry
The authors discuss forgotten contributions of Italian mathematician Paolo Pizzetti to the development of comparison geometry. In 1907, Pizzetti considered the two-dimensional case of the theorem comparing angles of two geodesic triangles with the same lengths lying on surfaces with different curvatures. Known authoritative historical sources provide very brief accounts of the origins and development of comparison geometry relating the theorem to the names of A. D. Alexandrov (1948) in the two-dimensional case and V. A. Toponogov (1959) in the \(n\)-dimensional Riemannian case. The authors argue that ``the proofs of Pizzetti and of Alexandrov and Toponogov cannot be meaningfully compared, given that, although the theorems are phrased similarly, the definitions of the terms involved are so different that the proofs could not have displayed any similarity'', yet Pizzetti's contributions to comparison geometry cannot be neglected. In addition to a brief biographical sketch, a short outline of Pizzetti's proof of the triangle comparison theorem is provided along with possible reasons for oblivion.
scientific article; zbMATH DE number 5925778
scientific article