of bounded extrinsic curvature was published in 1953. In 1954, J. Nash published the paper on С1-isometric immersions, which was improved by N. Kuiper in 1955. It follows from these studies that a Riemannian metric defined on a two-dimensional manifold, under very general assumptions, admits a realization on a С1-smooth surface in a three-dimensional Euclidean space. Moreover, this realization is carried out as freely as a topological immersion into the space of the manifold on which the metric is given. Hence it is clear that for С1-surfaces, even with a good intrinsic metric, it is impossible to preserve the connections between the intrinsic and extrinsic curvatures. Even in case if a С1-surface carries a regular metric of positive Gaussian curvature, then this does not imply the local convexity of the surface. This emphasizes the naturalness of the class of surfaces of bounded external curvature introduced by A. V. Pogorelov. A. V. Pogorelov solved Hilbert's fourth problem, set by D. Hilbert at the II International Congress of Mathematicians in Paris in 1900. He found all, up to isomorphism, realizations of the systems of axioms of classical geometries (Euclid, Lobachevsky and elliptic) if one omits the congruence axioms containing the concept of angle and supplement these systems with the axiom of "triangle inequality". A. V. Pogorelov was one of the first who has proposed (in 1970) a new idea in the construction of a cryoturbogenerator with superconducting field winding and took an active part in technical calculations and creation of corresponding industrial samples. Selected publications Geometry [translated from the Russian by Leonid Levant, Aleksandr Repyev and Oleg Efimov.]. Moscow: Mir Publishers (1987). . . See also Cauchy's theorem References Sources External links Website dedicated to Pogorelov and his work Biography – at the B. Verkin Institute for Low Temperature Physics and Engineering 1919 births 2002 deaths People from Korochansky District Ukrainian people of Russian descent Ukrainian mathematicians Soviet mathematicians Differential geometers National University of Kharkiv alumni Full Members of the USSR Academy of Sciences Full Members of the Russian Academy of Sciences Stalin Prize winners Textbook writers Laureates of the State Prize of Ukraine in Science and Technology