Hopf was born in Gräbschen, German Empire (now Grabiszyn [pl], part of Wrocław, Poland), the son of Elizabeth (née Kirchner) and Wilhelm Hopf. His father was born Jewish and converted to Protestantism a year after Heinz was born; his mother was from a Protestant family.[3][4]
After the war Hopf continued his mathematical education in Heidelberg (winter 1919/20 and summer 1920)[5] and Berlin (starting in winter 1920/21). He studied under Ludwig Bieberbach and received his doctorate in 1925.
Career
In his dissertation, Connections between topology and metric of manifolds (German: Über Zusammenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten), he proved that any simply connected complete Riemannian3-manifold of constant sectional curvature is globally isometric to Euclidean, spherical, or hyperbolic space. He also studied the indices of zeros of vector fields on hypersurfaces, and connected their sum to curvature. Some six months later he gave a new proof that the sum of the indices of the zeros of a vector field on a manifold is independent of the choice of vector field and equal to the Euler characteristic of the manifold. This theorem is now called the Poincaré–Hopf theorem.
In 1926 Hopf moved back to Berlin, where he gave a course in combinatorial topology. He spent the academic year 1927/28 at Princeton University on a Rockefeller fellowship with Alexandrov. Solomon Lefschetz, Oswald Veblen and J. W. Alexander were all at Princeton at the time. At this time Hopf discovered the Hopf invariant of maps and proved that the Hopf fibration has invariant 1. In the summer of 1928 Hopf returned to Berlin and began working with Pavel Alexandrov, at the suggestion of Courant, on a book on topology. Three volumes were planned, but only one was finished. It was published in 1935.
In 1929, he declined a job offer from Princeton University. In 1931 Hopf took Hermann Weyl's position at ETH, in Zürich. Hopf received another invitation to Princeton in 1940, but he declined it. Two years later, however, he was forced to file for Swiss citizenship after his property was confiscated by Nazis, his father's conversion to Christianity having failed to convince German authorities that he was an "Aryan".
Hopf, Heinz (1964), Selecta Heinz Hopf, Herausgegeben zu seinem 70. Geburtstag von der Eidgenössischen Technischen Hochschule Zürich, Berlin, New York: Springer-Verlag, MR0170777