Editing Heinz Hopf (section)

==Career==
In his dissertation, ''Connections between topology and metric of manifolds'' ([[German language|German]]: ''Über Zusammenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten''), he proved that any simply connected complete [[Riemannian manifold|Riemannian]] [[3-manifold]] of constant sectional [[curvature]] is globally [[Isometry|isometric]] to [[Euclidean space|Euclidean]], [[Elliptic geometry|spherical]], or [[hyperbolic space]]. He also studied the indices of zeros of [[vector field]]s on [[hypersurface]]s, 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]].

Hopf spent the year after his doctorate at the [[University of Göttingen]], where [[David Hilbert]], [[Richard Courant]], [[Carl Runge]], and [[Emmy Noether]] were working. While there he met [[Pavel Alexandrov]] and began a lifelong friendship.

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 [[James Waddell Alexander II|J. W. Alexander]] were all at Princeton at the time. At this time Hopf discovered the [[Hopf invariant]] of maps <math>S^3 \to S^2</math> 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 Zurich|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 [[Nazi Germany|Nazi]]s, his father's conversion to Christianity having failed to convince German authorities that he was an "[[Aryan]]".

In 1946/47 and 1955/56 Hopf visited the United States, staying at Princeton and giving lectures at [[New York University]] and [[Stanford University]]. He served as president of the [[International Mathematical Union]] from 1955 to 1958.<ref>{{Cite web|url=http://www.mathunion.org/organization/ec/ec-1952-2014/|title=International Mathematical Union (IMU): IMU Executive Committees 1952–2014|website=www.mathunion.org|language=en|archive-url=https://web.archive.org/web/20150108154651/http://www.mathunion.org/organization/ec/ec-1952-2014|archive-date=8 January 2015|url-status=dead|access-date=20 March 2017}}</ref>