Heinz Hopf
| Heinz Hopf | |
|---|---|
Heinz Hopf (on the right) in Oberwolfach, together with Hellmuth Kneser
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| Born |
19 November 1894 Gräbschen (near Breslau), Imperial Germany |
| Died | 3 June 1971 (aged 76) Zollikon, Switzerland |
| Nationality | German |
| Fields | Mathematics |
| Institutions | ETH Zürich |
| Alma mater | University of Berlin |
| Doctoral advisor |
Ludwig Bieberbach Erhard Schmidt |
| Doctoral students |
Beno Eckmann Hans Freudenthal Werner Gysin Friedrich Hirzebruch Heinz Huber Michel Kervaire Willi Rinow Hans Samelson Ernst Specker Eduard Stiefel James J. Stoker |
| Known for |
Hopf algebra Hopf bundle Hopf conjecture Hopf link H-space Hopf–Rinow theorem |
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
Hopf was born in Gräbschen, Germany (now Grabiszyn, 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.
Hopf attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age. In 1913 he entered the Silesian Friedrich Wilhelm University where he attended lectures by Ernst Steinitz, Kneser, Max Dehn, Erhard Schmidt, and Rudolf Sturm. When World War I broke out in 1914, Hopf eagerly enlisted. He was wounded twice and received the iron cross (first class) in 1918.
In 1920, Hopf moved to Berlin to continue his mathematical education. He studied under Ludwig Bieberbach, receiving his doctorate in 1925.
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 Riemannian 3-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.
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