Heinz Hopf was a German

Germans

The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 born in Gräbschen, Germany

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 (now Grabiszyn, part of Wrocław, Poland

Poland

Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

). He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium

Gymnasium (school)

A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

 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

Ernst Steinitz

Ernst Steinitz was a German mathematician.- Biography :Steinitz was born in Laurahütte , Silesia, Germany , the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D...

, Kneser

Kneser

Kneser is a surname and may refer to the following people:*Adolf Kneser , mathematician*Hellmuth Kneser , mathematician, son of Adolf Kneser*Martin Kneser , mathematician, son of Hellmuth Kneser...

, Max Dehn

Max Dehn

Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...

, Erhard Schmidt

Erhard Schmidt

Erhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905...

, and Rudolf Sturm. When World War I

World War I

World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...

 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

Berlin

Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

 to continue his mathematical education. He studied under Ludwig Bieberbach

Ludwig Bieberbach

Ludwig Georg Elias Moses Bieberbach was a German mathematician.-Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...

, 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

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 is globally isometric

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 to Euclidean, spherical, or hyperbolic space. He also studied the indices of zeros of vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s on hypersurface

Hypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a 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

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 is independent of the choice of vector field and equal to the Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 of the manifold. This theorem is now called the Poincaré-Hopf theorem.

Hopf spent the year after his doctorate at Göttingen

Göttingen

Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...

, where David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, Richard Courant

Richard Courant

Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...

, Carl Runge, and Emmy Noether

Emmy Noether

Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...

 were working. While there he met Paul Alexandrov and began a lifelong friendship.

In 1926 Hopf moved back to Berlin, where he gave a course in combinatorial topology

Combinatorial topology

In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...

. He spent the academic year 1927/28 at Princeton University

Princeton University

Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

 on a Rockefeller fellowship with Alexandrov. Solomon Lefschetz

Solomon Lefschetz

Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...

, Oswald Veblen

Oswald Veblen

Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...

 and J.W. Alexander were all at Princeton at the time. At this time Hopf discovered the Hopf invariant

Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.- Motivation :In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map\eta\colon S^3 \to S^2,...

 of maps .
and proved that the Hopf fibration has invariant 1.
In the summer of 1928 Hopf returned to Berlin and began working with Alexandrov, at the suggestion of Courant, on a book on topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. Three volumes were planned, but only one was finished. It was published in 1935.

In October 1928 Hopf married Anja von Mickwitz (1891–1967). The next year he declined a job offer from Princeton. In 1931 Hopf took Hermann Weyl

Hermann Weyl

Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

's position at ETH

ETH Zurich

The Swiss Federal Institute of Technology Zurich or ETH Zürich is an engineering, science, technology, mathematics and management university in the City of Zurich, Switzerland....

, in Zürich

Zürich

Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...

.

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

Nazi Germany

Nazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...

 authorities.

In 1946/47 and 1955/56 Hopf visited the United States, staying at Princeton and giving lectures at New York University

New York University

New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...

 and Stanford University

Stanford University

The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...

. He served as president of the International Mathematical Union

International Mathematical Union

The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...

 from 1955 to 1958. He received honorary doctorates from Princeton, Freiburg i. Br., Manchester, Sorbonne at Paris, Brussels, and Lausanne.

In memory of Hopf, ETH Zürich awards the Heinz Hopf Prize

Heinz Hopf Prize

The Heinz Hopf Prize is awarded every two years at ETH Zurich. The prize honours outstanding scientific work in the field of pure mathematics. It is named after the German mathematician Heinz Hopf , Professor of Mathematics at ETH from 1931 to 1965. The prize amount of 30,000 Swiss Francs The Heinz...

 for outstanding scientific work in the field of pure mathematics.

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