Wilhelm Paul Albert Klingenberg (28 January 1924 – 14 October 2010) was a German mathematician who worked on differential geometry and in particular on

closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flo ...

Life

Klingenberg was born in 1924 as the son of a Protestant minister. In 1934 the family moved to

Berlin Berlin ( , ) is the capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's List of cities in the European Union by population within ci ...

; he joined the

Wehrmacht The ''Wehrmacht'' (, ) were the unified armed forces of Nazi Germany from 1935 to 1945. It consisted of the ''Heer'' (army), the '' Kriegsmarine'' (navy) and the ''Luftwaffe'' (air force). The designation "''Wehrmacht''" replaced the previo ...

in 1941. After the war, he studied mathematics at the

University of Kiel Kiel University, officially the Christian-Albrecht University of Kiel, (german: Christian-Albrechts-Universität zu Kiel, abbreviated CAU, known informally as Christiana Albertina) is a university in the city of Kiel, Germany. It was founded in ...

, where he finished his Ph.D. in 1950 with , with a thesis in affine differential geometry. After some time as an assistant of Friedrich Bachmann, he worked in the group of

Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...

at the

University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vo ...

, where he defended his Habilitation in 1954. He then visited

Sapienza University of Rome The Sapienza University of Rome ( it, Sapienza – Università di Roma), also called simply Sapienza or the University of Rome, and formally the Università degli Studi di Roma "La Sapienza", is a public research university located in Rome, Ita ...

, working in the group of

Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algeb ...

and

Beniamino Segre Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry. Life and career He was born and studied in Turin ...

, after which he obtained a faculty position at the

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...

(with

Kurt Reidemeister Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...

), where he stayed until 1963. In 1954–55 Klingenberg spent a year at

Indiana University Bloomington Indiana University Bloomington (IU Bloomington, Indiana University, IU, or simply Indiana) is a public research university in Bloomington, Indiana. It is the flagship campus of Indiana University and, with over 40,000 students, its largest ca ...

; during this time he also visited

Marston Morse Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

. In 1956–58 he accepted invitations to the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...

Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...

. In 1962 he visited the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

as a guest of

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...

, who he knew from his time in Hamburg. Later he became a full (C4-) professor at the

University of Mainz The Johannes Gutenberg University Mainz (german: Johannes Gutenberg-Universität Mainz) is a public research university in Mainz, Rhineland Palatinate, Germany, named after the printer Johannes Gutenberg since 1946. With approximately 32,000 stud ...

, and in 1966 a full (C4-) professor at the

University of Bonn The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine ...

, a position he kept till his retirement in 1989. Klingenberg married Christine Klingenberg née Kob in 1953 and has two sons and a daughter. In 1966 he was an invited speaker at the quadrennial International Congress of Mathematicians in

Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 millio ...

; his talk was on "Morse theory in the space of closed curves".

Work

Klingenberg's area of work was geometry, especially differential geometry and

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

. Besides many articles he published several books. One of his major achievements was the proof of the

sphere theorem In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. ...

in joint work with

Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Las ...

in 1960: The sphere theorem states that a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

, simply connected Riemannian manifold with

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

contained in the interval (1, 4] is homeomorphism, homeomorphic to the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

Publications

* * * * *

References

* *

External links

Oberwolfach photos of Wilhelm Klingenberg
{{DEFAULTSORT:Klingenberg, Wilhelm 20th-century German mathematicians 21st-century German mathematicians Differential geometers 1924 births 2010 deaths People from Rostock Academic staff of the University of Göttingen Academic staff of Johannes Gutenberg University Mainz Academic staff of the University of Bonn German military personnel of World War II University of Kiel alumni

Life

Work

Publications

See also

References

External links