Erich Kähler
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Erich Kähler (; 16 January 1906 – 31 May 2000) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory.


Education and life

Erich Kähler was born in Leipzig, the son of a telegraph inspector Ernst Kähler. Inspired as a boy to be an explorer after reading books about
Sven Hedin Sven Anders Hedin, KNO1kl RVO,Wennerholm, Eric (1978) ''Sven Hedin – En biografi'', Bonniers, Stockholm (19 February 1865 – 26 November 1952) was a Swedish geographer, topographer, explorer, photographer, travel writer and illustrator ...
that his mother Elsa Götsch had given to him, the young Kähler soon focused his passion for exploration on astronomy. He is said to have written a 50-page thesis on fractional differentiation while still in high school, hoping that it would earn him a PhD. His teachers replied that he would have to attend university courses first. Kähler enrolled in the
University of Leipzig Leipzig University (german: Universität Leipzig), in Leipzig in Saxony, Germany, is one of the world's oldest universities and the second-oldest university (by consecutive years of existence) in Germany. The university was founded on 2 Decemb ...
in 1924. He read
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, met the mathematician
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
, and did research under the supervision of
Leon Lichtenstein Leon Lichtenstein (16 May 1878 – 21 August 1933) was a Polish-German mathematician, who made contributions to the areas of differential equations, conformal mapping, and potential theory. He was also interested in theoretical physics, publish ...
. Still fascinated by celestial mechanics, Kähler wrote a dissertation entitled ''On the existence of equilibrium solutions of rotating liquids, which are derived from certain solutions of the n-body problem'', and received his
doctorate A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''l ...
in 1928. He continued his studies at Leipzig for the following year, supported by fellowship from the Notgemeinschaft der Deutschen Wissenschaften, except for a research assistantship at the
University of Königsberg The University of Königsberg (german: Albertus-Universität Königsberg) was the university of Königsberg in East Prussia. It was founded in 1544 as the world's second Protestant academy (after the University of Marburg) by Duke Albert of Prussi ...
in 1929. In 1930 Kähler joined the Department of Mathematics at the University of Hamburg to work under the direction 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 taught ...
, writing a
habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
thesis entitled, "About the integrals of algebraic equations". He took a year in Rome to work with Italian geometers including Enriques, Castelnuovo,
Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
, Severi, and Segre in 1931-1932, which led him to publish his acclaimed work on what are now called Kähler metrics in 1932. Kähler returned to Hamburg after his year in Rome, where he continued to work until going to the
University of Konigsberg A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, t ...
in 1935, and was offered an ordinary professorship a year later. In 1938 he married his first wife Luise Günther. In the years leading up to
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
Kähler was a supporter of Hitler and of German nationalism, and reported that he volunteered for the German military in 1935, joined the navy in 1937, and the army on 24 August 1939 before the invasion of Poland. After being stationed at the
Saint-Nazaire submarine base The submarine base of Saint-Nazaire is one of five large fortified U-boat pens built by Germany during the Second World War in occupied Saint-Nazaire, France. Construction Before the Second World War, Saint-Nazaire was one of the largest ha ...
in German
Occupied France The Military Administration in France (german: Militärverwaltung in Frankreich; french: Occupation de la France par l'Allemagne) was an interim occupation authority established by Nazi Germany during World War II to administer the occupied zo ...
towards the end of the war, Kähler was captured by the Allies and taken to the prisoner of war camp at
Ile de Ré Ile may refer to: * iLe, a Puerto Rican singer * Ile District (disambiguation), multiple places * Ilé-Ifẹ̀, an ancient Yoruba city in south-western Nigeria * Interlingue (ISO 639:ile), a planned language * Isoleucine, an amino acid * Another ...
, and then to another camp in
Mulsanne Mulsanne () is a commune in the Sarthe department in the region of Pays de la Loire in north-western France. Population Motor racing The Circuit de la Sarthe, which is used in the sports car endurance race 24 Hours of Le Mans, features the lon ...
. Thanks to the French physicist
Frederic Joliot-Curie Frederic may refer to: Places United States * Frederic, Wisconsin, a village in Polk County * Frederic Township, Michigan, a township in Crawford County ** Frederic, Michigan, an unincorporated community Other uses * Frederic (band), a Japanese r ...
and mathematician
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
, Kähler was able to study mathematics during this time, receiving books and mathematics papers and working during his imprisonment. He was released in 1947. He reported that his oath to Hitler (as a civil servant) was important to him, and remained an apologist for the Third Reich decades later, in a 1988 interview with Sanford Segal. A former student reported in 1988 that he kept a Nazi navy flag in his office. After his release as a prisoner of war Kähler returned to the University of Hamburg to take up a temporary lectureship. He accepted a professorship in 1948 at his alma mater the University of Leipzig, filling a post that had been left open by the death of
Paul Koebe Paul Koebe (15 February 1882 – 6 August 1945) was a 20th-century German mathematician. His work dealt exclusively with the complex numbers, his most important results being on the uniformization of Riemann surfaces in a series of four papers in ...
in 1945. But in this same year, Soviet occupation authorities began transferring administrative in the region to German communist leaders, and from October 1949 the region was a part of newly-formed
East Germany East Germany, officially the German Democratic Republic (GDR; german: Deutsche Demokratische Republik, , DDR, ), was a country that existed from its creation on 7 October 1949 until its dissolution on 3 October 1990. In these years the state ...
. Kähler became increasingly unhappy with life in East Germany over the next decade, finally deciding to leave in 1958 to take up a lectureship at the
Technical University of Berlin The Technical University of Berlin (official name both in English and german: link=no, Technische Universität Berlin, also known as TU Berlin and Berlin Institute of Technology) is a public research university located in Berlin, Germany. It was ...
. There he was heralded as among the greatest living mathematicians, and his lectures overflowed with 600 students from engineering and the sciences. In 1964 he returned to the University of Hamburg to fill the post that opened when
Artin Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun, an Armenian given name * 15378 Artin, a main-belt asteroid See also

{{disambiguation, surname ...
died in 1962. His wife Luise became ill and died in 1970, and Kähler married his second wife Charlotte Schulze, who was the widow of his brother who had died in the war. Kähler remained at the University of Hamburg until his retirement in 1974. After retiring Kähler remained an active researcher, writing a number of important papers on the foundations of physics and the Poincaré group, as well as a number of philosophical papers.


Contributions

As a mathematician Kähler is known for a number of contributions: the Cartan–Kähler theorem on solutions of non-linear analytic differential systems; the idea of a
Kähler metric Kähler may refer to: ;People *Alexander Kähler (born 1960), German television journalist *Birgit Kähler (born 1970), German high jumper *Erich Kähler (1906–2000), German mathematician *Heinz Kähler (1905–1974), German art historian and arc ...
on
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s; and the
Kähler differentials Kähler may refer to: ;People *Alexander Kähler (born 1960), German television journalist *Birgit Kähler (born 1970), German high jumper *Erich Kähler (1906–2000), German mathematician *Heinz Kähler (1905–1974), German art historian and arc ...
, which provide a purely algebraic theory and have generally been adopted in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. In all of these the theory of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s plays a part, and Kähler counts as a major developer of the theory from its formal genesis with
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
.
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
s —
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s endowed with a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
and a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument s ...
so that the three structures are mutually compatible — are named after him. The
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
is named after Kummer, Kähler, and
Kodaira is a city located in the western portion of Tokyo Metropolis, Japan. , the city had an estimated population of 195,207 in 93,654 households, and a population density of 9500 persons per km². The total area of the city was . Geography Kodaira ...
. His earlier work was on
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
; and he was one of the forerunners of
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
theory, though his ideas on that were never widely adopted.


See also

*
Almost complex manifold In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...
* Complex Poisson manifold * Hyper-Kähler manifold * Kähler quotient * Hyperkähler quotient * Kähler–Einstein metric *
Nearly Kähler manifold In mathematics, a nearly Kähler manifold is an almost Hermitian manifold M, with almost complex structure J, such that the (2,1)-tensor \nabla J is skew-symmetric. So, : (\nabla_X J)X =0 for every vector field X on M. In particular, a Kähl ...
*
Quaternion-Kähler manifold In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(''n'')·Sp(1) for some n\geq 2. Here Sp(''n'') is the sub-group of SO(4n) con ...
* Special Kähler geometry


References


Sources


KÄHLER, Erich Ernst
International Who's Who. accessed September 3, 2006. * {{DEFAULTSORT:Kahler, Erich 1906 births 2000 deaths 20th-century German mathematicians Algebraic geometers Scientists from Leipzig People from the Kingdom of Saxony University of Königsberg alumni Leipzig University alumni Academic staff of Leipzig University Members of the German Academy of Sciences at Berlin Academic staff of the University of Hamburg