Erich Kähler
Encyclopedia
was a German
mathematician
with wide-ranging geometrical
interests.
Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig
. He held professorial positions in Königsberg
, Leipzig
, Berlin
and Hamburg
. Later in life he became interested in general philosophical issues.
As a mathematician he is known for a number of contributions: the Cartan–Kähler theorem on singular solution
s of non-linear analytic differential systems; the idea of a Kähler metric on complex manifold
s; and the Kähler differentials, which provide a purely algebraic theory and have generally been adopted in algebraic geometry
. In all of these the theory of differential form
s plays a part, and Kähler counts as a major developer of the theory from its formal genesis with Élie Cartan
.
Kähler manifold
s — complex manifold
s endowed with a Riemannian metric and a symplectic form so that the three structures are mutually compatible — are named after him.
The K3 surface
is named after Kummer
, Kähler, and Kodaira.
His earlier work was on celestial mechanics
; and he was one of the forerunners of scheme
theory, though his ideas on that were never widely adopted.
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...
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....
with wide-ranging geometrical
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
interests.
Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...
. He held professorial positions in Königsberg
Königsberg
Königsberg was the capital of East Prussia from the Late Middle Ages until 1945 as well as the northernmost and easternmost German city with 286,666 inhabitants . Due to the multicultural society in and around the city, there are several local names for it...
, Leipzig
Leipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...
, 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...
and Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...
. Later in life he became interested in general philosophical issues.
As a mathematician he is known for a number of contributions: the Cartan–Kähler theorem on singular solution
Singular solution
A singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...
s of non-linear analytic differential systems; the idea of a Kähler metric on complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s; and the Kähler differentials, which provide a purely algebraic theory and have generally been adopted in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. In all of these the theory of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s plays a part, and Kähler counts as a major developer of the theory from its formal genesis with Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
.
Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
s — complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s endowed with a Riemannian metric and a symplectic form so that the three structures are mutually compatible — are named after him.
The K3 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
is named after Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
, Kähler, and Kodaira.
His earlier work was on celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
; and he was one of the forerunners of scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
theory, though his ideas on that were never widely adopted.