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Felix Klein is best known for his work in non- euclidean geometry, for his work on the connections between geometry and group theory, and for results in function theory.
He received his doctorate from the University of Bonn in 1868, where he had studied mathematics and physics. His dissertaion was on line geometry and its applications to mechanics. In it he classified 2nd degree line complexes using Weierstrass's theory of elementary divisors.
After teaching at a number of universities Klein was appointed to a chair at the University of Göttingen in 1886. He taught at Göttingen until he retired in 1913.
Klein established a research establishment at Göttingen which served as a model for the best mathematical research centres. He introduced weekly discussion meetings, a mathematical reading room with a mathematical library. Klein brought Hilbert from Königsberg to join his research team at Göttingen in 1895.
The fame of the journal Mathematische Annalen is based on Klein's mathematical and management abilities. He set up a small team of editors who met regularly and made democratic decisions.
Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programme (1872), profoundly influenced mathematical development. It arose after joint work with his friend Sophus Lie.
Transformations play a major role in modern mathematics and Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. In this way the Erlanger Programme defined geometry so that it included both Euclidean geometry and non-Euclidean geometry.
Klein considered the action of the modular group on the complex plane and showed that it moves the fundamental region around to tessellate the plane. In 1879 he looked at the action of PSL(2,7), thought of as an image of the modular group, and obtained an explicit representation of a Riemann surface. He showed it had equation
x
y+yz+zx = 0 as a curve in projective space and its group of symmetries was PSL(2,7) of order 168.
Klein considered equations of degree greater than 4 and after producing similar results to Brioschi he developed a theory within automorphic functions, connecting algebraic and geometric results in his important 1884 book on the icosahedron.
A Klein bottle is an one-sided closed surface named after Klein. A Klein bottle cannot be constructed in Euclidean space. It is best pictured as a cylinder looped back through itself to join with its other end. However this is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible to construct a Klein bottle in non-Euclidean space.
Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society in 1912.
References elsewhere in this archive:
Tell me about Klein's part in the development of group theory
Tell me about his work on non-Euclidean geometry and on general relativity
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