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On the development of differential geometry in Estonia. (English) Zbl 1085.53001

Abel, Mati (ed.), FinEst Math 2002. Recent developments in mathematics. Proceedings of the 1st Finnish-Estonian colloquium, FinEst Math 2002, Tallinn, Estonia, May 27–29, 2002. Tartu: Estonian Mathematical Society (ISBN 9985-9444-3-7/pbk). Mathematics Studies (Tartu) 2, 20-41 (2004).
The author provides a concise survey of research in differential geometry in Estonia from the 1820s and the founding work of Martin Bartels (1769–1836), the same time teacher of Gauss in Braunschweig and of Lobachevsky in Kasan, to the present time and modern research in non-commutative geometry. Some biographical data are provided of the main protagonists. The preeminent role of the University of Tartu for the development of differential geometry in Estonia is evident from the time Bartels moved there from Kasan. Bartels’ invention of the method of moving frames for space curves predates the works of Frenet and Serret for some 20 years. His work at Tartu was followed by that of Ferdinand Minding (1806–1885), whose contributions were mainly in the theory of surfaces. He defined the geodesic curvature of a curve on a surface and laid the foundation of the theory of surface bending. One of the students of Minding’s was Karl Peterson (1828–1881), who continued the work on surface bending. In particular, he produced some equations of the surface using the coefficients of the first and the second fundamental forms (related to the Gauss-Codazzi equations), which determine the surface up to congruence. Other geometers of that period which merit recognition include Thomas Clausen (new types of squarable lunes of Hippocrates, a new proof of the Gauss’ theorem on the angle defect of the spherical geodesic triangle), Otto Staude, and Adolf Kneser.
Friedrich Schur (1856–1932) spent only four years at Tartu and is mostly known for the theorem (bearing his name) on the spaces of constant curvature and his significant contribution, alongside S. Lie, F. Engel, L. Maurer, and W. Killing, to the theory of transformation groups. His successor at Tartu, Leonid Lachtin studied Lobachevskian geometry, in particular its Poincaré model.
The modern period of differential geometry in Estonia begins in the 1950s and is largely influenced by the activities of Ülo Lumiste (born 1929). Lumiste’s research focused on several areas: 1. Minimal and ruled surfaces and their generalizations. 2. Semi-parallel and 2-parallel submanifolds. 3. General theory of connections on fibre bundles. 4. Connections in gauge theories. For example, he studied the minimal non-flat surfaces of constant Gaussian curvature in the elliptic space \(S_n(c),\) which he classified in dimensions \(n \neq 5\) and computed their Gaussian curvature, showing that such a surface is an orbit of a Lie group of isometries of the ambient elliptic space. Semi-parallel submanifolds (introduced by J. Deprez) are by definition those that satisfy the integrability condition for the parallelity of the second fundamental form, \(\bar\nabla h = 0.\) One of its subclasses is the class of 2-parallel submanifolds, which satisfy \(\bar\nabla^2 h = 0,\) studied by Lumiste and his students. It was proved by Lumiste (1990) that any semiparallel submanifold in a space form is the second order envelope of an orbit of a Lie group of isometries. This led to the study of symmetric orbits and their envelopes, whereby some of the orbits can be constructed by using mappings known in algebraic geometry (e.g., Segre mapping).
One large area of research that raised the interest of Lumiste and his students was the study of connections on fibre bundles, in particular on the bundle of subspaces of Euclidean and symplectic spaces (i.e. on various Grassmannians) and related non-abelian gauge field theories. The activities of the joint geometry-physics seminar at the University of Tartu in the 1970s generated a flurry of research in quantum Yang-Mills theory, supersymmetry, supermanifolds, and supergravity, in particular as related to the Faddeev-Popov ghost fields (two anticommuting generators of a certain infinite dimensional Grassmann algebra).
In quite recent times contributors to the development of differential geometry include Maido Rahula (born 1936) who obtained valuable results in the study of the structure of a jet space \(J_{n, m}\) which he applied to the geometric theory of differential equations and its singularities, as well as to gain information about various operators (such as Laplacian, Hessian, the curvature operator) on a manifold, their invariants and symmetries.
In the last section of the paper the author highlights some of his research done in collaboration with other differential geometers in the field of noncommutative geometry and quantum groups. An example includes the study of ternary Grassmann algebra and the related \(\mathbb Z_3\)-generalization of supersymmetry. The ternary Grassmann algebra is an associative algebra over complex numbers with generators \(\{\theta^A \}\) satisfying \(\theta^A \theta^B \theta^C = j \, \theta^B \theta^C \theta^A ,\) where \(j\) is a primitive cube root of unity. As a consequence, the products of more than three generators must vanish and a cube of a generator is equal to zero. This algebra admits both \(\mathbb Z_2\) and \(\mathbb Z_3\) grading in a natural way, which, together with the operations of the algebra, model bosonic and fermionic particles, depending on the number of quarks and antiquarks used. The supersymmetries and derivations of this algebra provide a non-commutative geometric model for Higgs fields.
Altogether, this work provides an informative, historic survey of differential geometry in Estonia through the works of major contributors and their students.
For the entire collection see [Zbl 1054.00009].

MSC:

53-03 History of differential geometry
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A35 Non-Euclidean differential geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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