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On admissible groups of diffeomorphisms

Rybicki, Tomasz (1997)

Proceedings of the 16th Winter School "Geometry and Physics"

The phenomenon of determining a geometric structure on a manifold by the group of its automorphisms is a modern analogue of the basic ideas of the Erlangen Program of F. Klein. The author calls such diffeomorphism groups admissible and he describes them by imposing some axioms. The main result is the followingTheorem. Let ( M i , α i ) , i = 1 , 2 , be a geometric structure such that its group of automorphisms G ( M i , α i ) satisfies either axioms 1, 2, 3 and 4, or axioms 1, 2, 3’, 4, 5, 6 and 7, and M i is compact, or axioms 1, 2,...

On embedding curves in surfaces

Bajguz, W. (2001)

Proceedings of the 20th Winter School "Geometry and Physics"

Contrary to a statement of Borsuk the author proves that every locally plane Peano continuum is embeddable into a 2-manifold.

On Gelfand-Zetlin modules

Drozd, Yu. A., Ovsienko, S. A., Futorny, V. M. (1991)

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0742.00067.]Let 𝔤 k be the Lie algebra 𝔤 l ( k , 𝒞 ) , and let U k be the universal enveloping algebra for 𝔤 k . Let Z k be the center of U k . The authors consider the chain of Lie algebras 𝔤 n 𝔤 n - 1 𝔤 1 . Then Z = Z k k = 1 , 2 , n is an associative algebra which is called the Gel’fand-Zetlin subalgebra of U n . A 𝔤 n module V is called a G Z -module if V = x V ( x ) , where the summation is over the space of characters of Z and V ( x ) = { v V ( a - x ( a ) ) m v = 0 , m 𝒵 + , a 𝒵 } . The authors describe several properties of G Z - modules. For example, they prove that if V ( x ) = 0 for some x ...

On geodesic mappings of special Finsler spaces

Bácsó, Sándor (1999)

Proceedings of the 18th Winter School "Geometry and Physics"

The author previously studied with F. Ilosvay and B. Kis [Publ. Math. 42, 139-144 (1993; Zbl 0796.53022)] the diffeomorphisms between two Finsler spaces F n = ( M n , L ) and F ¯ n = ( M n , L ¯ ) which map the geodesics of F n to geodesics of F ¯ n (geodesic mappings).Now, he investigates the geodesic mappings between a Finsler space F n and a Riemannian space ¯ n . The main result of this paper is as follows: if F n is of constant curvature K and the mapping F n ¯ n is a strongly geodesic mapping then K = 0 or K 0 and L ¯ = e ϕ ( x ) L .

On quasijet bundles

Tomáš, Jiří (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

In this paper a Weil approach to quasijets is discussed. For given manifolds M and N , a quasijet with source x M and target y N is a mapping T x r M T y r N which is a vector homomorphism for each one of the r vector bundle structures of the iterated tangent bundle T r [A. Dekrét, Casopis Pest. Mat. 111, No. 4, 345-352 (1986; Zbl 0611.58004)]. Let us denote by Q J r ( M , N ) the bundle of quasijets from M to N ; the space J ˜ r ( M , N ) of non-holonomic r -jets from M to N is embeded into Q J r ( M , N ) . On the other hand, the bundle Q T m r N of ( m , r ) -quasivelocities...

On sectioning multiples of the nontrivial line bundle over Grassmannians

Horanská, Ľubomíra (1998)

Proceedings of the 17th Winter School "Geometry and Physics"

Let G n , k ( G ˜ n , k ) denote the Grassmann manifold of linear k -spaces (resp. oriented k -spaces) in n , d n , k = k ( n - k ) = dim G n , k and suppose n 2 k . As an easy consequence of the Steenrod obstruction theory, one sees that ( d n , k + 1 ) -fold Whitney sum ( d n , k + 1 ) ξ n , k of the nontrivial line bundle ξ n , k over G n , k always has a nowhere vanishing section. The author deals with the following question: What is the least s ( = s n , k ) such that the vector bundle s ξ n , k admits a nowhere vanishing section ? Obviously, s n , k d n , k + 1 , and for the special case in which k = 1 , it is known that s n , 1 = d n , 1 + 1 . Using results...

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