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Peak functions on convex domains

Kolář, Martin (2000)

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

Let Ω n be a domain with smooth boundary and p Ω . A holomorphic function f on Ω is called a C k ( k = 0 , 1 , 2 , ) peak function at p if f C k ( Ω ¯ ) , f ( p ) = 1 , and | f ( q ) | < 1 for all q Ω ¯ { p } . If Ω is strongly pseudoconvex, then C peak functions exist. On the other hand, J. E. Fornaess constructed an example in 2 to show that this result fails, even for C 1 functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function on a...

Pontryagin algebra of a transitive Lie algebroid

Kubarski, Jan (1990)

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0699.00032.] It was previously known that for every principal fibre bundle P there is some corresponding transitive Lie algebroid A(P) - a vector bundle equipped with some structure like the structure of a Lie algebra in the module of sections. The author of this article shows that the Chern-Weil homomorphism of P is a notion of the Lie algebroid of P, i.e. knowing only A(P) of P one can uniquely reproduce the ring of invariant polynomials ( V g * ) I and the Chern-Weil...

Prolongation of vector fields to jet bundles

Kolář, Ivan, Slovák, Jan (1990)

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0699.00032.] In this interesting paper the authors show that all natural operators transforming every projectable vector field on a fibered manifold Y into a vector field on its r-th prolongation J r Y are the constant multiples of the flow operator. Then they deduce an analogous result for the natural operators transforming every vector field on a manifold M into a vector field on any bundle of contact elements over M.

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