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Some natural operations between connections on fibred manifolds

Doupovec, Miroslav, Vondra, Alexandr (1996)

Proceedings of the Winter School "Geometry and Physics"

Given a fibered manifold Y → X , a 2-connection on Y means a section J 1 Y → J 2 Y . The authors determine all first order natural operators transforming a 2-connection on Y and a classical linear connection on X into a connection on J 1 Y → Y . (The proof implies that there is no first order natural operator transforming 2-connections on Y into connections on J 1 Y → Y .) Using this result, the authors deduce several properties of characterizable connections on J 1 Y → X .

Some special geometry in dimension six

Čap, Andreas, Eastwood, Michael (2003)

Proceedings of the 22nd Winter School "Geometry and Physics"

Motivated by the study of CR-submanifolds of codimension  2 in  ℂ 4 , the authors consider here a 6 -dimensional oriented manifold  M equipped with a 4 -dimensional distribution. Under some non-degeneracy condition, two different geometric situations can occur. In the elliptic case, one constructs a canonical almost complex structure on  M ; the hyperbolic case leads to a canonical almost product structure. In both cases the only local invariants are given by the obstructions to integrability for these structures....

Space-time decompositions via differential forms

Fecko, Marián (1998)

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

The author presents a simple method (by using the standard theory of connections on principle bundles) of ( 3 + 1 ) -decomposition of the physical equations written in terms of differential forms on a 4-dimensional spacetime of general relativity, with respect to a general observer. Finally, the author suggests possible applications of such a decomposition to the Maxwell theory.

Special connections on smooth 3-web manifolds

Vanžurová, Alena (1996)

Proceedings of the 15th Winter School "Geometry and Physics"

For a three-web W of codimension n on a differentiable manifold M 2 n of dimension 2 n , the author studies the Chern connection and a family of parallelizing connections. The latter ones have a common property with the former: the web-distributions are parallel with respect to them.

Special Kaehler manifolds: A survey

Cortés, Vincente (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

This is a survey of recent contributions to the area of special Kähler geometry. A (pseudo-)Kähler manifold ( M , J , g ) is a differentiable manifold endowed with a complex structure J and a (pseudo-)Riemannian metric g such that i) J is orthogonal with respect to the metric g , ii) J is parallel with respect to the Levi Civita connection D . A special Kähler manifold ( M , J , g , ∇ ) is a Kähler manifold ( M , J , g ) together with a flat torsionfree connection ∇ such that i) ∇ ω = 0 , where ω = g ( . , J . ) is the Kähler form and ii) ∇ is symmetric. A holomorphic...

Spectral theory of invariant operators, sharp inequalities, and representation theory

Branson, Thomas (1997)

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

The paper represents the lectures given by the author at the 16th Winter School on Geometry and Physics, Srni, Czech Republic, January 13-20, 1996. He develops in an elegant manner the theory of conformal covariants and the theory of functional determinant which is canonically associated to an elliptic operator on a compact pseudo-Riemannian manifold. The presentation is excellently realized with a lot of details, examples and open problems.

Spinor equations in Weyl geometry

Buchholz, Volker (2000)

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

This paper deals with Dirac, twistor and Killing equations on Weyl manifolds with C -spin structures. A conformal Schrödinger-Lichnerowicz formula is presented and used to derive integrability conditions for these equations. It is shown that the only non-closed Weyl manifolds of dimension greater than 3 that admit solutions of the real Killing equation are 4-dimensional and non-compact. Any Weyl manifold of dimension greater than 3, that admits a real Killing spinor has to be Einstein-Weyl.

Supersymmetry, a biased review

Lindström, U. (2003)

Proceedings of the 22nd Winter School "Geometry and Physics"

In spite of the provocative title this is a run of the mill review of supersymmetry. The only thing which deserves some comment is that the author seems to think that coordinate free and coordinate dependent treatments belong to conflicting cultures. This is definitely not true. Coordinate free treatments concentrate one's mind on the geometry while coordinate dependent treatments are indispensable for computations producing numbers which can be compared with experimental values. Those who use the...

Symmetric algebras and Yang-Baxter equation

Beidar, K., Fong, Y., Stolin, A. (1997)

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

Let U be an open subset of the complex plane, and let L denote a finite-dimensional complex simple Lie algebra. A. A. Belavin and V. G. Drinfel’d investigated non-degenerate meromorphic functions from U × U into L ⊗ L which are solutions of the classical Yang-Baxter equation [Funct. Anal. Appl. 16, 159-180 (1983; Zbl 0504.22016)]. They found that (up to equivalence) the solutions depend only on the difference of the two variables and that their set of poles forms a discrete (additive) subgroup Γ of the...

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