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Isometric immersions and induced geometric structures

D‘Ambra, G. (1999)

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

In the paper under review, the author presents some results on the basis of the Nash-Gromov theory of isometric immersions and illustrates how the same results and ideas can be extended to other structures.

Isometry groups of k -curvature homogeneous pseudo-Riemannian manifolds

Gilkey, P., Nikčević, S. (2006)

Proceedings of the 25th Winter School "Geometry and Physics"

In 2005 Gilkey and Nikčević introduced complete ( p + 2 ) -curvature homogeneous pseudo-Riemannian manifolds of neutral signature ( 3 + 2 p , 3 + 2 p ) , which are 0 -modeled on an indecomposable symmetric space, but which are not ( p + 3 ) -curvature homogeneous. In this paper the authors continue their study of the same family of manifolds by examining their isometry groups and the isometry groups of their k -models.

Isospectral, non-isometric Riemannian manifolds

Schueth, Dorothea (1994)

Proceedings of the Winter School "Geometry and Physics"

The author gives a survey of the history of isospectral manifolds that are non-isometric discussing the work of Milnor, Vignéras, Sunada, and de Turck and Gordon. She describes the construction of continuous isospectral deformations as introduced by Gordon, Wilson, De Turck et al. She also discusses the construction of isospectral plane domains due to Gordon, Webb, and Wolpert. Some new examples of isospectral non-isometric manifolds are given.

Isotropy representation of flag manifolds

Alekseevsky, D. V. (1998)

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

A flag manifold of a compact semisimple Lie group G is defined as a quotient M = G / K where K is the centralizer of a one-parameter subgroup exp ( t x ) of G . Then M can be identified with the adjoint orbit of x in the Lie algebra 𝒢 of G . Two flag manifolds M = G / K and M ' = G / K ' are equivalent if there exists an automorphism φ : G → G such that φ ( K ) = K ' (equivalent manifolds need not be G -diffeomorphic since φ is not assumed to be inner). In this article, explicit formulas for decompositions of the isotropy representation for all flag manifolds...

Knit products of graded Lie algebras and groups

Michor, Peter W. (1990)

Proceedings of the Winter School "Geometry and Physics"

Let A = ⨁ k A k and B = ⨁ k B k be graded Lie algebras whose grading is in 𝒵 or 𝒵 2 , but only one of them. Suppose that ( α , β ) is a derivatively knitted pair of representations for ( A , B ) , i.e. α and β satisfy equations which look “derivatively knitted"; then A ⊕ B : = ⨁ k , l ( A k ⊕ B l ) , endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra A ⊕ ( α , β ) B . This graded Lie algebra is called the knit product of A and B . The author investigates the general situation for any graded Lie subalgebras A and B of a graded...

Lagrange functions generating Poisson manifolds of geodesic arcs

Klapka, Lubomír (2000)

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

Let X a smooth finite-dimensional manifold and W Γ ( X ) the manifold of geodesic arcs of a symmetric linear connection Γ on X . In a previous paper [Differential Geometry and Applications (Brno, 1995) 603-610 (1996; Zbl 0859.58011)] the author introduces and studies the Poisson manifolds of geodesic arcs, i.e. manifolds of geodesic arcs equipped with certain Poisson structure. In this paper the author obtains necessary and sufficient conditions for that a given Lagrange function generates a Poisson manifold...

Liftings of 1-forms to some non product preserving bundles

Doupovec, Miroslav, Kurek, Jan (1998)

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

Summary: The article is devoted to the question how to geometrically construct a 1-form on some non product preserving bundles by means of a 1-form on an original manifold M . First, we will deal with liftings of 1-forms to higher-order cotangent bundles. Then, we will be concerned with liftings of 1-forms to the bundles which arise as a composition of the cotangent bundle with the tangent or cotangent bundle.

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