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Content uploaded by Josef Janyška

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All content in this area was uploaded by Josef Janyška on Nov 06, 2015

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... It is well known (see [14], [17]) that the splitting of the tangent bundle to T M into the vertical and horizontal distributions, defined by the Levi Civita connection of g on M , and the corresponding Sasaki metric lead to an almost Kähler structure on T M. The results from [7] (see also [6], [11]), giving a general expression of the natural 1-st order lifts of the Riemannian metric g to T M , allow us to consider some interesting problems concerning the diagonal natural 1-st order almost Hermitian lifts of g to T M. The second author has studied some properties of a special natural 1-st order lift G of g and a natural almost complex structure J on T M (see [9], [10], [12], and see also [11], [13]). ...

... Denote by C = y i ∂ ∂y i the Liouville vector field on T M and by C = y i δ δx i the similar horizontal vector field on T M. Let a 1 , a 2 , b 1 , b 2 : [0, ∞) → R be some smooth functions. A natural 1-st order almost complex structure J of diagonal type on T M is given by (see [7]) ...

... Consider a diagonal 1-st order, natural F-metric G on T M (see [7], see also , [6], [11]), given by ...

Among all the natural almost Kählerian structures on the tangent bundle T M , we select those with the property that any holomorphic plane making a certain angle with Liouville vector field have the same curvature. Mainly, we prove that this happens only for those structures with constant holomorphic sectional curvature. Mathematics Subject Classification: 53C55, 53C15.

... Our idea of naturality is closely related to that of A. Nijenhuis, D. B. A. Epstein, P. Stredder and others (see [6] for the full references). We have used for our purposes the concepts and methods developed by D. Krupka [12] [13] and D. Krupka and V. Mikolášová [15]. See also I. Kolář, P. W. Michor and J. Slovák [6], and D. Krupka and J. Janyška [14] for the concept of naturality in general. ...

In this paper we prove that each g-natural metric on a linear frame bundle LM over a Riemannian manifold (M, g) is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define g-natural metrics on the orthonormal frame bundle OM and we prove the same invariance result as above for OM. Hence we see that, over a space (M, g) of constant sectional curvature, the bundle OM with an arbitrary g-natural metric G̃ is locally homogeneous.

... The fundamental differences between the geometry of the cotangent bundle and that of the tangent bundle of a Riemannian manifold, are due to the different construction of lifts to T * M , which cannot be defined just like in the case of T M (see [21]). The results from [4], [5], [6], [20], concerning the natural lifts, and the classification of the natural vector fields on the tangent bundle of a pseudo-Riemannian manifold, made by Janyška in [3], allowed the present author to introduce in the paper [1], a general natural almost complex structure J of lifted type on the cotangent bundle T * M , and a general natural lifted metric G defined by the Riemannian metric g on T * M (see the paper [9] by Oproiu, for the case of the tangent bundle). The main result from [1] is that the family of general natural Kähler structures on T * M depends on three essential parameters (one is a certain proportionality factor obtained from the condition for the structure to be almost Hermitian and the other two are coefficients involved in the definition of the integrable almost complex structure J on T * M ). ...

We study the conditions under which the cotangent bundle T*M of a Riemannian manifold (M, g), endowed with a Kählerian structure (G, J) of general natural lift type (see [4]), is Einstein. We first obtain a general natural Kähler-Einstein structure on the cotangent bundle T*M. In this case, a certain parameter, λ involved in the condition for (T*M, G, J) to be a Kählerian manifold, is expressed as a rational function of the other two, the value of the constant sectional curvature, c, of the base manifold (M, g) and the constant ρ involved in the condition for the structure of being Einstein. This expression of λ is just that involved in the condition for the Kählerian manifold to have constant holomorphic sectional curvature (see [5]). In the second case, we obtain a general natural Kähler-Einstein structure only on T0M, the bundle of nonzero cotangent vectors to M. For this structure, λ is expressed as another function of the other two parameters, their derivatives, c and ρ.

... Disentangling structures from geometric phenomena to their categorical formulation was a long process and it is described in [11]. It is well known that the differential invariant is defined as a G r n -equivariant mapping f : Y → Z from a G r n -manifold Y into a G r n -manifold Z ( see [4]), where G r n = inv J r 0 (R n , R n ) 0 (invertible r-th order jets from R n into R n with source and target in 0 = (0, . . . , 0)); G r n is a Lie group (called usually the jet group or the differential group), Y and Z are manifolds endowed with the left action of G r n and c Miroslav Kureš, 2015 4 MIROSLAV KURĚ S f (gy) = gf (y). ...

Fixed point subalgebras of some local algebras obtained as quotions of polynomial algebras over an arbitrary field F with respect to all F-algebra automorphisms are described.

... Are there sufficiently many such observables to separate points + on the physical phase space ¯ P of GR? The general mathematical context in which the answers must be sought is known as differential invariant theory [25] [29] [24]. Classical invariant theory is concerned with identifying functions on a G-space (a space with an action of a group G) that are invariant under the G-action, these are the usual invariants. ...

It is well known that General Relativity (GR) does not possess any
non-trivial local (in a precise standard sense) and diffeomorphism invariant
observables. We propose a generalized notion of local observables, which retain
the most important properties that follow from the standard definition of
locality, yet is flexible enough to admit a large class of diffeomorphism
invariant observables in GR. The generalization comes at a small price, that
the domain of definition of a generalized local observable may not cover the
entire phase space of GR and two such observables may have distinct domains.
However, the subset of metrics on which generalized local observables can be
defined is in a sense generic (its open interior is non-empty in the Whitney
strong topology). Moreover, generalized local gauge invariant observables are
sufficient to separate diffeomorphism orbits on this admissible subset of the
phase space. Connecting the construction with the notion of differential
invariants, gives a general scheme for defining generalized local gauge
invariant observables in arbitrary gauge theories, which happens to agree with
well-known results for Maxwell and Yang-Mills theories.

... The possibility to consider vertical, complete and horizontal lifts on the tangent bundle T M (see [18] ) leads to some interesting geometric structures, studied in the last years (see [1], [2], [3], [8], [9], [17]), and to interesting relations with some problems in Lagrangian and Hamiltonian mechanics. On the other hand, the na-tural lifts of g to T M (introduced in [5] and [6] ) induce some new Riemannian and pseudo- Riemannian geometric structures with many nice geometric properties (see [4], [5]). Professor Oproiu has studied some properties of a natural lift G, of diagonal type, of the Riemannian metric g and a natural almost complex structure J of diagonal type on T M (see [11], [12], [13], and see also [15], [16]). ...

We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold (M,g) and we find the conditions under which the Riemannian manifold (TM,G), where TM is the tangent bundle of M and G is the so-called general natural lifted metric of g, has constant sectional curvature.

This chapter contains a relatively complete theory of higher-order integral variational functionals with one-dimensional immersed submanifolds the subjects of variations.

We assume a vector bundle \(E\rightarrow M\) and the principal bundle PE of frames of \(E\). Let \(K\) be a general linear connection on \(E\) and \(\varLambda \) be a linear connection on \(M\). We classify all connections on \({\widetilde{W}}^2PE\) naturally given by \(K\) and \(\varLambda \).

In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W r FX of FX with structure group G nr. For a structure group of J r FX we find a suitable subgroup of G nr. We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r FX corresponds with the standard actions of differential groups on tensor spaces.

We consider a vector bundle E → M and the principal bundle PE of frames of E. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle PE into itself.

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