113 research outputs found
Compact Riemannian Manifolds with Homogeneous Geodesics
A homogeneous Riemannian space is called a geodesic orbit space
(shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of
the isometry group . We study the structure of compact GO-spaces and give
some sufficient conditions for existence and non-existence of an invariant
metric with homogeneous geodesics on a homogeneous space of a compact Lie
group . We give a classification of compact simply connected GO-spaces of positive Euler characteristic. If the group is simple and the
metric does not come from a bi-invariant metric of , then is one of
the flag manifolds or and
is any invariant metric on which depends on two real parameters. In
both cases, there exists unique (up to a scaling) symmetric metric such
that is the symmetric space or, respectively,
. The manifolds , are weakly symmetric spaces
Completely integrable systems: a generalization
We present a slight generalization of the notion of completely integrable
systems to get them being integrable by quadratures. We use this generalization
to integrate dynamical systems on double Lie groups.Comment: Latex, 15 page
Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics
We analyse the symmetries underlying nonassociative deformations of geometry
in non-geometric R-flux compactifications which arise via T-duality from closed
strings with constant geometric fluxes. Starting from the non-abelian Lie
algebra of translations and Bopp shifts in phase space, together with a
suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that
deforms the algebra of functions and the exterior differential calculus in the
phase space description of nonassociative R-space. In this setting
nonassociativity is characterised by the associator 3-cocycle which controls
non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists
to construct maps between the dynamical nonassociative star product and a
family of associative star products parametrized by constant momentum surfaces
in phase space. We define a suitable integration on these nonassociative spaces
and find that the usual cyclicity of associative noncommutative deformations is
replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star
product quantization on phase space together with 3-cyclicity, we formulate a
consistent version of nonassociative quantum mechanics, in which we calculate
the expectation values of area and volume operators, and find coarse-graining
of the string background due to the R-flux.Comment: 38 pages; v2: typos corrected, reference added; v3: typos corrected,
comments about cyclicity added in section 4.2, references updated; Final
version to be published in Journal of Mathematical Physic
A general method to construct invariant PDEs on homogeneous manifolds
Let M = G/H be an (n + 1)-dimensional homogeneous manifold and Jk(n,M) =: Jk be the manifold of k-jets of hypersurfaces of M. The Lie group G acts naturally on each Jk. A G-invariant partial differential equation of order k for hypersurfaces of M (i.e., with n independent variables and 1 dependent one) is defined as a G-invariant hypersurface E of Jk. We describe a general method for constructing such invariant partial differential equations for k>1. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup H(k-1) of the (k-1)-prolonged action of G. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space n+1 and in the conformal space n+1. Our method works under some mild assumptions on the action of G, namely: A1) the group G must have an open orbit in Jk-1, and A2) the stabilizer H(k-1) in G of the fiber Jk → Jk-1 must factorize via the group of translations of the fiber itself
A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics
The subject of investigations are the almost hypercomplex manifolds with
Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is
introduced such that the structure of these manifolds is parallel with respect
to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler
manifolds with respect to the first almost complex structure is of special
interest. It is proved that D has a D-parallel torsion and is weak if it is not
flat. Some curvature properties of these manifolds are studied.Comment: 18 page
Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles
We obtain the natural diagonal almost product and locally product structures
on the total space of the cotangent bundle of a Riemannian manifold. We find
the Riemannian almost product (locally product) and the (almost) para-Hermitian
cotangent bundles of natural diagonal lift type. We prove the characterization
theorem for the natural diagonal (almost) para-K\"ahlerian structures on the
total spaces of the cotangent bundle.Comment: 10 pages, will appear in Czechoslovak Mathematical Journa
Volume preserving multidimensional integrable systems and Nambu--Poisson geometry
In this paper we study generalized classes of volume preserving
multidimensional integrable systems via Nambu--Poisson mechanics. These
integrable systems belong to the same class of dispersionless KP type equation.
Hence they bear a close resemblance to the self dual Einstein equation. All
these dispersionless KP and dToda type equations can be studied via twistor
geometry, by using the method of Gindikin's pencil of two forms. Following this
approach we study the twistor construction of our volume preserving systems
Poisson structures on double Lie groups
Lie bialgebra structures are reviewed and investigated in terms of the double
Lie algebra, of Manin- and Gau{\ss}-decompositions. The standard R-matrix in a
Manin decomposition then gives rise to several Poisson structures on the
correponding double group, which is investigated in great detail.Comment: AmSTeX, 37 page
Holonomy of Einstein Lorentzian manifolds
The classification of all possible holonomy algebras of Einstein and vacuum
Einstein Lorentzian manifolds is obtained. It is shown that each such algebra
appears as the holonomy algebra of an Einstein (resp., vacuum Einstein)
Lorentzian manifold, the direct constructions are given. Also the holonomy
algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The
classification of the holonomy algebras of Lorentzian manifolds is reviewed and
a complete description of the spaces of curvature tensors for these holonomies
is given.Comment: Dedicated to to Mark Volfovich Losik on his 75th birthday. This
version is an extended part of the previous version; another part of the
previous version is extended and submitted as arXiv:1001.444
The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures
Newly introduced generalized Poisson structures based on suitable
skew-symmetric contravariant tensors of even order are discussed in terms of
the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are
expressed as conditions on these tensors, the cohomological contents of which
is given. In particular, we determine the linear generalized Poisson structures
which can be constructed on the dual spaces of simple Lie algebras.Comment: 29 pages. Plain TeX. Phyzzx needed. An example and some references
added. To appear in J. Phys.
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