198 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
Polyvector Super-Poincare Algebras
A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the
pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature
is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and
g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the
maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W.
Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of
spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all
the irreducible so(V)-submodules of W_0 are submodules of \Lambda V. We provide
a classification of such Lie (super)algebras for all dimensions and signatures.
The problem reduces to the classification of so(V)-invariant \Lambda^k V-valued
bilinear forms on the spinor module S.Comment: 41 pages, minor correction
Subspaces of a para-quaternionic Hermitian vector space
Let be a para-quaternionic Hermitian structure on the real
vector space . By referring to the tensorial presentation , we
give an explicit description, from an affine and metric point of view, of main
classes of subspaces of which are invariantly defined with respect to the
structure group of and respectively
Geometry of saccades and saccadic cycles
The paper is devoted to the development of the differential geometry of
saccades and saccadic cycles. We recall an interpretation of Donder's and
Listing's law in terms of the Hopf fibration of the -sphere over the
-sphere. In particular, the configuration space of the eye ball (when the
head is fixed) is the 2-dimensional hemisphere , which is called
Listing's hemisphere. We give three characterizations of saccades: as geodesic
segment in the Listing's hemisphere, as the gaze curve and as a piecewise
geodesic curve of the orthogonal group. We study the geometry of saccadic
cycle, which is represented by a geodesic polygon in the Listing hemisphere,
and give necessary and sufficient conditions, when a system of lines through
the center of eye ball is the system of axes of rotation for saccades of the
saccadic cycle, described in terms of world coordinates and retinotopic
coordinates. This gives an approach to the study the visual stability problem.Comment: 9 pages, 3 figure
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
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
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
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