290 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
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