4,899 research outputs found
The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces
We give a formula for the first eigenvalue of the Dirac operator acting on
spinor fields of a spin compact irreducible symmetric space
Parametrization of Pythagorean triples by a single triple of polynomials
It is well known that Pythagorean triples can be parametrized by two triples
of polynomials with integer coefficients. We show that no single triple of
polynomials with integer coefficients in any number of variables is sufficient,
but that there exists a parametrization of Pythagorean triples by a single
triple of integer-valued polynomials.Comment: to appear in J. Pure Appl. Algebr
A construction of symplectic connections through reduction
We give an elementary construction of symplectic connections through
reduction. This provides an elegant description of a class of symmetric spaces
and gives examples of symplectic connections with Ricci type curvature, which
are not locally symmetric; the existence of such symplectic connections was
unknown.Comment: 16 pages, Plain TeX fil
A Natural Basis of States for the Noncommutative Sphere and its Moyal bracket
An infinite dimensional algebra which is a non-decomposable reducible
representation of is given. This algebra is defined with respect to two
real parameters. If one of these parameters is zero the algebra is the
commutative algebra of functions on the sphere, otherwise it is a
noncommutative analogue. This is an extension of the algebra normally refered
to as the (Berezin) quantum sphere or ``fuzzy'' sphere. A natural indefinite
``inner'' product and a basis of the algebra orthogonal with respect to it are
given. The basis elements are homogenious polynomials, eigenvectors of a
Laplacian, and related to the Hahn polynomials. It is shown that these elements
tend to the spherical harmonics for the sphere. A Moyal bracket is constructed
and shown to be the standard Moyal bracket for the sphere.Comment: 18 pages Latex, No figures, Submitted to Journal of Mathematical
Physics, March 199
Extrinsic symplectic symmetric spaces
We define the notion of extrinsic symplectic symmetric spaces and exhibit
some of their properties. We construct large families of examples and show how
they fit in the perspective of a complete classification of these manifolds. We
also build a natural star-quantization on a class of examples
Special Symplectic Connections
By a special symplectic connection we mean a torsion free connection which is
either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary
signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a
connection with special symplectic holonomy. A manifold or orbifold with such a
connection is called special symplectic.
We show that the symplectic reduction of (an open cell of) a parabolic
contact manifold by a symmetry vector field is special symplectic in a
canonical way. Moreover, we show that any special symplectic manifold or
orbifold is locally equivalent to one of these symplectic reductions.
As a consequence, we are able to prove a number of global properties,
including a classification in the compact simply connected case.Comment: 35 pages, no figures. Exposition improved, some minor errors
corrected. Version to be published by Jour.Diff.Geo
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