6,627 research outputs found
Fractional analytic index
For a finite rank projective bundle over a compact manifold, so associated to
a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of
differential operators `acting on sections of the projective bundle' in a
formal sense. In particular, any oriented even-dimensional manifold carries a
projective spin Dirac operator in this sense. More generally the corresponding
space of pseudodifferential operators is defined, with supports sufficiently
close to the diagonal, i.e. the identity relation. For such elliptic operators
we define the numerical index in an essentially analytic way, as the trace of
the commutator of the operator and a parametrix and show that this is homotopy
invariant. Using the heat kernel method for the twisted, projective spin Dirac
operator, we show that this index is given by the usual formula, now in terms
of the twisted Chern character of the symbol, which in this case defines an
element of K-theory twisted by w; hence the index is a rational number but in
general it is not an integer.Comment: 23 pages, Latex2e, final version, to appear in JD
A Discrete Theory of Connections on Principal Bundles
Connections on principal bundles play a fundamental role in expressing the
equations of motion for mechanical systems with symmetry in an intrinsic
fashion. A discrete theory of connections on principal bundles is constructed
by introducing the discrete analogue of the Atiyah sequence, with a connection
corresponding to the choice of a splitting of the short exact sequence.
Equivalent representations of a discrete connection are considered, and an
extension of the pair groupoid composition, that takes into account the
principal bundle structure, is introduced. Computational issues, such as the
order of approximation, are also addressed. Discrete connections provide an
intrinsic method for introducing coordinates on the reduced space for discrete
mechanics, and provide the necessary discrete geometry to introduce more
general discrete symmetry reduction. In addition, discrete analogues of the
Levi-Civita connection, and its curvature, are introduced by using the
machinery of discrete exterior calculus, and discrete connections.Comment: 38 pages, 11 figures. Fixed labels in figure
A Generalized Montgomery Phase Formula for Rotating Self Deforming Bodies
We study the motion of self deforming bodies with non zero angular momentum
when the changing shape is known as a function of time. The conserved angular
momentum with respect to the center of mass, when seen from a rotating frame,
describes a curve on a sphere as it happens for the rigid body motion, though
obeying a more complicated non-autonomous equation. We observe that if, after
time , this curve is simple and closed, the deforming body \'{}s
orientation in space is fully characterized by an angle or phase .
We also give a reconstruction formula for this angle which generalizes R.
Montgomery\'{}s well known formula for the rigid body phase. Finally, we apply
these techniques to obtain analytical results on the motion of deforming bodies
in some concrete examples.Comment: 20 page
Fefferman's mapping theorem on almost complex manifolds
We give a necessary and sufficient condition for the smooth extension of a
diffeomorphism between smooth strictly pseudoconvex domains in four real
dimensional almost complex manifolds. The proof is mainly based on a reflection
principle for pseudoholomorphic discs, on precise estimates of the
Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost
complex manifolds
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