68 research outputs found
The Local Moduli of Sasakian 3-Manifolds
The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and
the local form of the metric and contact structure is presented. The local
moduli space can be parameterised by a single function of two variables and it
is shown that, given any smooth function of two variables, there exists locally
a Sasakian structure with scalar curvature equal to this function. The case
where the scalar curvature is constant (-Einstein Sasakian metrics) is
completely solved locally. The resulting Sasakian manifolds include ,
and , as well as the Berger spheres. It is also shown that
a conformally flat Sasakian 3-manifold is Einstein of positive scalar
curvature.Comment: 9 pages, RevTeX, no figure
Reflection of a wave off a surface
Recent advances in twistor theory are applied to geometric optics in
. The general formulae for reflection of a wavefront in a surface
are derived and in three special cases explicit descriptions are provided: when
the reflecting surface is a plane, when the incoming wave is a plane and when
the incoming wave is spherical. In each case particular examples are computed
exactly and the results plotted to illustrate the outgoing wavefront.Comment: 14 pages, AMS-LATE
On C-smooth Surfaces of Constant Width
A number of results for C-smooth surfaces of constant width in Euclidean
3-space are obtained. In particular, an integral inequality
for constant width surfaces is established. This is used to prove that the
ratio of volume to cubed width of a constant width surface is reduced by
shrinking it along its normal lines. We also give a characterization of
surfaces of constant width that have rational support function.
Our techniques, which are complex differential geometric in nature, allow us
to construct explicit smooth surfaces of constant width in ,
and their focal sets. They also allow for easy construction of tetrahedrally
symmetric surfaces of constant width.Comment: 14 pages AMS-LATEX, 5 figure
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