Wavefronts are defined by projected images of Legendrian submanifolds on Legendrian bundles. By Darboux's theorem, all Legendrian bundles are locally contact diffeomorphic to the canonical Legendrian bundle:
for some
, where
is the 1-jet bundle of functions in
variables with the contact structure defined by the canonical one-form
. Thus, it is enough to consider the canonical Legendrian bundle for local situations. An
-dimensional submanifold
in
is a Legendrian submanifold if
vanishes on
. The wavefront of
is defined by
. The theory of wavefronts describes stabilities of wavefronts generated by a hypersurface without boundary in a smooth manifold (see [2]).
The author constructs the theory of reticular Legendrian singularities that describes stabilities and genericities of wavefronts generated by a hypersurface with a boundary, a corner, or an
-corner in a smooth manifold (see [3]). Let
and
for each
σ⊂Ir
. Let
represent a germ of the union of
for all
. We call a map
→
germ a reticular Legendrian map if there exists a contact diffeomorphism germ
on
such that
. The wavefront of
is defined by the union of
for all
.
For the case
, the hypersurface that has a boundary, the wavefront of
is
. The sets
and
correspond to the wavefronts generated by the hypersurface and its boundary, respectively. A smooth function germ
defined on
is a generating family of
if
is
non-degenerate and
,
.
Then the wavefronts of
are given by
.
Typical wavefronts are defined by generic reticular Legendrian maps and their generating families are
stably reticular
-
-equivalent to one of the following:
In the case
:
In the case
:
[1] V. I. Arnold,
Singularities of Caustics and Wave Fronts, Dordrecht: Kluwer Academic Publishers, 1990.
[2] V. I. Arnold, S. M. Gusein–Zade, and A. N. Varchenko,
Singularities of Differential Maps I, Basel: Birkhäuser, 1985.
[3] T. Tsukada, "Genericity of Caustics and Wavefronts on an r-Corner", to appear in The Asian Journal of Mathematics, 2010.
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