Smooth atlases on regular zero sets of bundle sections #
For a section along a map from a Banach manifold, continuous at its zeros, regular Fredholm zeros
of constant index n form a manifold modelled on Fin n → 𝕜. The atlas is on the actual zero
set, not on an extended fiber-coordinate equation, which can have spurious zeros outside a
trivialization.
We choose a bundle trivialization at each zero, compose a source chart with implicit coordinates,
and restrict the source to the neighbourhood where the implicit-function coordinate map has
invertible derivative.
The inverse charts are then smooth throughout their targets. Chart transitions are inverse
charts followed by ambient source charts and linear projections of coordinate displacement,
so no global trivialization is needed. Atlas assembly follows levelSetChartedSpace and
isManifold_levelSet for ordinary maps. The inclusion into the parameter space is smooth
for this atlas and is an immersion with complement the fiber model.
Smoothness is stated in fiber coordinates and preferred source charts at zeros in the chosen
trivializations and regular implicit-coordinate neighbourhoods. It is required only for nonzero
differentiability order.
At order zero, chart continuity suffices. Only the base map needs to be continuous at the zeros;
no differentiability of that map or smooth bundle structure is needed once these coordinate
hypotheses are supplied. The source is a manifold modelled on a Banach space without boundary. No
second-countability hypothesis is imposed: IsManifold
asserts smooth compatibility of the atlas, not second countability.
References #
- D. McDuff, D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., Appendix A.3, for the regular-zero theorem for Fredholm sections.
The inverse preferred chart, included into the source manifold, is smooth throughout its target.
Transitions between the preferred charts of the zero set are smooth.
The actual zero set of a regular Fredholm section of constant index is a smooth manifold of dimension that index, with the preferred section-zero atlas.
The differential of the zero-set inclusion at a zero is the inclusion of the kernel of its coordinate derivative, read through the index model. Both tangent spaces use the preferred charts, so the ambient source-chart differential at its centre is the identity.
For the preferred section-zero atlas, the inclusion into the Banach source manifold is
an immersion with complement the fiber model F. Only the atlas's coordinate smoothness
hypotheses are needed; no smooth bundle structure or differentiability of the base map is
required.