The fiber of a projector field is the actual fixed subspace \(E_x=\{v:P(x)v=v\}\). To compare it with a nearby anchor fiber \(E_a\), the completion proof constructs
It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates.
Reading the drawing
Each panel shows a range direction against the same horizontal anchor. The determinant values decrease to zero at the final panel. The note calls for a new chart there while retaining the same fiber dimension.
Drawing labels and data 10 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
- \(0\)
- \(\det T=1\)
- \(\pi/6\)
- \(\det T=3/4\)
- \(\pi/3\)
- \(\det T=1/4\)
- \(\pi/2\)
- \(\det T=0\)
The ranges of \(P(\theta)\) are shown with a horizontal anchor.
At the last position, use a new anchor. The fiber still has dimension one.
An exact example of the domain restriction
For the rotating line and anchor \(P(0)\),
The chart fails at \(t=\pi/2\), but \(P(\pi/2)={\operatorname{diag}}(0,1)\) is a perfectly regular rank-one projector. Choosing a vertical anchor supplies a new chart there. What failed was the comparison with one reference fiber, not the fiber itself.
In general, the open sets where \(T_a(x)\) is invertible cover the base. The inverse chart maps and smooth overlap maps are constructed from the actual matrices. Their cocycle identities express consistent coordinate changes, rather than assuming a global basis exists.
The v0.6.0 supplement proves the intrinsic connection and its curvature using actual manifold derivatives. Its scope includes bounded idempotents, not only orthogonal ones. The illustration uses a finite-dimensional orthogonal projector to make the construction visible. Those simplifying choices do not restrict the recorded theorem.
Read this as
A smooth bundle is built from compatible local charts. The singularity of one chart determinant can be harmless, provided another anchor covers the same smooth fiber.