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A bracket-closed variable-rank family outside regular Frobenius
Statement refuted
Any variable-rank family of tangent subspaces satisfying a bracket-closure condition is covered by the regular Frobenius theorem.
Facts & Assumptions
Given: On , let
The rank is at the origin and elsewhere.
Counterexample
Any smooth vector field tangent to this family must vanish at the origin. [given] The Lie bracket of two such fields also vanishes at the origin, so the tangent fields are closed under bracket in this loose sense.
Nevertheless the family is not a smooth distribution of constant rank, so [given] it does not satisfy the hypotheses of the regular Frobenius theorem.
Hence variable-rank bracket closure does not place a family inside regular [given] Frobenius theory.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)