Alphabeta MathHow statement and proof provenance work
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Time-dependent vector fields and their evolution operators
Definition
Assume , so that carries its canonical smooth structure. Let be a smooth manifold, and let be an interval. A time-dependent vector field on over is a smooth map
such that for every . One often writes .
An evolution operator for is a family of maps
defined for pairs in some time domain, such that for each the curve solves
Depends on
Used by
- A regular continuation datum between Morse--Smale pairs Definition
- Time-dependent Hamiltonian vector field and flow Definition
- A time-dependent translation field and its evolution operator Example
- A compactly supported time-dependent field has a global time-one flow Lemma
- Compactness gives a compactly supported time-dependent velocity field Lemma
- Isotopic attaching embeddings give diffeomorphic handle attachments Lemma
- Isotopy extension for a compact source with boundary Lemma
- Smooth finite point motions extend to disk isotopies Lemma
- Smooth relative isotopy extension for disk arcs with puncture endpoints Lemma
- Smooth relative isotopy extension for finite disk arc systems Lemma
- Compactly supported time-dependent vector fields have global evolution on a compact time interval Theorem
- Differentiation of a pulled-back form along a time-dependent flow Theorem
- Time-dependent vector fields have local smooth evolution operators Theorem
Dependency tree · two levels
4 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
- Marius Crainic, Rui Loja Fernandes, and Ioan Marcut, Lectures on Poisson Geometry (standard reference, not scraped)
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)