Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Morse functions and excellent Morse functions
Definition
Let be a smooth manifold and let be smooth.
- The function is a Morse function when every critical point of is nondegenerate (Critical points and critical values of a smooth function, Nondegenerate critical points, nullity, index, and coindex).
- The function is an excellent Morse function when it is Morse and any two distinct critical points have distinct critical values.
Thus excellence is stronger than the Morse condition: it excludes repeated critical values, not repeated indices.
Depends on
Used by
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
- Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)
- Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)