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Every compact smooth manifold admits an excellent Morse function
Statement
Every compact smooth manifold admits an excellent Morse function.
Facts & Assumptions
Given: A compact smooth manifold .
Every smooth manifold embeds in some finite-dimensional Euclidean space (Every smooth manifold embeds in some finite-dimensional Euclidean space).
On a compact embedded manifold, a generic linear height function is Morse (For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions).
On a compact manifold, excellent Morse functions are dense among all smooth functions (On a compact smooth manifold, the excellent Morse functions form an open dense subset of the and hence topology).
Proof
By [L1], choose a smooth embedding . Then [L2] gives a Morse height function on that embedding.
Apply [L3] to the Morse function . Since excellent Morse functions are dense on the compact manifold , some excellent Morse function lies arbitrarily close to and in particular exists on .
Therefore every compact smooth manifold admits an excellent Morse function.
Depends on
- Every smooth manifold embeds in some finite-dimensional Euclidean space
- For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions
- On a compact smooth manifold, the excellent Morse functions form an open dense subset of the $C^2$ and hence $C^\infty$ topology
Used by
Nothing in the library uses this result yet.
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Sources
- Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)
- Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)