PlanetMath: Morse function

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Morse function

(Definition)

Let be a smooth manifold. A critical point of a map $u:M\to\mathbb{R}$ at $x\in M$ is called non-degenerate if the Hessian matrix (in any local coordinate system) at is non-degenerate.

A smooth function $u:M\to\mathbb{R}$ is called Morse if all its critical points are non-degenerate. Morse functions exist on any smooth manifold, and in fact form an open dense subset of smooth functions on (this fact is often phrased “a generic smooth function is Morse”).

"Morse function" is owned by bwebste.

(view preamble)

Also defines:

non-degenerate critical point

Log in to rate this entry.
(view current ratings)

Cross-references: subset, open, smooth function, non-degenerate, local coordinate, Hessian matrix, map, critical point, smooth manifold
There are 3 references to this entry.

This is version 5 of Morse function, born on 2003-08-21, modified 2005-02-10.
Object id is 4632, canonical name is MorseFunction.
Accessed 5237 times total.

Classification:

AMS MSC:

58E05 (Global analysis, analysis on manifolds :: Variational problems in infinite-dimensional spaces :: Abstract critical point theory )

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact