Morse lemma

E911360

mathematical theorem result in differential topology

Morse lemma is a fundamental result in differential topology that locally characterizes a non-degenerate critical point of a smooth function as being equivalent, via a coordinate change, to a quadratic form.

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All labels observed (1)

Label	Occurrences
Morse lemma canonical	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in differential topology ⓘ
appliesTo	non-degenerate critical points of smooth functions ⓘ smooth real-valued functions on manifolds ⓘ
assumes	non-degeneracy of the Hessian at the critical point ⓘ smoothness of the function ⓘ
characterizes	local behavior of smooth functions near non-degenerate critical points ⓘ non-degenerate critical points up to smooth coordinate change ⓘ
concerns	local normal form of functions ⓘ non-degenerate critical points ⓘ quadratic forms ⓘ smooth functions ⓘ
ensures	existence of a diffeomorphism sending the function to a quadratic form near the critical point ⓘ
field	differential geometry ⓘ differential topology ⓘ
hasConsequence	local product structure near non-degenerate critical points ⓘ normal form for smooth functions near non-degenerate critical points ⓘ
hasVariant	Morse lemma for Banach spaces NERFINISHED ⓘ Morse lemma for complex analytic functions NERFINISHED ⓘ parametrized Morse lemma ⓘ
historicalContext	developed in the context of Morse theory in the early 20th century ⓘ
holdsIn	finite-dimensional smooth manifolds ⓘ
implies	existence of coordinates in which the Hessian is diagonal with entries ±1 ⓘ local classification of non-degenerate critical points by index ⓘ non-degenerate critical points are isolated ⓘ
involvesConcept	Taylor expansion of smooth functions ⓘ diffeomorphism of neighborhoods ⓘ index of a critical point ⓘ signature of the Hessian ⓘ
namedAfter	Marston Morse NERFINISHED ⓘ
relatedTo	Hessian matrix ⓘ Morse function ⓘ Morse index ⓘ implicit function theorem ⓘ non-degenerate quadratic form ⓘ stable manifold theorem ⓘ
states	a smooth function near a non-degenerate critical point is equivalent to a quadratic form in suitable local coordinates ⓘ there exist local coordinates in which the function has no terms of order higher than two near a non-degenerate critical point ⓘ
typicalForm	f(x)=f(p)-x_1^2-\cdots-x_\lambda^2+x_{\lambda+1}^2+\cdots+x_n^2 in suitable coordinates ⓘ
usedFor	reducing nonlinear problems to quadratic ones near non-degenerate equilibria ⓘ simplifying local computations near critical points ⓘ
usedIn	Morse theory NERFINISHED ⓘ calculus of variations ⓘ critical point theory ⓘ local analysis of gradient flows ⓘ proofs of handle decomposition theorems ⓘ singularity theory for non-degenerate singularities ⓘ study of topology of manifolds via smooth functions ⓘ

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Full triples — surface form annotated when it differs from this entity's canonical label.

Morse Theory → centralResult → Morse lemma ⓘ

subject surface form: Morse theory