The critical set of a real function on a manifold #
The critical set TauCeti.criticalPoints I f of a function f : M β β is the zero locus of its
vector-valued manifold derivative mvfderiv I f, equivalently of mfderiv I π(β) f. On a
boundaryless manifold it is also the set of points x at which the coordinate expression of f
in the preferred extended chart at x has zero FrΓ©chet derivative. That is the criticality test
built into TauCeti.IsMorseOn, so a Morse function is exactly a smooth function all of whose
critical points are nondegenerate.
On a boundaryless manifold local extrema are critical points, so a continuous function on a
nonempty compact boundaryless manifold has a critical point, and so does each nonempty compact
sublevel of a continuous function. On a compact boundaryless manifold with at least two points it
has at least two critical points, a maximum and a minimum.
The critical set of a CΒΉ function is closed, and a nondegenerate critical point is isolated in
it, so a Morse function has finitely many critical points in every compact set; for a proper
function bounded below, this applies to every sublevel set. A compact band between regular
values therefore holds finitely many critical levels, along which properties of the band can be
proved one level at a time. Above a regular value, the critical points in a compact band stay a
positive distance away in value, so the band begins with a collar free of critical points.
On a compact manifold the values of a Morse function can be cut at regular values into bands
containing at most one critical value each: by finitely many monotone regular values, and by a
strictly increasing β€-indexed sequence of regular values, starting below and ending above all
values of f, into critical bands.
Restricting f to an open subset U keeps exactly the critical points lying in U, so the total
critical count is additive over a finite partition of M into open sets, such as its connected
components. In dimension zero every point is critical.
Critical points are counted, in total and by Morse index, with values in ββ: an infinite
critical set has count β€, never a junk natural number. The counts of a Morse function on a
compact manifold are finite, the counts by index add up to the total count, and replacing f by
-f exchanges the indices j and dim M - j. These counts are the coefficients of the Morse
polynomial in the Morse inequalities.
Main declarations #
TauCeti.criticalPoints: the critical set, withTauCeti.mem_criticalPoints_iff_mfderiv_eq_zero, on boundaryless manifoldsTauCeti.mem_criticalPoints_iff_fderiv_comp_extChartAt_symm_eq_zero, and on a normed spaceTauCeti.mem_criticalPoints_iff_fderiv_eq_zero.TauCeti.isMorseOn_iff_criticalPoints: Morse functions in terms of the critical set.TauCeti.criticalPoints_comp_subtypeValandTauCeti.totalCriticalCount_comp_subtypeVal: the critical points of the restriction offto an open submanifoldUare those offinU.IsLocalExtr.mem_criticalPoints,TauCeti.criticalPoints_nonemptyandTauCeti.exists_mem_criticalPoints_of_isCompact_sublevel: Fermat's theorem on a manifold and the existence of critical points on compact manifolds and in compact sublevels.TauCeti.isClosed_criticalPointsandTauCeti.IsManifoldNondegenerateCriticalPoint.eventually_notMem_criticalPoints: the critical set of aCΒΉfunction is closed, and nondegenerate critical points are isolated in it.TauCeti.exists_mem_Ioc_forall_notMem_criticalPointsandTauCeti.exists_mem_Ioc_isCompact_forall_mfderiv_ne_zero: a compact band starting at a regular value begins with a compact collar free of critical points.TauCeti.IsMorse.eventually_eq_of_mem_criticalPoints: the critical points of a Morse function are isolated.TauCeti.finite_criticalPoints_inter_of_forall_isManifoldNondegenerateCriticalPoint: aCΒΉfunction has finitely many critical points in a compact set where they are all nondegenerate.TauCeti.IsMorse.finite_criticalPoints_interandTauCeti.IsMorse.finite_criticalPoints_inter_preimage_Iic: finiteness on compact sets and on sublevel sets of proper functions bounded below.TauCeti.IsMorse.exists_criticalPoints_inter_preimage_Icc_subset: in a compact band aroundc, a Morse function has no critical values close tocother thancitself.TauCeti.IsMorse.exists_separating_levels: on a compact manifold, levels just belowf pand just abovef rseparate the critical values strictly between them.TauCeti.IsCriticalBandandTauCeti.IsCriticalBandCut: compact bands with at most one critical value, strictly inside, and sequences of values cuttingfinto such bands, which are strictly increasing (TauCeti.IsCriticalBandCut.strictMono), withTauCeti.exists_isCriticalBandCut_of_finite_image_criticalPointsandTauCeti.IsMorse.exists_isCriticalBandCut: on a compact manifold a critical-band cut exists for a function with finitely many critical values, such as a Morse function, starting below and ending above all values off.TauCeti.IsMorse.band_induction: a property of compact bands between regular values follows from its validity across bands without critical points, across single critical levels, and along triples of levels.TauCeti.IsMorse.exists_regular_sublevel_partition: on a compact manifold, regular values cutting the values of a Morse function into bands with at most one critical value each.TauCeti.criticalCountandTauCeti.totalCriticalCount: the extended counts of critical points, withTauCeti.sum_criticalCount_eq_totalCriticalCount,TauCeti.IsMorse.totalCriticalCount_ne_top,TauCeti.two_le_totalCriticalCountandTauCeti.IsMorse.criticalCount_neg.TauCeti.criticalCount_zero_ne_zeroandTauCeti.IsMorse.criticalCount_finrank_ne_zero: on a nonempty compact manifold the minimum is a critical point of index0and, for a Morse function, the maximum is one of index the dimension.TauCeti.exists_isMinOn_isMaxOn_of_totalCriticalCount_eq_two, withIsMinOn.eq_of_le_of_two_criticalPointsandIsMaxOn.eq_of_le_of_two_criticalPoints: a function with exactly two critical points is constant or has them as its unique minimum and maximum.TauCeti.totalCriticalCount_eq_finsum_connectedComponents: the total count is the sum of the counts in the connected components, when there are finitely many.TauCeti.totalCriticalCount_eq_sumandTauCeti.totalCriticalCount_eq_sum_connectedComponents: additivity of the total count over finite open partitions and over connected components.TauCeti.sum_range_neg_one_pow_mul_toNat_criticalCount_eq_sum,TauCeti.toNat_totalCriticalCount_modEq_sum_range_neg_one_pow_mul_toNat_criticalCountandTauCeti.IsMorse.sum_range_neg_one_pow_mul_toNat_criticalCount_neg: the signed countβα΅’ (-1)β± cα΅’is the sum of(-1)^Ξ»over the critical points, it has the parity of the number of critical points, and the signed counts offand-fdiffer by the factor(-1)α΅.
References #
- J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Β§2.
The critical set of a real-valued function on a manifold: the points where its manifold
derivative vanishes. No regularity is assumed; at a point where f is not differentiable the
manifold derivative is zero by convention, so such a point is critical.
Instances For
A point is critical when the vector-valued manifold derivative vanishes there.
A point is critical exactly when the manifold derivative mfderiv I π(β) f vanishes there.
On a normed space with its trivial model, a point is critical exactly when the FrΓ©chet derivative vanishes there.
The critical set is the zero locus of the manifold derivative mfderiv I π(β) f.
A point is regular for a real-valued function exactly when it is not critical.
A scalar value is regular exactly when it is not the value of a critical point.
On a manifold modelled on a zero-dimensional space every point is critical.
A function and its negation have the same critical points.
A value c is regular for -f exactly when -c is regular for f.
No critical point has a regular value b, so the critical points with values in [a, b) are
those with values in [a, b].
No critical point has a regular value a, so the critical points with values in (a, b] are
those with values in [a, b].
Composition with a local diffeomorphism #
Criticality under a local diffeomorphism. If Ο is a C^n local diffeomorphism at x
with 1 β€ n, then x is a critical point of f β Ο exactly when Ο x is a critical point of
f.
Restriction to an open submanifold #
A point of an open submanifold U is a critical point of the restriction of f to U exactly
when it is a critical point of f.
The critical points of the restriction of f to an open submanifold U are the critical
points of f lying in U.
Critical points read in the preferred chart #
A nondegenerate critical point is isolated among the critical points of f.
Where the coordinate expression of f in the preferred extended chart at x is differentiable,
x is a critical point exactly when that expression has zero FrΓ©chet derivative. No boundary
hypothesis is needed: range I is uniquely differentiable, so the derivative within it is the
FrΓ©chet derivative.
At a critical point, the coordinate expression of f in the preferred extended chart has zero
FrΓ©chet derivative. No boundary hypothesis is needed: where the expression is not differentiable
the FrΓ©chet derivative is zero by convention.
A nondegenerate critical point is a critical point.
Every critical point in s of a function which is Morse on s is nondegenerate.
Every critical point of a Morse function is nondegenerate.
On a boundaryless manifold, x is a critical point of f exactly when the coordinate
expression of f in the preferred extended chart at x has zero FrΓ©chet derivative there. No
regularity of f is needed: where f is not differentiable, neither derivative is.
A function is Morse on s exactly when it is smooth on s and each of its critical points in
s is nondegenerate.
Fermat's theorem on a manifold. A local extremum is a critical point.
A local minimum is a critical point.
A local maximum is a critical point.
If the only critical points of f are a global minimum xβ and a point xβ with a larger
value, then xβ is the only point at which f attains its minimum: any other such point would be
a third critical point.
If the only critical points of f are a global maximum xβ and a point xβ with a smaller
value, then xβ is the only point at which f attains its maximum: any other such point would be
a third critical point.
A continuous function on a nonempty compact boundaryless manifold has a critical point, for instance a point where it attains its minimum.
A function on a boundaryless manifold has a critical point in each nonempty compact sublevel
{f β€ t} on which it is continuous, for instance a point where it attains its minimum.
Closedness, isolation and finiteness #
The critical set of a CΒΉ function is closed.
A regular value has a collar without critical points above it. If f is CΒΉ, a < b,
the band fβ»ΒΉ[a, b] is compact and a is a regular value of f, then for some a' β (a, b] the
band fβ»ΒΉ[a, a'] contains no critical point of f.
A regular value has a compact collar without critical points above it. If f is CΒΉ,
a < b, the band fβ»ΒΉ[a, b] is compact and a is a regular value of f, then for some
a' β (a, b] the band fβ»ΒΉ[a, a'] is compact and f has nonzero derivative on it. This is the
form of TauCeti.exists_mem_Ioc_forall_notMem_criticalPoints consumed by the gradient-like flow.
Nondegenerate critical points in a compact set are finite in number. If every critical
point of a CΒΉ function f in a compact set K is nondegenerate, then K contains only finitely
many critical points of f. Nothing is assumed about the critical points outside K.
A Morse function has finitely many critical points in each compact set.
The critical points of a Morse function are isolated. Near any point x there is no
critical point of a Morse function other than x itself.
A Morse function on a compact manifold has finitely many critical points.
Levels separating the critical values. For a Morse function on a compact manifold and
a < b, there are levels a < cβ < cβ < b such that every critical value strictly between a
and b lies strictly between cβ and cβ.
Separating levels exist. For a Morse function on a compact manifold and points p and
r, there are levels cβ above f r and cβ below f p such that every critical value
strictly between f r and f p lies strictly between cβ and cβ, and no critical point lies on
the level cβ.
A proper Morse function bounded below has finitely many critical points in each sublevel set.
Critical values are isolated in a compact band. If the band fβ»ΒΉ[c - Ξ΄, c + Ξ΄] of a Morse
function is compact, then for some 0 < Ξ΅ β€ Ξ΄ the only critical points in the band
fβ»ΒΉ[c - Ξ΅, c + Ξ΅] are those on the level fβ»ΒΉ(c).
Induction over the critical levels of a compact band. Let f be a Morse function and
a β€ b regular values of f with compact band fβ»ΒΉ[a, b]. A property P s t of intervals holds
for [a, b] if it holds across every compact band fβ»ΒΉ[s, t] without critical points, across
some [c - Ξ΅, c + Ξ΅] inside any compact band fβ»ΒΉ[c - Ξ΄, c + Ξ΄] around a critical value
c β (a, b), and passes from [s, m] and [m, t] to [s, t] when fβ»ΒΉ[s, t] is compact.
The proof cuts the band just above its lowest critical value and inducts on the number of critical points in it. This is the induction along which Milnor's cell attachments across single critical levels assemble into statements about a whole band.
A regular partition of the values of a Morse function. For a Morse function f on a
compact boundaryless manifold, there are regular values s 0 β€ s 1 β€ β― β€ s (N + 1) with
s 0 < f β€ s (N + 1) on M, such that each band fβ»ΒΉ[s j, s (j + 1)] contains at most one
critical value.
Critical-band cuts #
The band fβ»ΒΉ[s, t] is a critical band of f when it is compact and contains at most one
critical value of f, which lies strictly between s and t (so in particular s < t).
The band is compact.
- exists_mem_Ioo_forall_eq : β c β Set.Ioo s t, β x β criticalPoints I f, f x β Set.Icc s t β f x = c
The band contains at most one critical value, lying strictly between
sandt.
Instances For
A critical-band cut of f is a sequence of values a : β€ β β such that each band
fβ»ΒΉ[a q, a (q + 1)] is a critical band of f: it is compact and contains at most one critical
value, lying strictly between a q and a (q + 1). Such a sequence is strictly increasing
(TauCeti.IsCriticalBandCut.strictMono), and the sublevels q β¦ {f β€ a q} form the
critical-band filtration of M.
- isCriticalBand (q : β€) : IsCriticalBand I f (a q) (a (q + 1))
Each band between consecutive cut values is a critical band.
Instances For
The ends of a critical band are in increasing order, since the band has a value strictly between them.
The values of a critical-band cut are strictly increasing.
The values of a critical-band cut are monotone.
Critical-band cuts exist on compact manifolds with finitely many critical values. A
continuous function f on a compact manifold with finitely many critical values has a
critical-band cut a : β€ β β which starts below and ends above all values of f, so that the
filtration q β¦ {f β€ a q} is finite.
Critical-band cuts exist for Morse functions on compact manifolds. A Morse function f on
a compact manifold has a critical-band cut a : β€ β β which starts below and ends above all values
of f, so that the filtration q β¦ {f β€ a q} is finite.
Counting critical points #
The number of critical points of f, in ββ; it is β€ when there are infinitely many.
Instances For
The number of critical points of f of Morse index j, in ββ; it is β€ when there are
infinitely many.
Instances For
The total critical count is the extended cardinality of the critical set.
The critical count in index j is the extended cardinality of the critical points of index
j.
The number of critical points of a given index is at most the number of critical points.
A function and its negation have the same number of critical points.
The number of critical points of the restriction of f to an open submanifold U is the
number of critical points of f in U.
The critical count is additive over components. If M has finitely many connected
components, the number of critical points of f is the sum over the components of the numbers of
critical points of f in each.
Additivity of the critical count. If finitely many pairwise disjoint open subsets U i
cover M, for instance the connected components of a compact manifold, then the number of
critical points of f is the sum of the numbers of critical points of its restrictions to the
U i.
The component sum formula for critical points. If M is locally connected with finitely
many connected components, for instance compact and boundaryless, then the number of critical
points of f is the sum over the components C of the numbers of critical points of the
restrictions of f to C, each component carrying its structure as an open submanifold.
A continuous function on a compact boundaryless manifold with at least two points has at least two critical points: a point of maximum and a point of minimum, which can coincide only for a constant function, all of whose points are critical.
A continuous function on a compact boundaryless manifold with exactly two critical points
attains its minimum at one of them, xβ, and its maximum at the other, xβ. Either
f xβ < f xβ, or f is constant, every point is critical, and M consists of xβ and xβ.
There are no critical points of index larger than the dimension.
The numbers of critical points of the indices 0, β¦, dim M add up to the number of critical
points.
The signed count βα΅’ (-1)β± cα΅’ of the critical points of f, by index, is the sum of (-1)^Ξ»
over the critical points, Ξ» being the Morse index, when the critical set is a finite set s.
Here n is any bound for the dimension of M.
Terms of index above the dimension of M do not contribute to a signed count
βα΅’ (-1)β± cα΅’ of critical points, so it may be truncated at dim M.
Parity of the critical count. If f has finitely many critical points, their number is
congruent modulo 2 to the signed count βα΅’ (-1)β± cα΅’, for any bound n of the dimension of
M.
Duality of critical counts. Replacing a Morse function by its negation exchanges the
numbers of critical points of index j and dim M - j.
Signed counts of f and -f. For a Morse function f on a manifold of dimension m, the
signed counts βα΅’ (-1)β± cα΅’ of the critical points of -f and of f differ by the factor (-1)α΅,
since the critical points of index i of -f are those of index m - i of f.
A Morse function on a compact manifold has finitely many critical points.
A Morse function on a compact manifold has finitely many critical points of each index.
Critical points of extreme index #
A CΒ² function on a nonempty compact boundaryless manifold has a critical point of index 0,
at a point of minimum.
A Morse function on a nonempty compact boundaryless manifold has a critical point of index the dimension of the manifold, at a point of maximum.