The rank of a map between manifolds #
The rank of a map f : M → N at a point x is the dimension (Module.finrank) of the image
of its manifold derivative df_x : T_x M → T_{f x} N (TauCeti.mfderivRank I J f x). At a point
where f is not differentiable the derivative is zero and so is the rank. As for any finrank, an
infinite-dimensional image also gives rank zero by convention; the image is finite-dimensional
whenever the source or the target is.
Into a manifold modelled on a finite-dimensional space E', the rank is at most dim E', and a
point is a regular point of f exactly when the rank there is maximal, equal to dim E'
(TauCeti.isRegularPoint_iff_mfderivRank_eq). The critical points are therefore the points of rank
loss (TauCeti.not_isRegularPoint_iff_mfderivRank_lt). On a manifold modelled on a
finite-dimensional space E, the rank and the dimension of the kernel of df_x add up to dim E
(TauCeti.finrank_ker_mfderiv_add_mfderivRank).
The rank does not depend on coordinates. It is computed in any pair of charts by the derivative of
the coordinate expression of f (TauCeti.mfderivRank_eq_finrank_range_fderivWithin), and it is
unchanged by composition on the left with maps of injective derivative and on the right with maps
of surjective derivative (TauCeti.mfderivRank_comp_of_injective and
TauCeti.mfderivRank_comp_of_surjective). Composition on either side with a diffeomorphism
preserves it without any differentiability assumption on f
(Diffeomorph.mfderivRank_diffeomorph_comp and Diffeomorph.mfderivRank_comp_diffeomorph).
For a C¹ map into a manifold modelled on a finite-dimensional space, the rank is lower
semicontinuous: it can only go up near a point
(TauCeti.eventually_mfderivRank_le_mfderivRank, TauCeti.lowerSemicontinuous_mfderivRank). So
each set {x | r ≤ rank_x f} is open (TauCeti.isOpen_setOf_le_mfderivRank); for r = dim E'
this is the openness of the set of regular points.
The rank stratifies the critical points of a smooth map, such as the singular points of a completely integrable system, where the rank of the family of integrals drops.
Main definitions #
TauCeti.mfderivRank I J f x: the rank offatx.
Main results #
TauCeti.isRegularPoint_iff_mfderivRank_eq: regular points are the points of maximal rank.TauCeti.mfderivRank_eq_finrank_range_fderivWithin: the rank read in arbitrary charts.TauCeti.mfderivRank_comp_of_injective,TauCeti.mfderivRank_comp_of_surjective,Diffeomorph.mfderivRank_diffeomorph_comp,Diffeomorph.mfderivRank_comp_diffeomorph: invariance of the rank under changes of coordinates.TauCeti.lowerSemicontinuous_mfderivRank,TauCeti.isOpen_setOf_le_mfderivRank: the rank of aC¹map is lower semicontinuous.
References #
- J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2013, Chapter 4 (the rank of a smooth map) and Exercise 4-1 (lower semicontinuity).
The rank of a map f at x: the dimension (Module.finrank) of the image of the
manifold derivative df_x : T_x M → T_{f x} N. It is zero at points where f is not
differentiable, and, by the convention of Module.finrank, also when that image is
infinite-dimensional.
Instances For
The rank of a map at a point where it is not differentiable is zero.
The rank of a map is at most the dimension of its source.
The rank of a map is at most the dimension of its target.
Rank–nullity for manifold derivatives. The dimension of the kernel of df_x and the rank
of f at x add up to the dimension of the source.
A map has rank zero at x exactly when its derivative there vanishes. The image of the
derivative only needs to be finite-dimensional, as it is when the source or the target is.
Regular points are the points of maximal rank. Into a manifold modelled on a
finite-dimensional space E', a point is a regular point of f exactly when the rank of f there
is dim E'.
Critical points are the points of rank loss. Into a manifold modelled on a
finite-dimensional space E', a point is not a regular point of f exactly when the rank of f
there is less than dim E'.
Invariance under changes of coordinates #
Composition on the left with an immersion preserves the rank. If g has injective
derivative at f x, then g ∘ f and f have the same rank at x.
Composition on the right with a submersion preserves the rank. If h has surjective
derivative at p, then f ∘ h has the same rank at p as f at h p.
Diffeomorphisms of the target preserve the rank. For a diffeomorphism h, the maps
h ∘ f and f have the same rank at every point. No differentiability of f is assumed.
Diffeomorphisms of the source preserve the rank. For a diffeomorphism h, the map f ∘ h
has the same rank at p as f at h p. No differentiability of f is assumed.
The rank read in charts. If f is differentiable at x, its rank there is the rank of the
derivative of its coordinate expression in any charts around x and f x.
Lower semicontinuity #
The rank can only go up nearby. At every point near a point x where f is C¹, the rank
of f is at least its rank at x.
The rank of a C¹ map is lower semicontinuous.
The points of rank at least r of a C¹ map form an open set.