Differentiable Functions and Maps

Function of a Real Variable > see analysis.

Function on a Manifold
$ Def: The function f : M → \(\mathbb R\) is k-differentiable at x ∈ M if, for for all (U, φ) such that x ∈ U, the function f \(\circ\) φ⁻¹ is k-differentiable at φ(x).

Mapping between Manifolds
$ Def: The map f : M → N is k-differentiable at x ∈ M if, for all (U, φ) such that x ∈ U and all (V, ψ) such that f(x) ∈ V, the function ψ \(\circ\) f \(\circ\) φ⁻¹ is k-differentiable at f(x).
* Pullback of a function: The contravariant functor *: (Man, Mor(Man)) → (Vec, Mor(Vec)), taking X → E:= {functions on X}, Y → F:= {functions on Y}, and, if f ∈ Hom(X, Y), f → f * ∈ Hom(F, E), defined by

f *g = g \(\circ\) f ,   for all g ∈ F .

* Pullback of a one-form: A contravariant functor from Man to Vec, as above, taking X → T*X, Y → T*Y, and, if f ∈ Hom(X, Y), f → f * Hom(T*Y, T*X), defined by (f *ω)v = ω(f' v) \(\circ\) f.
* Pullback of an r-form:

(f *ω)(v₁, v₂, ..., v_r):= ω(f_*v₁, f_*v₂, ..., f_*v_r) \(\circ\) f .

@ References: Golubitsky & Guillemin 73 [stable mappings and singularities]; > s.a. harmonic map; Singularities.

Transformations of a Manifold
$ Local pseudogroup: Given a point x₀ of a manifold X, a set of transformations of a neighborhood of x₀, σ_t: N(x₀) → X, t ∈ I ⊂ \(\mathbb R\), with composition law σ_t \(\circ\) σ_s = σ_t+s; Generator: The equation v(x) = dσ_t(x)/dt |_t=0 defines a unique vector field generating the 1-parameter local pseudogroup.

Differential of a Function
$ Def: The differential of a function f : M → \(\mathbb R\) is the 1-form df ∈ T*M such that \(\langle\)df, X\(\rangle\) = Xf, for any vector field X ∈ TM.

Differential of a Mapping
$ Def: Given a mapping f : X → Y between two differentiable manifolds, its differential at a point x ∈ X is the mapping f' or f_*: T_xX → T_f(x)Y given by (f' v)(h):= v(f*h), where h is a function on Y.
* Remark: One can push forward a vector, but not a vector field, usually.
* In category language: A covariant functor from pointed differentiable manifolds to vector spaces, that constructs everything for us; It associates (X, x) → T_xX, (Y, y) → T_yY, and, for f : X → Y such that y = f(x), f → f'.

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