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 xM if, for all (U, φ) such that xU, the function f \(\circ\) φ–1 is k-differentiable at φ(x).

Mapping between Manifolds
$ Def: The map f : MN is k-differentiable at xM if, for all (U, φ) such that xU 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 XE:= {functions on X}, YF:= {functions on Y}, and, if f ∈ Hom(X, Y), ff * ∈ Hom(F, E), defined by

f *g = g \(\circ\) f ,   for all gF .

* 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), ff * Hom(T*Y, T*X), defined by (f *ω)v = ω(f' v) \(\circ\) f.
* Pullback of an r-form:

(f *ω)(v1, v2, ..., vr):= ω(f*v1, f*v2, ..., f*vr) \(\circ\) 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, σt: N(x0) → X, t ∈ I ⊂ \(\mathbb R\), with composition law σt \(\circ\) σs = σt+s; Generator: The equation v(x) = dσt(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 : XY between two differentiable manifolds, its differential at a point xX is the mapping f' or f*: TxX → Tf(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) → TxX, (Y, y) → TyY, and, for f : XY such that y = f(x), ff'.


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