Differentiable
Functions and Maps| Differentiable
Functions and Maps |
Function of a Real Variable > see analysis.
Function on a Manifold
$ Def: The function f : M → R is k-differentiable
at x 
M if,
for all (U, 
)
such that x U,
the function f 
–1
is k-differentiable
at (x).
Mapping between Manifolds > s.a. harmonic
map.
$ Def: The map f : M → N is k-differentiable
at x M if,
for all (U, )
such that x U and
all (V, 
)
such that f(x) V,
the function
f –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 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 *
)v =
(f' v)
f.
* Pullback of an r-form:
(f *)(v1, v2,
..., vr):=
(f*v1, f*v2,
..., f*vr)
f .
Transformations of a Manifold
$ Local pseudogroup:
Given a point x0 of a manifold X,
a set of transformations of a neighborhood of x0, 
t: N(x0)
→ X, t I
R,
with composition law t s
= t+s; Generator:
The equation v(x) = dt(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 → R is the 1-form
df T*M such
that 
df,
X
= 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*:
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 : X → Y such that y = f(x), f → f'.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
26 jun 2008