Download why vector bundles?

SOME ILLUSTRATIONS TO
VECTOR BUNDLES
September 14, 2009
O.Knill
n
VECTOR BUNDLE λ U: E R
-1
U
π (x)
μ(r,v)
EU
E
v
n
R
φU =(π,λU)
trivialization
0
π
^
ο
bundle
projection
zero section
U
M
m-dim
manifold
some notation from Cliff Taubes notes for vector bundles
Propo: The smooth m+n dimensional manifold E has
-1
a vector space structure on each fibre π (p)
Proof:
1) on each fiber, there is a vector space structure
inherited from diffeo λ . Could depend on U however.
U
2) the R action μ on each fibre is compatible with
n
^
the scaling multiplication on R . The zero is o(p).
n
3) linearity of any map ψ on R follows from
t ψ(v)= ψ(tv). (linear algebra lemma)
4) apply 3) to
-1
U
to see that the vector
ψ=λ
U’
space structure does not depend on the chart
of the atlas.
ολ
n
Lemma: A smooth map ψ on R satisfying
t ψ(v)= ψ(tv) for all t,v is linear.
Proof: differentiate
t ψ(v)= ψ(tv)
to see that
ψ(v) = ψ(tv) . v
*
is independent of t. Call A =ψ(0)
*
ψ(v) = Av
This is linear and A does not
depend on v
Remark: not true between general vector spaces V
dimensions V V
W or in infinite
COCYCLE CONSTRUCTION
U
g
U’
g
g
U’’
M
g
U’,U
o
U,U’
U’,U’’
U ,U’’
: U’
: U’’ U’
Gl(n,R)
Gl(n,R)
: U’’ U
Gl(n,R)
U
bundle transition functions
g
U,U’’
o
g
U’’,U
cocycle
constraint
= ι
Define the bundle as an equivalence relation on
n
the disjoint union of U x R
(p,v) ~ (p’,v’)
if p=p’ and v’=g
U’,U
(p) v
Propo: The cocycle definition leads to the same
vector bundle E as discussed before.
Proof: Assume the cocycle condition:
π( (p,v) ) = p
μ(r, (p,v) ) = (p,rv)
λ( (p,v) ) = v
EU
form an atlas of E
with coordinate functions
(φ
,λ )
: Ε
U U
U
and transition functions: g
m
R xR
(x)
U,U’
n
TANGENT BUNDLE
M
U
g
U’
V
U’,U
: U’
Gl(n,R)
U
defined by
φ
φU’
U
φ (V)
U
ψ
= φU’
U’,U
φ (V)
U’
g
g
o φ -1
U’, U
U’,U
o
=
ψ
g
o
U,U’’
U’,U’ *
g
U’’,U
= ι
cocycle constraint
by chain rule
U
ψ
o
U,U’
ψ
o
U’,U’’
ψ
U’’,U
= ι
VECTOR BUNDLE EXAMPLES
Trivial bundle
Moebius bundle
Tangent bundle
Cotangent bundle
Normal bundle
Tautological bundle
Matric cocycle bundle
A trivial bundle
of the torus
TAUTOLOGICAL BUNDLE
Tangent bundle
of torus
WHY VECTOR BUNDLES?
“Bring Vector space back in picture” (Cliff)
Construct manifolds from given manifolds
Understand manifolds (i.e. find invariants by
imposing more structure)
Important in physics: i.e. vector fields, dynamics,
geodesic flow, particle physics
FIBER BUNDLE EXAMPLES
Unit tangent bundle (needs a metric on each fibre)
Hopf fibration
Billiard phase space
Frame bundles
Cocycles
Klein bottle (circle bundle over circle)
Visualiation by Thomas Banchoff (Brown University), 1990. Povray adapted from Paul MyLander, 2008
EXPLANATIONS
Dimensions-math.org