SOME ILLUSTRATIONS TO VECTOR BUNDLES

September 14, 2009O.Knill

ο̂

E

M

π

zero section

bundleprojection

m-dimmanifold

Rn-1π (x)

U

EU

λU

μ(r,v)

v

some notation from Cliff Taubes notes for vector bundles

: E R U

n

φU

=(π,λ )Utrivialization

0

VECTOR BUNDLE

Propo: The smooth m+n dimensional manifold E has a vector space structure on each fibre π (p)-1

Proof: 1) on each fiber, there is a vector space structure inherited from diffeo λ

U

2) the R action μ on each fibre is compatible with 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)

. Could depend on U however.

n

^

U’ψ = λ λο

U-14) apply 3) to to see that the vector

space structure does not depend on the chart of the atlas.

Lemma: A smooth map ψ on R satisfying n

t ψ(v)= ψ(tv) is linear.

t ψ(v)= ψ(tv)Proof: differentiate

to see that

ψ(tv)*

is independent of t. Call A =

.ψ(v) v=

ψ(0)*

ψ(v) = AvThis is linear and A does not

depend on v

Remark: not true between general vector spaces V W or in infinite dimensions V V

for all t,v

COCYCLE CONSTRUCTION

U U’

MU’’

Gl(n,R)gU,U’

U:g

U’,U’’: U’’

U’ U’ Gl(n,R)

gU ,U’’

: U’’ U Gl(n,R)

gU’,U

gU,U’’

gU’’,U

o o = ι cocycle constraint

bundle transition functions

Define the bundle as an equivalence relation on the disjoint union of U x Rn

(p,v) ~ (p’,v’) if p=p’ and v’=g (p) vU’,U

Propo: The cocycle definition leads to the samevector bundle E as discussed before.

Proof: Assume the cocycle condition:

π( (p,v) ) = pμ(r, (p,v) ) = (p,rv)λ( (p,v) ) = v

EUform an atlas of E

with coordinate functions

φ U λ U( , ) Ε: U Rm x R n

and transition functions: g (x)U,U’

TANGENT BUNDLE

U U’Gl(n,R)g

U’,UU: U’

gU’,U

gU,U’’

gU’’,U

o o = ι

cocycle constraint by chain rule

M

φ φU U’

φ

V

U(V) φ

U’(V)

φU-1oφ

U’ψ =

U’,U

gU’, U

ψ U’,U’

=*

ψ U’’,U

ψ U’,U’’

ψ U,U’

o o = ι

defined by

VECTOR BUNDLE EXAMPLESTrivial bundle

Moebius bundle

Tangent bundle

Cotangent bundle

Normal bundle

Tautological bundle

Matric cocycle bundle

A trivial bundleof the torus

TAUTOLOGICAL BUNDLE

Tangent bundleof 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