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PMATH 955: Assignment 1

Due: Wednesday, 10 February, 2010

1. Tangent and normal bundles of submanifolds ofRn. Recall that a smoothk-dimensional submanifold ofRn is a subset M Rn with the followingproperty: given any point p M, there exists an open neighbourhood Wof p in Rn such that

M W = {(x1, , xn) | F(x1, . . . , xn) = 0},

where F : W Rn Rnk is a smooth map whose Jacobian matrixJac(F) := (Fi/xj) has rank (n k) at every point x M W.

(a) DefineTpM := ker(Jac(F)(p))

to be the tangent space of M at p M W, and let

T M :=

pM

TpM

be the tangent bundle of M. Show that T M is a smooth rank-kvector bundle on M.

Hint: Recall that any smooth k-dimensional submanifold of Rn islocally the graph of a smooth map f : Rk Rnk, (x1, . . . , xk) (f1(x1, . . . , xk), . . . , f nk(x1, . . . , xk)) and can therefore locally be ex-pressed as the zero set ofF : Rn Rnk, (x1, . . . , xk, y1, . . . , ynk) (y1 f1(x1, . . . , xk), . . . , ynk fnk(x1, . . . , xk)).

(b) DefineNpM := spanR(F1(p), . . . , Fnk(p))

to be the normal space of M at p M W, and let

N M :=

pM

NpM

be the normal bundle of M. Show that N M is a rank-(n k) vectorbundle on M.

2. Stereographic projection. Let Sn = {x21 + + x2n+1 = 1} be the unit

sphere in Rn+1. Let N = (0, . . . , 0, 1) be the north pole in Sn, andS = N be the south pole. Define the stereographic projection from thenorth pole : Sn\{N} Rn by

(x1, . . . , xn+1) =(x1, . . . , xn)

1 xn+1.

Let (x) = (x) for x Sn\{S}.

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(a) For any x Sn\{N}, show that (x) is the point where the linethrough N and x intersects the linear subspace where xn+1 = 0,

identified with Rn in the obvious way (see Figure 1.13, p.28, in Leesbook). Similarly, show that (x) is the point where the line throughS and x intersects the same subspace. (The map is therefore calledstereographic projection from the south pole.)

(b) Show that is bijective with inverse

1(u1, . . . , un) =(2u1, . . . , 2un, |u|

2 1)

|u|2 + 1,

where |u| is the distance from the origin to u = (u1, . . . , un).

(c) Compute the transition map 1 and verify that the atlas consist-ing of the two charts (Sn\{N}, ) and (Sn\{S}, ) defines a smoothn-manifold structure on Sn. (The coordinates defined by and are called stereographic coordinates.)

(d) Use the atlas obtained in (c) to find explicit local trivialisation of thetangent bundle T Sn over Sn\{N} and Sn\{S}. Also, compute thecorresponding transition function.

(e) Show that, for n odd, the tangent bundle T Sn has at least onenowhere vanishing section. In other words, there exists at least onetangent vector field with no zeroes on an odd-dimensional sphere.

3. Tautological line bundle onPn. Let p be any point in n-dimensional pro-jective space Pn, and denote by lp the line in Rn

+1 corresponding to p: ifp = [x1 : : xn+1], then lp is the line in Rn+1 passing through the originand (x1, , xn+1). We define

1n :=

pPn

{p} lp = {(p,v) Pn Rn+1 | v lp}.

(a) Prove that 1n is a line bundle over Pn, called the tautological line

bundle overPn. In particular, find explicit local trivialisations of 1nand compute the corresponding transition functions.

(b) Show that 1n has no nowhere vanishing continuous sections.

(c) Conclude that the tautological line bundle 1n is not isomorphic tothe trivial line bundle Pn R on Pn.

4. Mobius strip. In this exercise, we show that the Mobius strip M is a non-trivial fibre bundle over S1. Recall that the Mobius strip can be described

as a surface in R3

parametrised by the two patches

1(, t) = ((1 t sin(/2))cos , (1 t sin(/2))sin , t cos(/2)),

with (, t) R1 := (0, 2) (, ), and

2(, t) = ((1 t sin(/2))cos , (1 t sin(/2))sin , t cos(/2)),

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with (, t) R2 := (, ) (, ), for any fixed constant 0 < < 0.5.In other words, M = U1 U2, where U1 := 1(R1) and U2 := 2(R2).

Note that points on M of the form 1(, 0) and 2(, 0) correspond tounit circle S1 in the xy-plane, and consider the open cover {U1, U2} ofS

1,where U1 := S

1\{(1, 0, 0)} and U2 := S1\{(1, 0, 0)}.

(a) Prove that the Mobius strip is a smooth fibre bundle over S1 withfibre F = (, ). In particular, show that 1 and 2 induce naturallocal trivialisations 1 and 2 of M over U1 and U2, respectively,whose transition function is

g12 = 1 12 : (U1 U2) R (U1 U2) R

(p,v) (p,(p)v),

where : U1 U2 GL(1,R) is defined by

(p) =

1, if the y-coordinate of p is > 0,1, if the y-coordinate of p is < 0.

(b) The infinite Mobius strip or Mobius bundle is defined as the quo-tient space

M := ([0, 1] R) / ,

where the equivalence relation is (0, t) (1, t) for all t R. Showthat

p : M S1

[(x, t)] (cos(2x), sin(2x), 0)

is a smooth line bundle over S1. Prove, in particular, that M admitsthe following local trivialisations

1 : p1(U1) U1 R

[(x, t)] ((cos(2x), sin(2x), 0), t)

and

2 : p1(U2) U2 R

[(x, t)]

((cos(2x), sin(2x), 0), t), if 0 x < 1/2,((cos(2x), sin(2x), 0), t), if 1/2 < x 1,

with transition function 1 12 (p,t) = (p,(p)t) on (U1 U2) R,

where is the map defined in (a).

(c) Prove that the Mobius strip and the Mobius bundle are diffeomorphic

as fibre bundles.(Note: Two smooth fibre bundles p1 : E1 B1 and p2 : E2 B2are said to be diffeomorphic if there exist smooth maps g : B1 B2

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and G : E1 E2 such that p2 G = g p1, i.e., the following diagramcommutes

E1 G E2

p1

p2B1

gB2,

and such that G maps any fibre p11 (b1) of p1 diffeomorphically ontothe fibre p12 (g(b1)) of p2.)

(d) Prove that any continuous section of the Mobius bundle has at leastone zero.

(e) Show that the Mobius bundle is notisomorphic to the trivial line bun-dle S1 R on S1; conclude that the Mobius strip is not diffeomorphicto the trivial fibre bundle S1 (, ).

5. Bundle isomorphisms. Let : E M and : E M be two smoothrank-k vector bundles over a smooth manifold M. Suppose that {U}Ais an open cover of M such that both E and E admit local trivialisationsover each U. Let {} and {

} be the transition matrices determinedby the local trivialisations of E and E, respectively. Show that E and E

are smoothly isomorphic (over M) if and only if for each there exists asmooth map : U GL(k,R) such that

(p) = (p)1(p)(p), p U U .

6. (a) Show that the real projective line P1 is diffeomorphic to S1.

(Hint: Use the standard atlas on P1, whose open sets are V1 = {x1 =0} and V2 = {x2 = 0}, and the atlas you found in question 1.(d) onS1 to construct an explicit diffeomorphism from P1 to S1.)

(b) Show that the tautological line bundle 11 on P1 is isomorphic, as a

vector bundle, to the Mobius bundle.

7. Operations on vector bundles.

Let : E M and : E M be two vector bundles of rank r and r,respectively, over the manifold M. Consider local trivialisations (U, a)and (U,

) for Eand E, respectively, and their corresponding transition

functions and , respectively, on overlaps U U .

(a) Let E E :=pM

Ep E

p, where Ep E

p is the direct sum of

the vector spaces Ep and E

p. Show that E E

is a rank-(r + r

)vector bundle on M, called the direct sum of E and E, with localtrivialisations and transition functions

00

.

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(b) Let E =pM(Ep)

, where (Ep) is the dual of the vector spacesEp. Show that E

is a rank-r vector bundle on M, called the dual

bundle of E. In particular, show that if

: EU U Rr

is a local trivialisation of E, then

(t)1 : EU U (R

r)

is a local trivialisation for E; moreover, the corresponding transitionfunctions of E are (t)

1.

Note: ()t denotes matrix transposition.

(c) Tensor products. Let E E :=pMEp E

p, where Ep E

p is thetensor product of the vector spaces Ep and Ep over R. Show thatE E is a rank-rr vector bundle on M, called the tensor product ofE and E, with local trivialisations and transition functions

: U U GL(rr,R).

(d) Pullback bundle. Let f : N M be a smooth map between a smoothmanifold N and M. We define the pullback bundle fE of the vectorbundle : E M by setting

(fE)p := Ef(p)

and

f

E :=

pM(f

E)p = {(p,v) N Rr

| v Ef(p)}.

Show that fE is a rank-r vector bundle on M. In particular, verifythat the local trivialisations {} of E induce local trivialisations

: fEf1U U R

r

(p,v) (f(p), v)

of fE, and compute their transition functions.

8. Cartesian products. Let M and M be smooth manifolds.

(a) Show that the Cartesian product M M is a smooth manifold.

(Hint: Recall that M M is endowed with the product topology inwhich open sets of the form UU, with U M and U M open,forms a basis for the topology. Show that given atlases (U, ) and(U, ) of M and M

, respectively, (U U, ) is an atlasfor M M.)

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(b) The Cartesian productof vector bundles : E M and : E M

is defined as

E E :=

(p,p)MM

Ep Ep .

Show that E E is a vector bundle over M M.

(c) Show that the tangent bundle T(M M) of M M is isomorphicto the Cartesian product T M T M.

(d) Recall that a smooth manifold is called parallelizable if its tangentbundle is trivial. Prove that if M and M are parallelizable, then sois M M.

(e) Let T = S1 S1 be the n-torus, consisting of the Cartesianproduct of n copies of the circle S1. Show that T is parallelizable.

9. Constructions of vector bundles(a) Let : E M be a smooth vector bundle of rank-k over a smooth

manifold M. Suppose {U}A is an open cover of M, and for each A we are given smooth local trivialisations :

1(U) U R

k of E with corresponding transition matrices : U U GL(k,R). Show that the following identity is satisfied for all, , A:

(p) (p) = (p), p U U U. ()

This identity is known as the cocycle rule.

(b) Let M be a smooth manifold and let {U}A be an open cover ofM.Suppose for each , A we are given a smooth map : U GL(k,R) such that the cocycle rule () is satisfied for all , , A.Show that there is a smooth rank-k vector bundle E M withsmooth local trivialisations :

1(U) URk whose transition

matrices are the given maps .

Hint: Define an appropriate equivalence relation onA(U R

k),and use the bundle construction lemma.

Bonus The Hairy Ball Theorem. Coming soon....

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