Midterm Exam Problem 1. Let M be a smooth . Suppose V and W are smooth n i ∂ nector fields on M and ω is a smooth one-form on M. Let V = Pi=1 V ∂xi and n i ∂ n i W = Pi=1 W ∂xi , and ω = Pi=1 ωidx be the coordinate expressions for V , W and ω in terms of some smooth local coordinates (xi) for M. For a smooth f ∈ C∞(M), we define

(*) LV f = Vf.

(i) Show that LV W =[V,W] has the coordinate expression

n ∂Wj ∂V j ∂ L W = X (V i − W i ) . V ∂xi ∂xi ∂xj i,j=1

(ii) The one-form LV ω is defined by the product rule

LV (ω(W )) = (LV ω)(W )+ω(LV W ), for all smooth vector fields W,

in which ω(W ) is simply a smooth function and LV (ω(W )) is defined by (*). Deduce that LV ω has the coordinate expression

n ∂ω ∂V i L ω = X (V i j + ω )dxj . V ∂xi i ∂xj i,j=1

k (iii) Let D be a connection on M with the Christoffel symbol Γij . Define the 1-form DV W by the product rule

DV (ω(W )) = (DV ω)(W )+ω(DV W ), for all smooth vector fields W.

Show that the coordinate expression for DV W is

n i i j k DV ω = X(V ∂iωk − V ωjΓik)dx . j=1 .

Problem 2. Let M be a smooth manifold.

(1) Show that the L :Γ(TM) × Γ(TM) → Γ(TM), with (X, Y ) 7→ LX Y (the Lie ), is not a connection. (2) Show that there is a vector field on R2 that vanishes along the x1-axis, but whose ∂ 1 with respect ∂x1 does not vanish on the x -axis.

Problem 3. Prove that the of any abelian Lie is abelian.

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Problem 4. Notice the following

Definition. A vector field W is said to be invariant under a flow θ if (θt)∗Wp =

Wθt(p) for all (t, p) in the domain of θ. Let V and W be smooth vector fields on M, with flows θ and ψ, respectively. Show that the following are equivalent: (1) [V,W]=0. (2) LV W =0. (3) LW V =0. (4) W is invariant under the flow of V . (5) V is invariant under the flow of W . (6) θt ◦ ψs = ψs ◦ θt whenever either side is defined. ⇒ (Hint: Show (2) (4) by considering X(t)=(θ−t)∗(Wθt(p)) and deriving that 0 d X (t0)=(θ−t )∗ (θ−s)∗Wθ (θ (p)) =(θ−t )∗(LV W ), making the change of 0 ds s t0 0 s=0 variables t = t0 + s.)

Problem 5. Let V1, ··· ,Vk be smooth independent vector fields on an open O of a smooth manifold M. Choose a smooth chart (U, (xi)) centered at p. By rearranging the coordinates if necessary, we may assume that the vectors

∂ ∂ (V1 , ··· ,Vk , , ) p p ∂xk+1 ∂xn p p

span TpM. Let θi denote the flow of Vi for i =1, ··· ,k. Then there exist ε>0 and

a neighborhood W of p such that the composition (θk)tk ◦ (θk−1)tk−1 ◦···◦(θ1)t1 is defined on W and maps W into U whenever |t1|, ···|tk| <ε. Let

S = {(uk+1, ··· ,un):(0, ··· , 0,uk+1, ··· ,un) ∈ W },

and define ψ :(−ε, ε)k × S → U by

1 k k+1 n k+1 n ψ(u , ··· ,u ,u , ··· ,u )=(θk)uk ◦···◦(θ1)u1 (0, ··· , 0,u , ··· ,u ).

≤ ≤ ∂ ··· (a) Suppose that [Vi,Vj ]=0for1 i, j k. Show that ψ∗ ∂ui = Vi, for i =1, ,k. (Hint: Use (6) in Problem 4). ∂ ∂ (b) Observe that ψ∗ i = i for i = k +1, ··· ,n. ∂u 0 ∂x p Use this, (1) and the theorem to show that ψ is a diffeomorphism in a neighborhood of 0 if [Vi,Vj ] = 0 for 1 ≤ i, j ≤ k. (c) Show that the following are equivalent:

(1) [Vi,Vj ] ≡ 0, ∀1 ≤ i, j ≤ k,inO. (2) There exist smooth coordinates (u1, ··· ,un) in a neighborhood of each point O ≡ ∂ ··· of such that Vi ∂ui , i =1, ,k. 3

Problem 6. Let D0 and D1 be two connections on a smooth manifold M. 2 (a) Show that the difference between them defines a 1-tensor field A by 1 0 A(X, Y )=DX Y − DX Y, for X, Y ∈ Γ(TM), called the difference tensor. Thus, if D0 is any linear connection on M, the of all connections is precisely 2 {D0 + A : A is a smooth  -tensor}. 1 (b) Show that D0 and D1 determine the same geodesics iff their difference tensor is antisymmetric; i.e. A(X, Y )=−A(Y,X), for X, Y ∈ Γ(TM). (c) Show that D0 and D1 have the same tensor iff their difference tensor are symmetric; i.e. A(X, Y )=A(Y,X), for X, Y ∈ Γ(TM).

Problem 7. For any fixed R>0, the following Riemannian are all mutually isometric. n (1) Hyperbolic Model HR is the “upper sheet” ({τ>0}) of the two-sheeted hyperboloid in Rn+1 defined in coordinates (ξ1, ··· ,ξn,τ) by the equation 2 2 2 1 ∗ n+1 n+1 τ −|ξ| = R , with the metric hR = ı m, where ı : HR ,→ R is inclusion, and m is the Minkowski metric on Rn+1. n n (2) Poincar´eModel BR is the of radius R in R , with the metric given in coordinates (u1, ··· ,un)by (du1)2 + ···+(dun)2 h2 =4R4 . R (R2 −|u|2)2 n n (3) Poincar e Half-space Model UR is the upper half-space in R defined in coordinates (x1, ··· ,xn,y)by{y>0}, with the metric (dx1)2 + ···+(dxn−1)2 + dy2 h3 = R2 . R y2 To construct an between the two metrics given in (2) and (3), we first construct an explicit diffeomorphism

n n κ : BR → UR. Writing the coordinates on the ball as (u1, ··· ,un−1,v)=(u, v), κ can be written as 2R2u R2 −|u|2 − v2 κ(u, v)=(x, y)= ,R . |u|2 +(v − R)2 |u|2 +(v − R)2 It is straightforward to check that its inverse is 2R2x |x|2 + |y|2 − R2 κ−1(x, y)=(u, v)= ,R . |x|2 +(y + R)2 |x|2 +(y + R)2 Hence κ is a diffeomorphism, called the generalized Cayley transform. ∗ 3 2 Sow that κ hR = hR. 4

Problem 8. Let M(n, R) denote the vector space of n × n matrices with real entries. For a A ∈ M(n, R), let AT be its transpose and let

O(n)={A ∈ GL(n, C),AT A =1}, the . SL(n)={A ∈ GL(n, C),AT A =1}, the special . SO(n)={A ∈ O(n) : det A =1}, the special oethogonal group.

(1) Let S(n, R) denote the set of symmetric n × n matrices. Define Φ : GL(n, R) → S(n, R)by

Φ(A)=AT A.

Let A ∈ O(n) be arbitrary. To compute the pushforward

Φ∗ : TAGL(n, R) → TΦ(A)S(n, R),

we choose arbitrarily n × n matrix B and consider the γ(t)=A + tB. 0 0 T T (i) Show that Φ∗B =Φ∗γ (0) = (Φ ◦ γ) (0) = B A + A B.

(ii) Show that the identity matrix In is a regular value of Φ; i.e. show that Φ∗ : TAGL(n, R) → TΦ(A)S(n, R) is surjective for each A ∈ O(n). −1 (iii) Show that O(n)=Φ (In) is an embedded of GL(n, R). (2) Let det : GL(n, R) → R \{0} denote the function. i (i) Using the matrix entries (Xj ) as global coordinates on GL(n, R), show that

∂ −1 i i (det X) = (det X)(X )j . ∂Xj

(Hint: Expand det X by minors along the ith column and use Cramer’s rule.) (ii) Show that the differential of the determinant function is

−1 d(det)X (B) = (det X)tr(X B)

∼ i for X ∈ GL(n, R) and B ∈ TX GL(n, R) = M(n, R), where trX = Pi Xi is the of X. (iii) Show that det : GL(n, R) → R \{0} is a . (Hint: d(det)AA =6 0 for each A ∈ GL(n, R).) (iv) Show that SL(n, R) is an embedded submanifold of GL(n, R). (v) Show that Lie(SL(n, R)) =∼ {A ∈ M(n, R), trA =0}. (3) (i) Show that SO(n, R) is an embedded Lie submanifold of GL(n, R). (ii) Show that Lie(SO(n)) =∼ Lie(O(n)) ∩ Lie(SL)(n, R). 5

Problem 9. Let M(n, C) be the vector space of all the n×n matrices of complex entries. Let GL(n, C) is the of all complex n×n matrices of nonvanishing determinant. Let A∗ = AT be the conjugate transpose of A and

U(n)={A ∈ GL(n, C),A∗A =1}, the , SU(n)={A ∈ U(n) : det A =1}, the , SL(n, C)={A ∈ GL(n, C) : det A =1}, the complex .

(1) Show that

Lie(U(n)) =∼{A ∈ M(n, C): A∗ + A =0}, Lie(SL(n, C)) =∼{A ∈ M(n, C), trA =0}, Lie(SU(n)) =∼Lie(U(n)) ∩ Lie(SL)(n, C).

(2) (i) Show that the following matrices are a basis for Te of Lie(SU(2)):

1 0 i 1 0 −1 1 i 0 J =   ,J=   ,J=   . 1 2 i 0 2 2 10 3 2 0 −i

(ii) Show that this Lie algebra has the brackets

[J1,J2]=J3, [J2,J3]=J1, [J3,J1]=J2.

(iii) Show that cos(t/2) i sin(t/2) exp(tJ )=  1 i sin(t/2) cos(t/2)

(2) (i) Show that the following matrices are a basis for Te of Lie(SO(3)):

 00 0  001  0 −10 L1 = 00−1 ,L2 = 000,L3 = 100.  01 0  −100  000

(ii) Show that this Lie algebra has the brackets

[L1,L2]=L3, [L2,L3]=L1, [L3,L1]=L2.

(iii) Show that  10 0 exp(sL1)= 0 cos s − sin s .  0 sin s cos s 

(3) Show that exp(tJ1) is a double covering of exp(sL1). (4) Show that for all t1, t2 and t3 ∈ R, the map

t : exp(t1J1 + t2J2 + t3J3) 7→ exp(t1L1 + t2L2 + t3L3)

is a double covering of SO(3) by SU(2). 6

Problem 10. Definition. A (smooth) k-dimensional is a pair of smooth manifolds E (the total space) and M (the base), together with a surjective (smooth) map π : E → M (the projection), satisfying the following conditions: −1 (1) Each set Ep = π (p) (called the fiber of E over p) is endowed with the structure of a vector space. (2) For each p ∈ M, ∃a nbhd U of p and a (diffeomorphism) ϕ : π−1(U) → U × Rk, called a (smooth) local trivilization of E such that the following diagram commutes. ϕ π−1(U) −−−−→ U × Rk   π π1 y y U U (where π1 is the projection onto the first factor). k (3) The restriction of ϕ to each fiber ϕ : Ep →{p}×R is a linear . Let I =[0, 1] ⊂ R be the interval, and let p : I → S1 be the quotient map p(x)=e2πix, which identifies the two endpoints of I. Consider the “inifinite strip” I × R, and let

π1 : I × R → I be the projection on the first factor. Let ∼ be the equivalence relation on I × R that identifies each point of (0,y) in the fiber over 0 with the point (1, −y) in the fiber over 1; in other words, the right-hand edge is given a half-twist to turn it upside-down, and then is glued to the left-hand edge. Let E =(I × R)/ ∼ denote the resulting quotient space, and let q : I × R → E be the quotient map. Because p◦π1 is constant on each equivalence class, it descends to a continuous map π : E → S1. q I × R −−−−→ E   π1 π y p y I −−−−→ S1 Show that this makes E into a smooth vector bundle over S1, (called the M¨obius bundle). (Hint: Construct a local trivilization whose domain includes the fiber where the gluing takes place.)

Problem 11. Define an action of Z on R2 by n · (x, y)=(x + n, (−1)ny)

(1) Show that the action is smooth, free, and proper. (2) Let E = R2/Z denote the quotient manifold. Show that E is smoothly isomorphic to the M¨obiusbundle constructed in Problem 10. 7

Problem 12. Definition. Let E → M be a vector bundle. If U ⊂ M is an , lo- cal sections σ1, ··· ,σk of E over U are said to be independent if their values σ1(p), ··· ,σk(p) are linearly elements of Ep, ∀p ∈ U. They are said to span E if their values span Ep, ∀p ∈ U. A local frame for E over U is an ordered k-tuple (σ1, ··· ,σk) of independent local sections over U that span E. This local frame (σ1, ··· ,σk) is called a global frame if U = M.

Definition. If M is a smooth manifold, we use the term local frame for M to mean a local frame for the TM, or in other words as an n-tuple of independent vector fields span TM over some open subset of M.Aglobal frame for M is defined similarly. A smooth manifold M is said to be parallizable if it admits a smooth global frame. Show that every Lie group is parallizable.