Differential Geometry - homework problems

1. Coordinate charts on S2 via stereographic projection

Let M = (x, y, z) ∈ R3 : x2 + y2 + z2 = 1, and let U = p ∈ M : z 6= 1. Then letφ : U → R2 map (x, y, z)→ (u, v), where (u, v) is the intersection of the line joining thetwo points (0, 0, 1) and (x, y, z) ∈M with the equatorial plane z = 0. The coordinatesof this intersection are labeled (u, v). Find u, v as functions of x, y, z. This chart coversthe entire sphere except for the north pole. The local coordinates here are (u, v).

A second chart, with local coordinates (u, v) can be obtained via stereographic projec-tion from the south pole (0, 0,−1). Find (u, v) as functions of x, y, z and then computethe patching function giving u, v as functions of u and v.

If w = u + iv and w = u + iv what is w as a function of w? What’s wrong with thisand how and why should it be fixed?

2. The Lie group SU(2) := A2×2 complex matrices s.t. A†A = I, det(A) = 1. Showthat SU(2) can be put in 1-1 correspondence with S3. Hint: if

A =

(a bc d

)∈ SU(2),

show that the A−1 = A† ⇒ c and d can be written in terms of a and b. Then use thecondition det(A) = 1.

3. Show that the map sending ei ∈ V → ei ∈ V ∗ is not canonical - i.e., the map dependson the choice of basis.

4. Show that V ∼= V ∗∗ and that the isomorphism is canonical.

5. It is false that any differential 1-form µ can be written locally as µ = df for somesmooth fundtion f . Why?

6. It is not difficult to show that if the vectorfield ξ is smooth on M , x0 ∈ M , andt → Φ(t, x0) is the unique solution to the initial value problem dx/dt = ξ; x(0) = x0,which exists for −δ < t < δ, then there exists a neighborhood U(x0) and a positivenumber ε ≤ δ such that solutions to the IVP exist for all x ∈ U and for all t ∈ (−ε, ε).This is proven in Math 850, for instance, or you can look it up in any text on ODEs.So we get a map Φ : (−ε, ε) × U → M defined by: Φ(t, x) is the point of the solutioncurve to the ODE reached after time t starting with initial condition x. The map φ isdifferentiable and it’s called the flow of the ODE.

Show that if M is compact, there exists a single positive number ε such that the flowΦ exists for t ∈ (−ε, ε), ∀x ∈M .

Given this, show that the domain of Φ is R ×M . The flow exists for all time at allpoints. Such a flow is called complete.

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7. Find the orbits of ξ = x(1 − x) on R (there are 5). Find the flow of ξ on (0, 1) andshow that it’s complete.

8. Let a = x + iy, b = u + iv, and take (a, b)T ∈ C2 such that |a|2 + |b|2 = 1 andtherefore lies on the sphere S3. For each such (a, b)T , define Φ(t, (a, b)T ) = eit(a, b)T .This is a flow. The corresponding vector field is found by diferentiating with respectto t : ξ(a, b)T = d/dt

(Φ(t, (a, b)T )

)|t=0= i(a, b)T . In terms of the real coordinates

(x, y, u, v), you can check it’s orthogonal to the radius vector and hence tangent tothe sphere. Clearly the orbits of the vectorfield ξ are circles on S3 (great circles, infact). These S1s form a fibration of the 3-sphere, known as the Hopf fibration. Thismeans they fill out the sphere and are non-intersecting, which all follows from thefact that they are orbits of a vector field on a compace manifold. In fact, projectingthe point (a, b)T ∈ S3 to the complex projective line (P1(C) ≈ S2) via (a, b) → [a, b](homogeneous coordinates) gives us a smooth surjection π : S3 → S2. (There is noholomorphic structure here; we are just using complex numbers for their convenience.)Evidently, for any p = [a, b] ∈ S2, π−1(p) is an entire orbit.

This structure π : S3 → S2 is called the Hopf bundle. It exhibits S3 as a circle bundleover S2. This bundle is not trivial ; that is, S3 is not diffeomorphic to S2×S1 (because,for instance, their fundamental groups are different).

9. Let ξ and η be two smooth vector fields on M . For all smooth real-valued functions fon M define

• X(f) = ξ(η(f)), and

• Y (f) = ξ(η(f))− η(ξ(f)).

Show that X is not a vector field. Show that Y is a vector field. It’s commonlydenoted by [ξ, η] and called the commutator of the two vector fields. It’s alsoknown as the Lie derivative of η with respect to ξ.

• Show that the commutator is (a) bilinear in its two arguments (b) skew-symmetric(i.e., [ξ, η] = −[η, ξ]), and (c) satisfies Jacobi’s identity: for any ξ, η, τ,

[[ξ, η], τ ] + [[η, τ ], ξ] + [[τ, ξ], [η] = 0.

10. Connections:

(a) Show that ∇X∇YZ −∇Y∇XZ −∇[X,Y ]Z = Ω(X, Y )Z, where in any local basissα of sections of the bundle, Ω(X, Y )Z = Ωα

β(X, Y )Zβsα.

(b) If p ∈ M, eα is a basis for Ep, and γ is a closed, piecewise smooth curve atp, parallel propagate eα around the loop by requiring that DXeα = 0, whereX(t) = γ(t). Writing e for the column matrix (e1, e2, . . . , ek)

t, the result can beexpressed as e · g, for some g ∈ GL(k,R). Do this for every closed loop at p.Show that the resulting subset of GL(k,R) is in fact a subgroup, known as theholonomy group of the connection at p.

(c) Show that the holonomy group is well-defined: that, up to isomorphism, it isindependent of the basis e and the basepoint p. (Assume M is connected.)

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(d) If H denotes the holonomy group, let H0 be the subgroup of H generated by thecontractible loops. Show that H0 is normal in H. What is H/H0?

(e) Show that H0 = id ⇐⇒ Ω = 0 ⇐⇒ parallel propagation is locally indepen-dent of the path.

(f) The connection D is flat if parallel propagation is independent of the path in M .Show that if D is flat, then M is parallelizable: there exists a global C∞ framefield.

11. Affine connections:

(a) If ωij = Γijkduk in some local coordinate U , and ω′ij = Γ′ijkdv

k in V , and U ∩V 6= ∅,find Γ′ijk in terms of the Γijk in the intersection.

12. The frame bundle

(a) The frame bundle of an n-dimensional differentiable manifold M is the set of allpairs

P =

(p, e) : p ∈M, e = e1, e2, . . . , en,

where e is a set of n linearly independent vectors (a “frame”) in TpM . Show thatP has the structure of a principle bundle with structure group G = GL(n,R).The local product structure is obtained by referring everything in a coordinatechart U to the basis vectors ∂/∂ui : 1 ≤ i ≤ n.

(b) Show that TM is the vector bundle associated to P under the standard represen-tation (matrix multiplication) of G on Rn. Similarly, the bundle T ∗M arises fromthe representation g → (g−1)t acting via matrix multiplication on Rn.

13. Calculus of variations:

(a) Fundamental lemma: Let f be continuous on [a.b] and suppose that, for all con-tinuous functions g vanishing at a and b,∫ b

a

f(x)g(x) dx = 0.

Then f = 0 on the interval.

Notation: A more satisfactory way to write out variational problems in onedimension is to assume that there’s a local extremum at γ0, and write γ(s, t), fora one-parameter variation of γ0. That is

• γ(0, t) = γ0(t),∀t, and

• γ(s, a) = γ0(a), γ(s, b) = γ0(b) (All the curves t→ γ(s, t) = γs(t) satisfy thesame boundary conditions.

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In class, we wrote γ(s, t) = γ(t) + sη(t), but this is unnecessarily restrictive; forinstance, it doesn’t allow computation of a meaningful second derivative wrt s.

The standard notation is

δγ(t) =∂γ(s, t)

∂s|s=0.

Any appearance of the symbol δ means differentiate wrt s and set s = 0.

So in the more standard notation, where we use local coordinates x(s, t) for the 1-parameter family of curves¡ the derivation of the Euler-Lagrange equations lookslike

δJ [γ0] = 0, at an extremum, and this means

δ

∫ b

a

L(t, x, x) dt =

∫ b

a

(∂L∂x

δx+∂L

∂xδx)dt

=

∫ b

a

(∂L∂x− d

dt

∂L

∂x

)δx dt

= 0 for all admissible curves δx(t).

where in the second step we’ve integrated by parts and used the BCs to get ridof d/dt

((∂L/∂x)δx

). Then the fundamental lemma leads to the vanishing of the

integrand (Euler Lagrange eqns).

(b) Let x(t) be a curve in R3 traversed by a point particle with mass m. Then theaction integral for this system is

J [x] =

∫ t=b

t=a

L(x, x) dt

J [x] =

∫ t=b

t=a

1

2mx•x− V (x) dt,

where L is the kinetic enegy minus the potential energy of the particle evaluatedon the path. Show that δJ = 0 is the same as Newton’s second law. (Write outthe equation for a general 1-parameter variation, and then use the fact that ifit holds for general variations, it must hold in particular for variations involvingonly the x-coordinate, or only the y-coordinate . . . . You should get three secondorder ODEs.)

(c) In a Riemannian manifold (M, g) with the unique, metric and torsion-free con-nection, write

ds2 = gabdxadxb so that

δ(ds2) = 2dsδds, and therefore

δ(ds) =δ(ds2)

2ds

Use this to compute δ∫γds and show that this vanishes if and only if γ is an

affinely parametrized geodesic.

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