DIFFERENTIAL GEOMETRY

Perugia Aug 07

Assignment Week 4 and 5

(1) (a) Show that the curvature of the metric dr^{2} + f^{2}(r, θ)dθ^{2} is given by

−\frac{1}{f} \frac{∂^{2}f}{∂r^{2}}

.

(b) On a surface we can introduce polar coordinates (r, θ) at p by

(r, θ) → exp_{p}(r cos θ e_{1} + r sin θ e_{2})

where e_{1}, e_{2} is an orthonormal basis of T_{p}M. Show that the metric in polar coordinates takes on the form dr^{2} + f^{2}(r, θ)dθ^{2}.

(c) Show that locally every function on a surface is the curvature of some metric.

(2) (a) Show that the curvature of the metric f(x, y)(dx^{2} +dy^{2}) is given by \frac{−△ log f}{2f}

where △g(x, y) = \frac{∂g}{∂x^{2}} + \frac{∂g}{∂y^{2}} . These coordinates are called isothermal coordinates and one can show that they always exist locally.

(b) Show that the curvature of hyperbolic space \frac{dx^{2}+dy^{2}}{y^{2}} is −1.

(3) Compute the curvature of the metric ds^{2} = \frac{dx^{2}+dy^{2}}{y}. Is it complete?

(4) Show that on a 3-dimensional manifold the Ricci curvature determines the sectional curvature. In particular, show that a 3-manifold with constant Ricci curvature has constant sectional curvature.

(5) For f ∈ C^{∞}(M) define grad f by ⟨grad f, X⟩ = df(X). Show that if | grad f| = 1, then all integral curves of grad f are minimizing geodesics. Give some examples of such functions.

(6) Let M be a Riemannian manifold.

(a) Show that the limit of minimizing geodesics is minimizing in the following sense: If γ_{i} and γ are geodesics with γ_{i}(0) → γ(0) and γ′_{i}(0) → γ′(0) show that γ_{i}(t) → γ(t) for all t and if γ_{i} is minimizing, then so is γ.

(b) If M is non-compact and complete, show that there exists at least one geodesic ray from any point, i.e. a normal geodesic γ: [0, ∞] → M with d(γ(0), γ(t)) = t for all t > 0.

(c) Give an example of a non-compact manifold which contains no line, i.e. a normal geodesic γ: [−∞, ∞] → M with d(γ(t), γ(s)) = |t − s|for all t, s.

(7) (a) DoCarmo p. 103 # 1.

(b) Show that ⟨A, B⟩ = −\frac{1}{2} tr AB defines a biinvariant metric on SO(n). Show that ⟨A, B⟩ = −\frac{1}{2} tr AB̄ defines a biinvariant metric on U(n).

(c) (After you have done p. 103 # 1) show that ⟨A, B⟩ = −\frac{1}{2} tr AB on SO(3) has constant curvature \frac{1}{4}.

(d) show that ⟨A, B⟩ = −\frac{1}{2} tr AB on SU(2) has constant curvature 1.

(e) Show that the unit quaternions S^{3} and SU(2) are isomorphic.

(f) Show that SU(2) = S^{3} is a two fold cover of SO(3) (Hint: use conjugation with quaternions).

(g) Show that S^{3} × S^{3} is a two fold cover of SO(4)

(8) Show that if M^{n} ⊂ ℝ^{n+1}, n ≥ 3, has constant positive sectional curvature, then it

is a portion of a round sphere. What if n = 2 ?

Hint: First show that all points are umbilic, i.e. all principal curvatures are the same.

(9) DoCarmo p.81 # 5.

(10) If J(t) is a Jacobi field along a geodesic γ with J(0) = 0, show that |J(t)| = t − \frac{t^{3}}{6} sec(σ) + o(t^{3}) where σ is the two plane spanned by J′(0) and γ′(0).

(11) For a metric of the form dr^{2} + f^{2}(r, θ)dθ^{2} show that f is the absolute value of a Jacobi field and conclude again (much simpler proof) that its curvature is −\frac{1}{f} \frac{∂^{2}f}{∂r^{2}} .