DIFFERENTIAL GEOMETRY

Perugia

Exam II Aug 29

(1) Let M be a manifold and γ: [0, a] → M be a curve with γ(t) = p for all t and some p ∈ M. Let V (t) be a vector field along γ. Show that \frac{DV}{dt} = V′ where V′ is the ordinary derivative in T_{p}M.

(2) Consider the hyperboloid in 3 space x^{2} + y^{2} − z^{2} = 1.

(a) Find at least two geodesics on it.

(b) Describe the general behavior of a geodesic.

(c) Show that it has only one closed geodesic.

(3) Let f: M → N be a local isometry and σ ⊂ T_{p}M a 2-plane. Show that the sectional curvature of σ in M is the same as the sectional curvature of df_{p}(σ) in N.

(4) Let J be a Jacobi field along a geodesic γ in M. If there exist t_{0} ≠ t_{1} with J(t_{0}) = J(t_{1}) = 0, show that J is orthogonal to γ′(t) for all t.

(5) Let γ be a geodesic in M.

(a) Show that tγ′(t) is a Jacobi field on M.

(b) Find a one parameter family of geodesics such that the variational vector field is equal to J.

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

is a portion of a round sphere. Is this also true if n = 2 ?

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