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HW Set 1 for wed

Kiselev:
52, 53, 55, 60, 61, 67, 69, 75, 79, 89, 96, 97

HW Set 2 for wed
104, 105, 106, 107, 113, 114, 141, 143, 145, 150

HW Set 3 for wed
153, 154, 155, 158, 159, 178, 179, 180

HW Set 4 for wed
From taste of topology:
2.1: 1,2 2.3: 2,3

HW Set 5 for wed
1) Sea (X, d) un espacio metrico y sea (X, L_d) un length space induced by d. Demuestra que: ∀x, y ∀γ : d(x, y) ≤ L(γ) where γ is a path from x to y.

2) Let X = ℝ² and d_T the taxicab metric: d_T(p₁, p₂) = |x₁ − x₂| + |y₁ − y₂| for p₁ = (x₁, y₁) and p₂ = (x₂, y₂). Describe the minimal geodesics from p₁ to p₂ for L_{d_T}.

3) Let X = ℝ², d the Euclidean metric and d_P the Paris railroad metric with respect to P ∈ ℝ². So: d_P(p₁, p₂) = d(p₁, P) + d(p₂, P), where d is the Euclidean metric, and p₁, p₂ ∈ ℝ².

Describe the minimal geodesics from p₁ to p₂ for L_{d_P}.

4) Sea (ℝ², L) un length space. Vamos a llamar un mapeo inyectivo γ : ℝ → ℝ² una recta, si cado segmento γ|_[a, b] de γ es un geodesico minimal al respecto de L. Cuales son rectas en (ℝ², L_{d_P}) y cuales son rectas en (ℝ², L_{d_T})?

5) Con ese definición de las rectas se cumplen los axiomas de geometría de Euclid para (ℝ², L_{d_T}) y para (ℝ², L_{d_P})? Justify.

HW set 6 for wed
0) Show that for the Eucledian metric d on ℝ² parallel transport along a segment from A to B is just rigid translation by the map ℝ² → ℝ², C ↦ C + (B − A). 1) Let ϕ : (X, d_X) → (Y, d_Y) be a surjective isometry of metric spaces, so that ∀x, y ∈ X : d_X(x, y) = d_Y(ϕ(x), ϕ(y)). Show that

ϕ is continuous.
ϕ is injective.
ϕ⁻¹ : (Y, d_Y) → (X, d_X) is also an isometry.

2) For ϕ as in problem 1. Show that γ : [a, b] → X is a minimal geodesic in (X, L_{d_X}) if and only if ϕ ∘ γ is a minimal geodesic in (Y, L_{d_Y}).

3) Extra credit: Suppose that X, Y are surfaces in ℝ³ and d_X, d_Y the natural metrics as discussed in class. For ϕ : X → Y as above, show that ϕ preserves angles between geodesics.

4) Extra credit: Use the above to show that for ϕ as in problem 3), the holonomy around a closed piece-wise geodesic curve γ in X coincides with the holonomy around a closed piece-wise geodesic curve ϕ ∘ γ in Y. Conclude that the zero curvature condition is preserved by an isometry as in problem 3).