my email: yashasavelyev@gmail.com
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, Ld)
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
dT the
taxicab metric: dT(p1, p2) = |x1 − x2| + |y1 − y2|
for p1 = (x1, y1)
and p2 = (x2, y2).
Describe the minimal geodesics from p1 to p2 for LdT.
3) Let X = ℝ2,
d the Euclidean metric and
dP the
Paris railroad metric with respect to P ∈ ℝ2. So: dP(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 LdP.
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, LdP)
y cuales son rectas en (ℝ2, LdT)?
5) Con ese definición de las rectas se cumplen los axiomas de geometría
de Euclid para (ℝ2, LdT)
y para (ℝ2, LdP)?
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, dX) → (Y, dY)
be a surjective isometry of metric spaces, so that ∀x, y ∈ X : dX(x, y) = dY(ϕ(x), ϕ(y)).
Show that
ϕ is continuous.
ϕ is injective.
ϕ−1 : (Y, dY) → (X, dX) is also an isometry.
2) For ϕ as in problem 1.
Show that γ : [a, b] → X
is a minimal geodesic in (X, LdX)
if and only if ϕ ∘ γ
is a minimal geodesic in (Y, LdY).
3) Extra credit: Suppose that X, Y are surfaces in ℝ3 and dX, dY
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).