Problem 3. (Dense orbit on torus) Throughout this problem, you may need some number theory results, which you only need to state and do not have to give a proof. A torus is the product manifold T2 := S' x S'. Here S' is the unit circle in R2. (1) Let [0, 1] x [0, 1] be the unit square in R2. We introduce the equivalent relation (x, 1) ~ (x, 0) for re [0, 1], and (0, y) ~ (1, y) for y E [0, 1]. Show that [0, 1] x [0, 1]/ ~ is diffeomorphic to T2. (2) It is natural to define a map 4 : R2 - T2 using the identification in (1). Let [x] be the largest integer that is less or equal to x, and define {x} := x - [x]. Define "(x, y) = (x}, {y}). Show this can be defined as a smooth map from R2 - T2. (3) Show n is a local diffeomorphism. (4) Let a E R. Show that the map y : R - T2 defined by y(t) = (t, at) is an immersion. (5) Show that the image of y is dense in T2 if and only if a is irrational. (6) Show that y is an embedding if and only if a is a rational number