a1 - 2014 Fall Math 5041 Assignment 1 Due Name In problems...

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2014 Fall Math 5041 Assignment 1. Due: Sept 26, 2014 Name: In problems 1 and 2, we will show explicitly that R P 3 is null-cobordant. 1. a) Consider the set of quanternions H = { x = a + bi + cj + dk | a, b, c, d R , i 2 = j 2 = k 2 = - 1 , ij = k, jk = i, ki = j } . It is clear that H is a real vector space. We define the conjugate of x = a + bi + cj + dk to be x = a - bi - cj - dk . Show that x 7→ || x || = x x defines a norm on H . b) Show that H \ { 0 } is a group, and the unit sphere U = { x H | || x || = 1 } is a subgroup. Remark: U is (isomorphic to) the so-called SU (2 , C ) group. c) Show that the action of U on H by conjugation g · x 7→ gxg - 1 (1) preserves the norm, and V = { a + bi + cj + dk H | a = 0 } is an invariant subspace. d) Show that the stabilizer of the action of U on V is 1 } . Use this result to conclude that the special orthogonal group SO (3 , R ) = { A Mat 3 × 3 ( R ) | AA T = I } is diffeomorphic to R P 3 . Remark: This is the well-known fact that SU (2) is the universal double cover of SO (3). 2. Consider the subset of T R 3 consisting of the vectors tangent to the 2-sphere S 2 R 3 of unit length.
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