aem570-w10-hw6 - v x || = 1 be the circle bundle of S 2...

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AA/EE/ME 570: Manifolds and Geometry for Systems and Control Homework #6 Due: Thursday March 4, 11:30am All problems have equal value. 1. Let M be a mnaifold with TM its tangent bundle. (a) Using teh definition of the tangent space, define natural operations of vector addition and scalar multiplcation on T x M , and verify that these operations make T x M an R -vector space for each x M . (b) Is T x M a normed vector space? If so, what is the norm? 2. Let S = { ( x, y ) R 2 | x 2 - y 2 = 1 } . Show that the two charts φ 1 : { ( x, y ) S x > 0 } → R , φ ± ( x, y ) = y define a manifold structure on the disconnected set S . 3. Let Q ( S 2 ) = { ( v x ) T S 2 |||
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Unformatted text preview: v x ) || = 1 } be the circle bundle of S 2 . Prove that Q ( S 2 ) is a submanifold of T S 2 of dimension three. 4. Let ρ : R × S n → S n and σ : R n +1 × S n → S n be trivial vector bundles. Show that T S n ⊕ ( R × S n ) ∼ = ( R n +1 × S n ) . Hint: Realize ρ as the vector bundle whose one-dimensional fiber is the normal to the sphere. 5. Submit a one-page introduction to the topic of your project (e.g. motivation, description of the system and setting, relation to the material in the course). 1...
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