# HW201sol - Differential Geometry 2 Homework 01 1 Let V be a...

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Unformatted text preview: Differential Geometry 2 Homework 01 1. Let V be a finite-dimensional real vector space on which [ •|• ] is an inner product of type (- + ... +). Fix R > 0 and select a point a in one component H of the hyperboloid { z ∈ V : [ z | z ] =- R 2 } . Let B = B R ( a ⊥ ) be the ball of radius R in a ⊥ and calculate explicitly the pullback of the canonical Riemannian metric h on H via the inverse F : B → H of stereographic projection from- a . 2. Let ( •|• ) be a (positive-definite) inner product on the finite-dimensional real vector space V of which S is the unit sphere. Fix a ∈ S and let ψ = ψ a : S- { a } → a ⊥ be stereographic projection from a . Choose n ∈ S and determine explicitly the image under ψ of the ‘equatorial hypersphere’ n ⊥ ∩ ( S- { a } ). Solutions (1) By direct calculation, F : B → H ⊂ V acts on w ∈ B to produce F ( w ) =- a + f ( w )( w + a ) where f ( w ) = 2 R 2 R 2- [ w | w ] whence if also u ∈ a ⊥ then F w ( u ) = f w ( u )( w...
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HW201sol - Differential Geometry 2 Homework 01 1 Let V be a...

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