Unformatted text preview: Diﬀerential Geometry 2
Homework 01 1. Let V be a ﬁnitedimensional 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 ] = −R2 }.
Let B = BR (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 (positivedeﬁnite) inner product on the ﬁnitedimensional
real vector space V of which S is the unit sphere. Fix a ∈ S and let
ψ = ψa : 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}). 1 ...
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
 Spring '09
 Robinson

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