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Saturday, 20 May 2017

The exponential map in the plane and in the sphere

A geodesic in a plane P writes as γ(s)=p+sv, pP and vTpP=P. Then the exponential map is simply exp(v)=p+v
that is, the exponential map at p is a translation on P with translation vector p. Let us observe that the exp is a diffeomorphism in the whole TpP and thus the normal neighborhood of p is the very plane P.

In the unit sphere S2  a geodesic is γ(s)=cos(s)p+sin(s)v/|v|, where pS2 and vTpS2. Then exp(v)=cos(1)p+sin(1)v/|v|. We want to study when exp is one-to-one. By the property γ(t;p,λv)=γ(λt;p,v), we have exp(λv)=γ(t). If |v|=1, for λ=π, γ(π;p,λv)=p. This implies that the exponential map exp is one-to-one in the ball Bπ(0) of radius π.

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