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Charts for S2:

Stereographic projection:

UN =Sn {N= (1, 0,..., 0)}.

N :UNRn,N(x) =

x

(1x0)

US=Sn {S= (1, 0,..., 0)}.

S: USRn,

S(x) = x

(1+x0)

Pre-atlas for Sn: A1 = {(UN, N), (US, S)},

Projection:

Uzp = {(x,y,z) S2 |z >0},

zp : Uzp {(x, y) | x2 +y2 0},

yp : Uyp {(x, z) | x2 +z2 0},xp: Uxp {(y, z) | y

2 +z2

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:

G(p, N) = {[g] | gsmooth :UN, for some Uopen such that p UM}

G(p) = G(p,R)

Let : I M where I= an interval R,(0) =p. Note [] G[0, M]

Directional derivative of [g] in direction [] =

Dg=d(g )

dt |t=0 R

Tp(M) = {v : G(p) R | v is linear and satisfies the Leibniz rule }

v Tp(M) is called a derivation

Given a chart (U, ) atpwhere(p) = 0, thestandard basisforTp(M) = {v1,...,vm},wherevi = Di and for some >0, i : (, ) M,i(t) =

1(0,...,t,..., 0)

Ifv Tp(M), then v = mi=1aivi where ai = v([i ])

Suppose fsmooth :MN, f(p) =q.

The tangent (or differential) map,

dfp : TpMTqN

dfp(v) = the derivation which takes [g] G(q) to the real number v([g f])

I.e., dfp takes the derivation (v : G(p) R) TpM

to the derivation in TqNwhich takes

the germ [g] G(q) to the real number v([g f]).