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7/22/2019 Cartas Para S2
1/2
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
7/22/2019 Cartas Para S2
2/2
:
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]).