174 V.M. Buchstaber
For eachUαthere is fixed a homeomorphismφα:Uα→Rnproviding thelocal
coordinatesx^1 α,...,xnαinUα. Therefore, there are two sets of coordinates on
each intersectionUα∩Uβ, namely, for everyx∈Uα∩Uβwe have
φα(x)=(x^1 α,...,xnα)andφβ(x)=(x^1 β,...,xnβ).Thecoordinate transformationis given by the set of smooth functions:
xjα=fαj(x^1 β,...,xnβ),j=1,...,n;
xkβ=gβk(x^1 α,...,xnα),k=1,...,n,where the compositionf◦gis the identity transformation ofRn.Asmooth
mapF:M→Nbetween two manifolds is given by smooth functions in any
local coordinate system.
Suppose we are given a curve segmentx=x(τ)∈M, a≤τ ≤b,on
a manifoldM. The part of the curve belonging to a coordinate regionUαis
described by the set ofparametric equations
xjα=xjα(τ),j=1,...,n.Thevelocity(ortangent)vectorat a pointx=x(τ) is given by
x ̇=( ̇x^1 α,...,x ̇nα).In the intersectionUα∩Uβwe can write the parametric equationsxα(τ)and
xβ(τ) in the two coordinate systems. Using the coordinate transformation
formulae we obtain
xjα(τ)=fαj(
x^1 β(τ),...,xnβ(τ))
.
Therefore,
x ̇jα=∑
k(
∂fαj
∂xkβ)
x ̇kβ.Ann-dim manifoldMnis calledorientableif there is a coordinate covering
Mn=∪αUαsuch that the coordinate transformations satisfy
det(
∂fαj
∂xkβ)
> 0
for allx∈Uα∩Uβ.
A smooth manifold Mn that can be smoothly embedded in Rn+1 is
orientable. It follows that a 2-dim manifold is embeddable inR^3 if and only
if it is orientable.
Atangentvector to ann-dim manifoldMnis by definition the velocity
vector of some smooth curve. In any system of local coordinatesxjαthe tangent