3-1 Coordinates relating to manifolds 115
Example 3.1 (Circle S
1
r , one-dimensional manifold, topology).
First, we present the topology on S
1
r . Second, we present the topology on φ(S
1
r ). The “one-sphere” S
1
r
(circle of radius r) is defined as the manifold
S
1
r :=
x ∈ R
2 x
2 + y
2 = r
2 , r ∈ R
+ , r > 0
, U := S
1
r /{x = +r} .
(3.1)
(i) Topology on S
1
r .
The topology on S
1
r is defined by the Euclidean metric, namely the distance function of the ambient
space R
3 , i. e.
d(x 1 , x 2 ) := x 1 − x 2 2 .
(3.2)
Along the orthonormal base {e 1 , e 2 O} attached to the origin O, the center of the “one-sphere”,
we define a Cartesian coordinate system {x, y} such that x
2
2 = x
2 + y
2 = r
2 > 0. A point P
of the “one-sphere” is orthogonally projected on the x axis such that p = π(P ). Refer to Fig. 3.1
for an illustration. The unit vector e x := x/ x 2 and the unit base vector e 1 include the angle
λ = (e x , e 1 ), an element of the open interval λ ∈ {R 0 < λ < 2π}.
Φ(x) = λ(x) ⇔ Φ
−1 (x) : x(λ) = e 1 r cos λ + e 2 r sin λ ,
(3.3)
Φ
−1 (x) = r
cos λ
sin λ
,
(3.4)
d(x 1 , x 2 ) = r
(cos λ 1 − cos λ 2 ) 2 + (sin λ 1 − sin λ 2 ) 2 =
= r
√
2
1 − (cos λ 1 cos λ 2 + sin λ 1 sin λ 2 ) =
= r
√
2
1 − cos(λ 1 − λ 2 ) .
(3.5)
We apologize for our sloppy notation x(λ) meaning x = κ(λ), but introduced for economical reason:
save extra symbols.
λ(x) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
arctan(y/x)
forx > 0
arctan(y/x) + π for x < 0
(π/2) sgn y
for x = 0 and y = 0
undefined
for x = 0 and y = 0
.
(3.6)
Note that {λ = 0 or 2π} or, equivalently, {x = r, y = 0} is the exceptional point which is not curved
by the angular parameter λ!
(ii) Topology on Φ(S
1
r ).
The topology on Φ(S
1
r ) is defined by the Euclidean metric, namely the distance function
d(y 1 , y 2 ) := y 1 − y 2 2 = |λ 1 − λ 2 | .
(3.7)
End of Example.