Page 211 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes or, equivalently,
E (t) = cos((t))E (t) + sin((t))E (t)
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E (t) = sin((t))E (t) + cos((t))E (t) .
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From these two representations, we may derive an important formula:
'
E (t) . E (t) = E (t) . E (t) – '(t)
'

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Expressed in the form of differentials, without specifying parameters, this formula becomes:
dE E = dE E – d.
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Since E (t) = T(t) × E (t), we have:
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E (t) . E (t) = [E (t), E (t), T(t)]
'
'
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or, in differentials:
dE E = –[dE ,E ,T] .
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More generally, if z(t) is a unit vector in the normal space at x(t), then we may define a function
w(t) = –[z’(t), z(t),T(t)]. This is called the connection function of the unit normal bundle. The
corresponding differential form w = –[dz, z,T] is called the connection form of the unit normal
bundle.
A vector function z(t) such that |z(t)| = 1 for all t and z(t) . x’(t) = 0 for all t is called a unit normal
vector field along the curve x. Such a vector field is said to be parallel along x if the connection
function w(t) = –[z’(t), z(t),T(t)] = 0 for all t. In the next section, we will encounter several unit
normal vector fields naturally associated with a given space curve. For now, we prove some
general theorems about such objects.
Proposition 2. If E (t) and E (t) are two unit normal vector fields that are both parallel along the
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curve x, then the angle between E (t) and E (t) is constant.
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Proof. From the computation above, then:
'
E (t)· (–E (t) × T(t)) = E (t)· (–E (t) × T(t)) – '(t).
'

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But, by hypothesis,
'
'
E (t)· (–E (t)xT(t)) = 0 = E (t)(–E (t) × T(t))
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so it follows that ’(t) = 0 for all t, i.e., the angle (t) between E (t) and E (t) is constant.
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Given a closed curve x and a unit normal vector field z with z(b) = z(a),
we define
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(x, z) =   [z'(t),z(t),T(t)]dt   [dz,z,T].
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If z is another such field, then
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
(x, z) – (x, z) =   [z'(t),z(t),T(t)] [z'(t),z(t),T(t)]dt
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
=    '(t)dt   [ (b)   (a)].
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