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3. The Unit Normal Bundle and Total Twist

Consider a curve X(t) with X'(t) 0 for all t. A vector Z perpendicular to the tangent vector X'(t0) at X(t0) is called a normal vector at X(t0). Such a vector is characterized by the condition ZX(t0) = 0, and if |Z| = 1, then Z is said to be a unit normal vector at X(t0). The set of unit normal vectors at a point X(t0) forms a great circle on the unit sphere. The unit normal bundle is the collection of all unit normal vectors at X(t) for all the points on a curve X.
At every point of a parametrized curve X(t) at which X'(t) 0, we may consider a frame E2(t), E3(t), where E2(t) and E3(t) are mutually orthogonal unit normal vectors at X(t). If E2(t), E3(t) is another such frame, then there is an angular function (t) such that
E2(t) = cos((t))E2(t) - sin((t))E3(t)
E3(t) = sin((t))E2(t) + cos((t))E3(t),
or, equivalently,
E2(t) = cos((t))E2(t) + sin((t))E3(t)
E3(t) = sin((t))E2(t) + cos((t))E3(t).
From these two representations, we may derive an important formula: E2'(t) E3(t) = E2'(t)E3(t) - '(t). Expressed in the form of differentials, without specifying parameters, this formula becomes dE2E3 = dE2E3 - d.
Since E3(t) = T(t) x E2(t), we have E2'(t) E3(t) = -[E2'(t), E2(t), T(t)], or, in differentials, dE2E3 = - [ dE2, E2, T ].
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 1: If E2(t) and E2(t) are two unit normal vector fields that are both parallel along the curve X, then the angle between E2(t) and E2(t) is constant.
Proof: From the computation above, then E2'(t) (-E2(t) x T(t)) = E2'(t)(-E2(t) x T(t)) - '(t). But, by hypothesis, E2'(t) (-E2(t) x T(t)) = 0 = E2'(t) (-E2(t) x T(t)), so it follows that '(t) = 0 for all t, i.e., the angle (t) between E2(t) and E2(t) is constant.
Given a closed curve X and a unit normal vector field Z with Z(b) = Z(a), we define (X, Z) = -(1/2)[Z'(t),Z(t),T(t)]dt = -(1/2) [dZ, Z. T]. If Z is another such field, then (X, Z) - (X, Z) = -(1/2) { [Z'(t),Z(t),T(t)] - [Z'(t),Z(t),T(t)] }dt = -(1/2)'(t)dt = -(1/2)[ (b) - (a) ].
Since the angle (b) at the end of the closed curve must coincide with the angle (a) at the beginning, up to an integer multiple of 2, it follows that the real numbers (X, Z) and (X, Z) differ by an integer. Therefore the fractional part of (X, Z) depends only on the curve X and not on the unit normal vector field used to define it. This common value (X) is called the total twist of the curve X. It is a global invariant of the curve.
Proposition 2: If a closed curve lies on a sphere, then its total twist is zero.
Proof: If X lies on the surface of a sphere of radius R centered at the origin, then |X(t)|2 = X(t)X(t) = R2 for all t. Thus X'(t)X(t) = 0 for all t, so X(t) is a normal vector at X(t). Therefore Z(t) = X(t)/R is a unit normal vector field defined along X, and we may compute the total twist by evaluating (X, Z) = -(1/2)[Z'(t),Z(t),T(t)]dt. But [Z'(t),Z(t),T(t)] = [X'(t)/R, X(t)/R, T(t)] = 0 for all t since X'(t) is a multiple of T(t). In differential form notation, we get the same result: [dZ, Z, T ] = (1/R2) [ X'(t), X(t), T(t) ]dt = 0. Therefore (X, Z) = 0, so the total twist of the curve X is zero.
Remark 1: W. Scherrer proved that this property characterized a sphere, i.e. if the total twist of every curve on a closed surface is zero, then the surface is a sphere.
Remark 2: T. Banchoff and J. White proved that the total twist of a closed curve is invariant under inversion with respect to a sphere with center not lying on the curve.
Remark 3: The total twist plays an important role in modern molecular biology, especially with respect to the structure of DNA.

Exercises

  1. Let X be the circle X(t) = ( R cos(t), R sin(t), 0 ), where R is a constant > 1. Describe the collection of points X(t) + Z(t) where Z(t) is a unit normal vector at X(t).
  1. Let be the sphere of radius R > 0 about the origin. The inversion through the sphere S maps a point X to the point X = R2X/ |X|2. Note that this mapping is not defined if X = 0, the center of the sphere. Prove that the coordinates of the inversion of X = (x1, x2, x3) through S are given by xi = R2xi/(x12 + x22 + x32). Prove also that inversion preserves point that lie on the sphere S itself, and that the image of a plane is a sphere through the origin, except for the origin itself.
  1. Prove that the total twist of a closed curve not passing through the origin is the same as the total twist of its image by inversion through the sphere S of radius R centered at the origin.
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