Page 209 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes The unit tangent vectors emanating from the origin form a curve T(t) on the unit sphere called
the tangential indicatrix of the curve x. To calculate the length of the tangent indicatrix, we form
the integral of |T’(t)| = (t)s’(t) with respect to t, so the length is (t)s’(t)dt = (s)ds. This significant
integral is called the total curvature of the curve x.
Up to this time, we have concentrated primarily on local properties of curves, determined at
each point by the nature of the curve in an arbitrarily small neighborhood of the point. We are
now in a position to prove our first result in global differential geometry or differential geometry
in the large.

By a closed curve x(t), a  t  b, we mean a curve such that x(b) = x(a). We will assume moreover
that the derivative vectors match at the endpoints of the interval, so x’(b) = x’(a).
Theorem 1 (Fenchel’s Theorem): The total curvature of a closed space curve x is greater than or
equal to 2.
(s)ds  2
The first proof of this result was found independently by B. Segre in 1934 and later independently
by H. Rutishauser and H. Samelson in 1948. The following proof depends on a lemma by
R. Horn in 1971:

Lemma 1. Let g be a closed curve on the unit sphere with length L < 2. Then there is a point m on
the sphere that is the north pole of a hemisphere containing g.
To see this, consider two points p and q on the curve that break g up into two pieces g and g of
1
2
equal length, therefore both less than . Then the distance from p to q along the sphere is less
than  so there is a unique minor arc from p to q. Let m be the midpoint of this arc. We wish to
show that no point of g hits the equatorial great circle with m as north pole. If a point on one of
'
the curves, say g , hits the equator at a point r, then we may construct another curve g by
t
1
rotating g one-half turn about the axis through m, so that p goes to q and q to p while r goes to
1
'
the antipodal point r’. The curve formed by g and g has the same length as the original
1
t
curve g, but it contains a pair of antipodal points so it must have length at least 2, contradicting
the hypothesis that the length of g was less than 2.
From this lemma, it follows that any curve on the sphere with length less than 2 is contained in
a hemisphere centered at a point m. However if x(t) is a closed curve, we may consider the
differentiable function f(t) = x(t) . m. At the maximum and minimum values of f on the closed
curve x, we have
.
.
0 = f’(t) = x’(t) m = s’(t)T(t) m
so there are at least two points on the curve such that the tangential image is perpendicular to m.
Therefore the tangential indicatrix of the closed curve x is not contained in a hemisphere, so by
the lemma, the length of any such indicatrix is greater than 2. Therefore, the total curvature of
the closed curve x is also greater than 2.

1
Corollary 1. If, for a closed curve x, we have (t)  for all t, then the curve has length L  2R.
R
Proof.






L = ds   R (s)ds = R  (s)ds 2 R
Fenchel also proved the stronger result that the total curvature of a closed curve equals 2 if and
only if the curve is a convex plane curve.

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