GEODESIC LINES ON A CONVEX SURFACE OF
POSITIVE GAUSS CURVATURE
J. LIBERMAN
Abstract. This is a translation with modernized notation, of the paper
Les lignes geodesiques sur une surface convexe de courbure de Gauss
positive, C.R. (Doklady) de l’Acad. Sci. URSS (N.S.) 33, (1941), 9–11.
1. Translation of the paper
In this note we consider the metric g of positive Gauss curvature and the
geodesic lines for this metric. We show the following:
Theorem 1. There exists a positive number N such that every geodesic line
segment having length larger than N must have self-intersections.
To simplify the proof, we will assume in the following that the metric g
is analytic.
Proof. Let us take a sequence of positive numbers εi > 0 (i = 1, 2, . . .)
tending to zero. We will corresponding to each i take a finite covering Ci
of our space by collections of open sets of diameter less than ε. The number
of elements in the covering Ci will be denoted by Ni . Let ρ2 be the finite
upper bound of all the geodesics line segments without conjugate points. By
virtue of the positive Gauss curvature, such a bound exists.
Assume that our theorem is not true. Then there exists a sequence of
geodesics line segments xi (s) (0 ≤ s ≤ 8Ni , parametrized by arc length,
that do not have self-intersections. By Dirichlet’s principle there exists for
each i two integers ai and a0i and an element Ui of the covering Ci such that
(1) 0 < ai < a0i < 2Ni ,
(2) xi (4ai ) ∈ Ui and xi (4a0i ) ∈ Ui .
It follows from (2) that the distance between the points xi (4ai )
and xi (4a0i ) is smaller than i . Let us choose a sequence i1 < i2 < . . .
such that:
(3) The points xij (4aij ) converges to a fixed point x when j goes to
infinity.
(4) The directions of the geodesic lines xij at s = 4aij tend towards a
fixed direction l at the point x.
Denote by tj (s) the segment −p ≤ s ≤ p of the geodesic xij (4aij +
s), by t1j (s) the segment −p ≤ s ≤ p of the geodesic xij (4a0ij + s).
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By the choice of the geodesics xij (s), and the numbers aij and a0ij
we have:
(I) The segments tj (s) et t1j (s) (−p ≤ s ≤ p, j = 1, 2, . . .) do not have
self-intersections.
(II) When j goes to infinity, the points tj (0) et t1j (0) tend to the point
x, and the directions to l at the point x.
It follows from (I) and (III) that one has (III):
(III) When j goes to infinity, the directions of the segments t1j (s), −p ≤
s ≤ p tend for s = 0 to the direction l at the point x.
Let us take for at the point x a geodesic t0 in the direction l and choose
the paramatrization for this geodesic in a such a way that one has t0 (0) = x,
and for −p ≤ s ≤ p the points tj (s) tend to t0 (s). In a small neighborhood
U of the geodesic t0 (s) one may introduce a rectangular coordinate system
such that: the curve v ≡ 0 coincides with t0 (s), the coordinates (s, 0) and
the square of the line element is expressed by the formula
(1.1)
ds2 = A2 (uj v)du2 + dv 2 .
For j sufficiently large the segments tj (s) and t1j (s) (−p ≤ s ≤ p) can be
defined by the equations v = vj (s) and v = vj1 (s) as well as by the inverse
equations s = sj (u) and s = s1j (u) such that sj (0) → 0 and s1j (0) → 0 when
j → ∞. For large j such a definition is unique.
There exists an ε with 0 < ε < ν2 , such every coordinate point (u, v) (for
−p − ε ≤ u ≤ p + ε, −ε ≤ v ≤ ε) belongs to U . It is clear that for large
enough j one has −ε ≤ vj (u) ≤ ε and −ε ≤ vj1 (u) ≤ ε if −p − ε ≤ p + ε.
In the following we will be considering the line element (1.1) defined in the
coordinates (u, v) and we will assume that −p−ε ≤ u ≤ p+ε and −ε ≤ v ≤ ε.
−1
We will denote by tj (s) the point (u−1
j (s), v = vj (uj (s))).
In all the other cases we will consider there are analogs for t1j (s). By
the choice of geodesics tj (s) and t1j (s), their segments −p − ε ≤ u ≤ p + ε
don’t have an intersection. Denote by zj (s) (for − 2ε ≤ s ≤ 2s the geodesic
line segment such that zj (0) = tj (0) are perpendicular to tj (s) at the point
s = 0. It is clear that for large enough j there exists precisely one point
(uj , vj ) such that (uj , vj ) ∈ zj (s) (for 2ε ≤ s ≤ 2ε ) and (uj , vj ) ∈ tj (s) (for
−p ≤ s ≤ p). It is obvious that (uj , vj ) → (0, 0) when j → ∞. By (1) one
may assume that for j large enough, if (uj , vj ) = zj (s), one has s > 0.
Let sj and s0j be such that zj (s) = t1j (s0j ). It is clear that sj → 0 and
s0j → 0 for j → ∞. In the following we will assume that the angle between
the segments zj (s) (for 0 ≤ s ≤ sj ) and t1j (s) (for −p ≤ s ≤ s0j ) are not
larger than π2 . It is clear that it is easy to arrange that this condition be
fulfilled.
Denote by yj (s) (for −p ≤ s ≤ s, j = 0, 1, 2, . . .) the solution to the differential equation y 00 (s) + kj (s) = 0 [where kj (s) denotes de Gauss curvature of
the metric (1.1) at the point tj (s)] with the initial conditions yj (0) = 1 and
yj0 (0) = −1. Denote by yj 1 (s) (for −p ≤ s ≤ p, j = 0, 1, 2, . . .) the solution to
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the differential equation y 00 (s)+kj 1 (s) = 0 [kj (s) denotes de Gauss curvature
of the metric (1.1) at the point t1j (s); t10 (s) coincides with t0 (s)], with the
initial conditions yj 1 (0) = 1 and yj0 1 (0) = 1 (s00 is assumed zero). Denote by
qj (j = 0, 1, . . .) the largest numbers q < 0 such that yj (q) = 0, and denote
by qj0 the smallest of the numbers q > 0 such that yj (q) = 0. Denote by
qj 1 the largest of the numbers q < s0j such that yj 1 (0). By positivity of the
Gauss curvature, such numbers exist. If not, one would have that
p
p
p
(1.2)
− ≤ q01 < q0 < 0 < q00 ≤
and |qj | ≤
(j = 0, 1, . . .)
2
2
2
(1.3)
qj 1 → q01 ,
qj → q0 ,
qj0 → q00
when
j → ∞.
It follows from (1) and (2) that there exists a sequence tj (s) (0 ≤ s ≤ p) of
geodesic line segments such that:
(a) tj (0) = tj (qj );
(b) the angle ϕj between the segments tj (s) (0 ≤ s ≤ p) and tj (s)
(qj ≤ s ≤ p) tend to zero as j → ∞.
(c) for large enough j there exists s00j (0 ≤ s00j ≤ p) such that tj (s00j ) =
t1j (s0j ). It is obvious that s00j → −q0 when j → ∞.
It follows from (b) that for sufficiently large j and for s in the segment
− q20 ≤ s ≤ −q0 + q20 one has for the distance ρ(s) from the point tj (s) to
the geodesic tj (s) (−p ≤ s ≤ p) the relation
(1.4)
ρ(s) =
yj (qj + s)
ϕj + ϕ2j Mj (s),
yj0 (qj )
where the numbers Mj (s) and the derivatives Mj0 (s) are in absolute values
less than a fixed constant M .
But also for the angle ψj between the segments τj (s) (0 ≤ s ≤ s00j and
zj (s) (0 ≤ s ≤ sj ) one has the relation
(1.5)
sin ψj =
yj0 (qj + s00j )
ϕj + ϕ2j Mj0 (s00j ).
yj0 (qj )
Since the functions yj (s) are convex and positive in the interval qj < s <
y 0 (0)
1
< 0. As s00j → −q0 and qj → q0
one has yj0 (qj ) > 0 and y00(qj ) = − y0 (q
0
0 0)
when j → ∞, one has qj + s00j → 0 when j → ∞.
Thus for large enough j one has
qj0 ,
yj0 (qj + s00j )
1 y00 (0)
<
< 0.
yj0 (qj )
2 y00 (q0 )
But it also follows from (1.5) that for sufficiently large j one has sin ψj < 0.
Thus, for j large enough one has
π
(1.6)
ψj ≥ .
2
3
It follows from (4), and the fact that the angle between the segments zj (s)
(0 ≤ s ≤ sj ) and t1j (s) (qj ≤ s ≤ 0) is at most π2 , that for j large enough
q
+q
the segment t1j (s) ( 012 0 ≤ s ≤ s0j ) intersects either the segment tj (s)
(qj ≤ s ≤ 0) or τj (s) (0 ≤ s ≤ s00j ).
If as by the hypothesis the segments t1j (s) (−p ≤ s ≤ p) and tj (s) (−p ≤
s ≤ p) do not intersect, one sees that for j sufficiently large the segments
q +q
t1j (s) ( 012 0 ≤ s ≤ s0j ) and τj (s) (qj ≤ s ≤ 0) do not intersect.
But this contradicts that for j large enough one has by (1) and (2) the
inequality
q0 − q01
q01 + q0
− qj 1 >
.
2
2
The theorem is proven.
(Translated from French by Niels M. Møller.)
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