Reports of the Academy of Sciences of the USSR

1. Volume 120, No. 4

MATHEMATICS

V. A. TOPONOGOV

RIEMANNIAN SPACES OF CURVATURE BOUNDED BELOW BY A POSITIVE NUMBER

(Presented by Academician S. L. Sobolev on 11 XII 1957)

In our previous note \((^1)\), for \(m\)-dimensional twice continuously differentiable spaces \(R_k^m\) with a complete metric, whose curvature is bounded below by the number \(k\), the following theorem was formulated:

Theorem 1. In \(R_k^m\), the angles of an arbitrary triangle \(ABC\) composed of shortest arcs are not smaller than the corresponding angles of the triangle \((ABC)_k\) in the plane \(R_k\) of constant curvature \(k\) with the same side lengths.

For the case \(k=0\), the proof of Theorem 1 is given in \((^1)\). For the case when \(k<0\) or \(k>0\), but \(AB+AC+BC<2\pi/\sqrt{k}\), the proof of Theorem 1 is obtained by a simple modification of the proof for \(k=0\).

In the present article the case \(k>0\) is considered. In this case, for the triangle \(ABC\), two possibilities remain to be considered:

I. \(AB+AC+BC=2\pi/\sqrt{k}\).

II. \(AB+AC+BC>2\pi/\sqrt{k}\).

It is proved that in \(R_k^m\) \((k>0)\) case II is impossible; for case I, Theorem 1 is proved, which thereby is established in full. At the same time, a number of other geometric results are obtained.

Since Theorem 1 is valid for the case \(AB+AC+BC<2\pi/\sqrt{k}\), by passing to the limit we obtain:

Theorem 2. If in the space \(R_k^m\) there exists a triangle the sum of whose side lengths is equal to \(2\pi/\sqrt{k}\), and none of the sides is equal to the sum of the other two, then all angles of this triangle are equal to \(\pi\), so that the triangle is a closed shortest curve.

Theorem 3. In the space \(R_k^m\), the sum of the side lengths of a triangle does not exceed \(2\pi/\sqrt{k}\).

Proof. Suppose that the theorem is false. Then there exists a triangle \(ABC\) for which \(AB+AC+BC>2\pi/\sqrt{k}\). Let \(X(t)\) and \(Y(t)\) be variable points of \(AB\) and \(AC\) \((0\le t\le 1)\); there exists a set \(M\) of values of \(t\) for which
\[ AX(t)+AY(t)+X(t)Y(t)=2\pi/\sqrt{k}. \]

Two cases are possible:

I. There is at least one \(t=t_0\in M\) for which \(X(t_0)Y(t_0)<\pi/\sqrt{k}\).

II. For every \(t\in M\), \(X(t)Y(t)=\pi/\sqrt{k}\).

The first case is impossible by virtue of Theorem 2, the non-branching condition for shortest arcs, and our assumptions.

In the second case \(M\) contains only one value \(t=t_1\). Hence

for \(t>t_1\),
\[ AX(t)+AY(t)+X(t)Y(t)>2\pi/\sqrt{k}. \]
Using Theorem 1 for the case where the sum of the side lengths of the triangles is strictly less than \(2\pi/\sqrt{k}\), and the nonbranching condition for shortest arcs, by passage to the limit one can prove that, for \(t>t_1\), the lines \(AX(t)Y(t)A\) are closed geodesics. Since the closed geodesics \(AX(t)Y(t)A\) for \(t>t_1\) have the common part \(X(t_1)AY(t_1)\), they all coincide. In particular, \(AX(1)Y(1)A\) coincides with \(AX(t)Y(t)A\) for any \(t>t_1\). But for values \(t>t_1\) sufficiently close to \(t_1\), the length of \(AX(t)Y(t)A\) differs arbitrarily little from \(2\pi/\sqrt{k}\). Hence the length of the line \(ABCA\) is equal to \(2\pi/\sqrt{k}\). The contradiction obtained proves the theorem.

From Theorem 3 it follows easily:

Theorem 4. In the space \(R_k^m\) there exists no shortest arc of length greater than \(\pi/\sqrt{k}\).

Theorem 4, unlike the other theorems of this note, can also be proved by classical methods (with the aid of the Jacobi equations).

If in the space \(R_k^m\) \((k>0)\) a triangle \(ABC\) is given for which
\[ AB+AC+BC=2\pi/\sqrt{k} \]
and one of the sides is equal to \(\pi/\sqrt{k}\), then the corresponding triangle \((ABC)'_k\) of the plane of constant curvature \(k\) is not uniquely determined. In this case we agree to understand by the triangle \((ABC)'_k\) the degenerate triangle, i.e., the triangle in which two angles are equal to zero. With this convention in mind, one can easily derive Theorem 1 in full from Theorem 2.

Theorem 5. If in the space \(R_k^m\) there exists a shortest arc of length \(\pi/\sqrt{k}\), then \(R_k^m\) is the \(m\)-dimensional sphere of radius \(1/\sqrt{k}\).

Proof. Let the length of the shortest arc \(AB\) be \(\pi/\sqrt{k}\). From Theorems 3 and 2 it follows easily that any geodesic \(p\) passing through the point \(A\) contains \(B\) and is a closed geodesic of length \(2\pi/\sqrt{k}\).

Let \(X(t)\) be a variable point of \(p\), where \(t\) is the arc length, and let
\[ X(0)=A; \]
then
\[ X(\pi/\sqrt{k})=B \]
and
\[ X(t)=X(t+2\pi/\sqrt{k}) \quad (0\le t<\infty). \]

Using Sturm’s comparison theorem, one can prove that the curvature of the space \(R_k^m\) at any point of the geodesic \(p\) and in any two-dimensional direction tangent to \(p\) is equal to \(k\), and that the nearest point on \(p\) conjugate to \(X(t)\) is \(X(t+\pi/\sqrt{k})\).

Moreover, we shall now prove that the arc
\[ l=X(t)X(t+\pi/\sqrt{k}) \]
of the geodesic \(p\) is a shortest arc for every \(t\). Indeed, suppose that \(l\) is not a shortest arc. Then there is a greatest value \(t=t_0\), \(t<t_0<t+\pi/\sqrt{k}\), such that the arc
\[ l'=X(t)X(t_0) \]
of the geodesic \(p\) is a shortest arc. Surround the arc
\[ X(t-\varepsilon)X(t+\pi/\sqrt{k}-\varepsilon) \]
by the central field of extremals \(\mathfrak A\) with center at the point \(X(t-\varepsilon)\), where \(\varepsilon\) is a sufficiently small number. Then the shortest arcs
\[ X(t)X(t+s), \]
where \(s=t_0-t+\sigma\) and \(\sigma\) is sufficiently small, do not belong to the field of extremals \(\mathfrak A\). Therefore the limiting shortest arc
\[ \overline{X(t)X(t_0)} \]
of the shortest arcs
\[ X(t)X(t+s) \]
also does not belong to the field \(\mathfrak A\), and consequently does not coincide with the arc \(l'\).

On the other hand, applying Theorem 2 to the triangle
\[ \overline{X(t)X(t_0)}B \]
(or to the triangle \(X(t)X(t_0)A\)), we are convinced that the angle between \(\overline{X(t)X(t_0)}\) and \(X(t)B\) is equal to \(\pi\); therefore, by the nonbranching condition for shortest arcs, \(\overline{X(t)X(t_0)}\) coincides with \(l'\). Thus, \(l\) is a shortest arc. Applying the preceding considerations to the shortest arc \(l\) (replacing the points \(A\) and \(B\), respectively, by the points \(X(t)\) and \(X(t+\pi/\sqrt{k})\)), we prove that the curvature of the space \(R_k^m\) at any point and in any two-dimensional direction is equal to \(k\). Theorem 5 is proved, since it is easy to see that \(R_k^m\) is homeomorphic to a sphere.

Theorem 5 admits the following generalization.

Theorem 6. If in the space \(R_k^m\) there exists a triangle \(ABC\) the sum of whose side lengths is equal to \(2\pi/\sqrt{k}\), then \(R_k^m\) is an \(m\)-dimensional sphere of radius \(1/\sqrt{k}\).

Proof. By Theorem 5, it is enough to prove that in \(R_k^m\) there exists a shortest curve of length \(\pi/\sqrt{k}\). Therefore we may assume that none of the sides of the triangle \(ABC\) is equal to the sum of the other two. Take on the shortest curve \(BC\) a point \(D\) such that \(ABD = ACD = \pi/\sqrt{k}\). Join the point \(A\) to the point \(D\) by a shortest curve \(AD\). We shall prove that \(AD = \pi/\sqrt{k}\). Indeed, if this is not so, then the sums of the side lengths of the triangles \(ABD\) and \(ADC\) are less than \(2\pi/\sqrt{k}\). Hence it follows that

\[ \text{angle } A'B'D' < \pi;\qquad \text{angle } A'C'D' < \pi, \tag{1} \]

where \(A'B'D'\) is the angle of the triangle \((ABD)'_k\), and the angle \(A'C'D'\) is the angle of the triangle \((ACD)'_k\).

Construct the quadrilateral \(A'B'C'D'\), fitting together along equal sides the triangles \((ABD)'_k\) and \((ACD)'_k\). In the quadrilateral so obtained, the angle \(D'\), by Theorem 1, is not greater than \(\pi\); on the other hand, \(D'\) cannot be equal to \(\pi\), since then, by Theorem 2, all the remaining angles of the quadrilateral would also have to be equal to \(\pi\), contrary to (1). Thus the quadrilateral \(A'B'C'D'\) has three angles \(B'\), \(C'\), and \(D'\) less than \(\pi\); but from simple geometric considerations it follows easily that in the plane \(R_k\) there do not exist convex quadrilaterals the sum of whose side lengths is equal to \(2\pi/\sqrt{k}\) and in which three angles are less than \(\pi\). The contradiction obtained proves the theorem.

Institute of Radiophysics and Electronics
of the West-Siberian Branch
of the Academy of Sciences of the USSR

Received
29 XII 1957

CITED LITERATURE

1. V. A. Toponogov, DAN, 115, 672 (1957).