V. A. TOPONOGOV

THEOREMS ON SHORTEST PATHS IN NONCOMPACT RIEMANNIAN SPACES OF POSITIVE CURVATURE

(Presented by Academician A. D. Aleksandrov on 18 VIII 1969)

In the works of A. V. Pogorelov and W. Klingenberg, for simply connected compact Riemannian spaces (V^m) the following theorem was proved:

If in a Riemannian space (V^m) the Riemannian curvature (K) at every point and in every two-dimensional direction does not exceed (k_1>0) and is positive for (m=2p) (not less than (k_1/4) for (m=2p+1)), then every arc of a geodesic line of length not greater than (\pi/\sqrt{k_1}) is a shortest path in (V^m) ((^{2-4})).

For noncompact Riemannian spaces an analogous theorem had been obtained only for spaces with a pole ((^5)).

In the present work arbitrary complete noncompact Riemannian spaces are considered.

Denote by (R^m(R_0^m)) a complete (m)-dimensional twice continuously differentiable Riemannian space whose curvature at every point and in every two-dimensional direction is positive (nonnegative).

Theorem 1. If the Riemannian curvature of (R^m) at every point and in every two-dimensional direction does not exceed the number (k_1>0), then every arc of a geodesic line whose length does not exceed (\pi/\sqrt{k_1}) is a shortest path in (R^m).

Theorem 2. If the Riemannian curvature of (R_0^m) at every point and in every two-dimensional direction does not exceed some number (k_1>0), then there exists a number (\rho>0) such that every arc of a geodesic line of length not greater than (\rho) is a shortest path in (R_0^m).

In Theorem 2 the number (\rho) cannot be estimated in terms of (k_1), as follows from the example of W. Klingenberg ((^1,\ \text{p. }230)).

The proof of Theorems 1 and 2 rests essentially on the fundamental Lemmas 1 and (1^0).

1. We shall call a ray (p(A)) in the space (R^m(R_0^m)) with initial point (A) a half-geodesic issuing from (A), every arc of which is a shortest path in (R^m(R_0^m)).

Since (R^m(R_0^m)) is a complete and noncompact space, through every point of (R^m(R_0^m)) there passes at least one ray. The distance between points (A) and (B) will be denoted by (AB).

If (ABC) is a triangle in (R^m(R_0^m)) made up of shortest paths, then by (A'B'C') we shall denote the triangle in the Euclidean plane whose sides are equal to the corresponding sides of the triangle (ABC).

Let (AB) be a shortest path, (p_1(A)) a ray with initial point at (A), and (p_2(B)) a ray with initial point at (B) in (R^m). Denote by (\alpha) the angle between (p_1(A)) and the shortest path (AB), and by (\beta) the angle between (p_2(B)) and the shortest path (BA).

Lemma 1. If on the ray (p_1(A)) in the space (R^m) there exists a sequence of points (Q_n) such that the shortest paths (BQ_n) converge to (p_2(B)), and if neither of the rays (p_1(A)) and (p_2(B)) is a part of the other, then (\alpha+\beta>\pi).

Proof. Denote by (\beta_n) the angle in the triangle (ABQ_n) at the vertex (B), and by (\alpha'n) and (\beta'_n) the angles of the triangle (A'B'Q'_n) at the vertices (A') and (B'). Choose (n_0) so large that the triangle (ABQ) is a nondegenerate triangle. Take the ball (T) with center at the point (A) and radius
[
r=2(AQ_{n_0}+AB+BQ_{n_0}).
]
Denote by (k_0) the minimum of the Riemannian curvature over all points of the ball (T) and over all two-dimensional directions. Apply Theorems 1 and 6 of [6] to the triangle (ABQ_{n_0}); it follows from them that, first, the perimeter of the triangle does not exceed (2\pi/\sqrt{k_0}), and, second, the angle (\alpha) in the triangle (AQ_{n_0}B) is not less than the angle (\alpha''{n_0}) in the triangle (A''Q''B), at the vertex (A''). Comparing now the angle (\alpha''}B''), constructed on the two-dimensional sphere of radius (1/\sqrt{k_0}) with side lengths the same as those of the triangle (AQ_{n_0{n_0}) of the triangle (A''Q''B'') with the angle (\alpha'{n}) of the triangle (A'B'Q'_n), we obtain
[
\alpha \geq \alpha+\delta,
\tag{1}
]
where (\delta) is some positive number that can be estimated from below in terms of (k_0) and the side lengths of the triangle (ABQ_{n_0}). By Theorem 5 of [6],
[
\alpha'n \leq \alpha' n\geq n_0.}\quad \text{for
\tag{2}
]
Moreover, by Theorem 6 of [6],
[
\beta_n \geq \beta'n.
\tag{3}
]
Finally,
[
\lim(\alpha'n+\beta'_n)=\pi.
\tag{4}
]
From (1), (2), (3), and (4) it follows that
[
\alpha+\beta=\alpha+\lim\beta_n
\geq \lim_{n\to\infty}(\alpha'_n+\beta'_n)+\delta
=\pi+\delta>\pi,
]
as was required to prove.

Lemma (1^0). If on the ray (p_1(A)) of the space (R_0^m) there exists a sequence of points (Q_n) such that the shortest paths (BQ_n) converge to (p_2(B)), then (\alpha+\beta\geq\pi).

The proof of Lemma (1^0) is based on the same Theorems 1 and 5 of [6] and is much simpler than the proof of Lemma 1.

Lemma 2 follows from Lemma 1.

Lemma 2. In (R^m) there exists no closed geodesic line.

2. Proof of Theorems 1 and 2. Suppose that Theorems 1 and 2 are false. Then there exists a nondegenerate bigon (\gamma) of perimeter less than (2\pi/\sqrt{k_1}). Let (A) be one of the vertices of (\gamma). Denote by (\gamma(A)) a bigon of least length among all nondegenerate bigons with vertex at the point (A). By the usual methods of the calculus of variations (see, for example, [4] or [7]) one can show that (\gamma(A)) is a geodesic loop with vertex at the point (A). Let (T(R)) be the closed ball with center at the point (A) and radius (R). Denote by (\gamma(R)) ((\gamma_0(R))) the geodesic loop in (R^m) ((R_0^m)) of least length among all geodesic loops whose vertices lie in (T(R)), by (Q(R)) ((Q_0(R))) the vertex of (\gamma(R)) ((\gamma_0(R))), and by (f(R)) ((f_0(R))) the length of (\gamma(R)) ((\gamma_0(R))).

By Lemma 2, the function (f(R)) is strictly monotone, while the function (f_0(R)) tends to zero as (R\to\infty), by the assumption that Theorem 2 is false.

In both cases one can choose a sequence of numbers (R_n\to\infty) such that: 1) the shortest paths (AQ(R_n)) ((AQ_0(R_n))) converge to some ray (p(A)) ((p_0(A))); 2) the lower left derivatives of the function (f(R)) ((f_0(R))) at the points (R_n) are strictly negative.

We now choose (n_0) so large that the angle (\alpha) between (p(A)) ((p_0(A))) and (AQ(R_n)) ((AQ_0(R_n))) is less than (\pi/4):

[
\alpha < \pi/4 .
\tag{5}
]

Divide the geodesic (\gamma(R_n)) ((\gamma_0(R_n))) by points (A_1,A_2,\ldots,A_{k+1}=Q(R_{n_0})) in such a way that any segment (A_iA_{i+1}) ((i=1,\ldots,k)) of the loop (\gamma(R_n)) ((\gamma_0(R_n))) is a shortest curve in (R^m) ((R_0^m)). On the ray (p(A)) choose a sequence of points (P_m) such that, for any (i), the shortest curves (A_iP_m) as (m\to\infty) converge to certain rays (p_i(A_i)) ((p_i^0(A_i))). Extend the rays (p_i(A_i)) ((p_i^0(A_i))) by geodesics (\sigma_i) ((\sigma_i^0)) in the directions opposite to the directions of the rays (p_i) and (p_i^0). On each geodesic (\sigma_i) ((\sigma_i^0)), lay off from the points (A_i) segments of length (\delta); at the endpoints (B_i(\delta)) of these segments, as vertices, construct a polygon (\bar\gamma(\delta)) ((\bar\gamma_0(\delta))), whose perimeter we denote by (p(\delta)) ((p_0(\delta))).

By Lemma 1 (Lemma (1^0)), for sufficiently small (\delta),

[
p(\delta) < f(R_{n_0}) \qquad (p_0(\delta)=f_0(R_n)+o(\delta)).
]

Next, by the usual methods of the calculus of variations one can show the existence of a geodesic loop (\tilde\gamma(\delta)) ((\tilde\gamma_0(\delta))) with vertex at the point (B_1(\delta)), whose length does not exceed (p(\delta)) ((p_0(\delta))).

On the other hand, from Theorem 6 of [6] and (5) it follows that the angle between the shortest curve (Q(R_{n_0})A) and (\sigma_1) ((Q_0(R_{n_0})) and (\sigma_1^0)) is less than (\pi/4), and consequently, for sufficiently small (\delta), (A_1B_1(\delta)<AA_1). Hence (f(R)<f(R_{n_0})) ((f_0(R)\le f_0(R_{n_0})+o(\delta))) for (R_{n_0}-\delta\le R<R_{n_0}), whence, in turn, it follows that the left lower derived number for the function (f(R)) ((f_0(R))) at the point (R_{n_0}) is not less than zero, contrary to the choice of the number (n_0).

Thus Theorems 1 and 2 are proved.

Institute of Mathematics

Siberian Branch of the Academy of Sciences of the USSR

Novosibirsk

REFERENCES

(^{1}) D. Gromoll, W. Klingenberg, W. Meyer, Riemansche Geometrie im Grossen, Berlin, 1968.
(^{2}) W. Klingenberg, Ann. Math., 69, 654 (1959).
(^{3}) W. Klingenberg, Comm. Math. Helv., 35, 47 (1961).
(^{4}) I. A. Pogorelov, Mat. sborn., 18 (60), 181 (1946).
(^{5}) I. A. Sokolenko, DAN, 134, No. 5 (1960).
(^{6}) V. A. Toponogov, UMN, 14, issue 1 (85) (1959).
(^{7}) V. A. Toponogov, Sibirsk. matem. zhurn., 8, No. 5 (1967).