Mathematics

V. A. TOPONOGOV

EXTREMAL THEOREMS FOR RIEMANNIAN SPACES OF CURVATURE BOUNDED ABOVE

(Presented by Academician A. D. Aleksandrov, 12 V 1968)

Let \(R^m(k_0,k_1)\) \(\bigl(R_{k_1}^m\bigr)\) be an \(m\)-dimensional compact, four-times continuously differentiable, simply connected Riemannian space whose Riemannian curvature \(K\) at every point and in every two-dimensional direction satisfies the inequalities \(k_0 < K \le k_1\) \((K \le k_1)\). In addition, consider the class of Riemannian spaces \(R_{k_1}^m(a)\), defined by the conditions:

(1) \(R_{k_1}^m(a)\) is a Riemannian space \(R_{k_1}^m\) for \(k_1 > 0\);

(2) the length of any closed geodesic in \(R_{k_1}^m(a)\) is not less than \(2\pi/\sqrt{k_1}\).

The class of spaces \(R_{k_1}^m(a)\) is nonempty.

As follows from \((1\text{–}3)\), the spaces \(R^m(0,k_1)\) for \(m=2p\) and the spaces \(R^m(1/4\,k_1,k_1)\) for \(m=2p+1\) belong to the class \(R_{k_1}^m(a)\).*

We note that condition (2) is equivalent to conditions \((2^*)\) and \((2^{**})\):

\((2^*)\) The perimeter of any nondegenerate polygon in \(R_{k_1}^m(a)\), composed of geodesics, is not less than \(2\pi/\sqrt{k_1}\).

\((2^{**})\) Any arc of a geodesic in \(R_{k_1}^m(a)\) whose length does not exceed \(\pi/\sqrt{k_1}\) is shortest.

In this paper the extremal cases of conditions (2), \((2^*)\), and \((2^{**})\) are considered.

Theorem 1. If in \(R_{k_1}^m(a)\) the index of a closed geodesic \(\gamma\) of length \(2\pi/\sqrt{k_1}\) is equal to \(q\), then in \(R_{k_1}^m(a)\) there exists a \((q+1)\)-dimensional totally geodesic surface, isometric to a \((q+1)\)-dimensional sphere of radius \(1/\sqrt{k_1}\), containing \(\gamma\).

Theorem 2. If in \(R_{k_1}^m(a)\) there exists a nondegenerate polygon \(\gamma\), composed of geodesics, with perimeter \(2\pi/\sqrt{k_1}\), then in \(R_{k_1}^m(a)\) there exists a two-dimensional totally geodesic surface, isometric to some polygon on a sphere of radius \(1/\sqrt{k_1}\), whose boundary coincides with \(\gamma\).

Theorem 3. If the diameter of \(R_{k_1}^m(a)\) is equal to \(\pi/\sqrt{k_1}\), then there exists a number \(q\), equal either to 2, or to 4, or to 8, or to \(m\), such that for every point \(P \in R_{k_1}^m(a)\) and every vector \(\lambda\) at the point \(P\) there exists a \(q\)-dimensional totally geodesic surface \(F(P,\lambda)\), isometric to a \(q\)-dimensional sphere of radius \(1/\sqrt{k_1}\), passing through the point \(P\), whose tangent plane contains \(\lambda\). Two surfaces \(F(P,\lambda_1)\) and \(F(P,\lambda_2)\) either coincide or have no common points except \(P\).

From Theorem 1, since the index of any closed geodesic in \(R^{2p}(0,k_1)\) is not less than \(1\) \((^2)\), it follows

Theorem 4. If in \(R^{2p}(0,k_1)\) there exists a closed geodesic \(\gamma\) of length \(2\pi/\sqrt{k_1}\), then in \(R^{2p}(0,k_1)\) there exists a two-dimensional totally geodesic—

* The result announced by the author of this article in \((^4)\), asserting that \(R^m(0,k_1)\) for \(m=2p+1\) belongs to the class \(R_{k_1}^m(a)\), is incorrect. At the International Congress of Mathematicians in Moscow (1966), V. Klingenberg communicated to the author an example refuting the announced result.

surface isometric to a two-dimensional sphere of radius \(1/\sqrt{k_1}\), containing \(\gamma\).

Theorem 4 for \(m=2\) was proved by W. Klingenberg in \((^5)\).

Further, from Morse’s comparison theorem one can obtain that the index of any closed geodesic in \(R^m(1/4 k_1, k_1)\) is not less than \((m-1)\) \((^6)\). Therefore, from Theorem 1 it follows that

Theorem 5. If in \(R^m(1/4 k_1, k_1)\) there exists a closed geodesic of length \(2\pi/\sqrt{k_1}\), then \(R^m(1/4 k_1, k_1)\) is isometric to an \(m\)-dimensional sphere of radius \(1/\sqrt{k_1}\).

Theorem 5 was previously proved by the author and, independently, by Tsukamoto \((^7,^8)\).

Finally, using the last assertion of Theorem 3, one can prove

Theorem 6. If the diameter of the space \(R^m_{k_1}(a)\), homeomorphic to a sphere, is equal to \(\pi/\sqrt{k_1}\), then \(R^m_{k_1}(a)\) is isometric to an \(m\)-dimensional sphere of radius \(1/\sqrt{k_1}\).

In fact, the Riemannian spaces satisfying the conditions of Theorem 3 apparently must be isometric to symmetric spaces of rank 1, but the author has not succeeded in proving this assertion for all dimensions, only for \(m=3,4,5\).

The following assertion, proved by M. Berger \((^9)\), also speaks in favor of our hypothesis.

If the diameter of \(R^m(1/4 k_1, k_1)\) is equal to \(\pi/\sqrt{k_1}\), then \(R^m(1/4 k_1, k_1)\) is isometric to a symmetric space of rank 1.

Let us indicate the idea of the proof of Theorem 1.

Lemma 1. Under the conditions and notation of Theorem 1, any two points \(P\) and \(Q\) of the geodesic \(\gamma\), separated from each other along \(\gamma\) by the distance \(\pi/\sqrt{k_1}\), are conjugate to each other with multiplicity \(q\).

The proof of Lemma 1 is obtained from condition \((2^*)\) and from the known lemmas of the calculus of variations.

From Lemma 1 it is already not difficult to obtain, by induction,

Lemma 2. Under the conditions and notation of Theorem 1, there exists an arc \(\sigma\) of the geodesic \(\gamma\) of length greater than \(\pi/\sqrt{k_1}\) and a \(q\)-parameter family of parallel vector fields \(\nu\) along \(\sigma\) such that the Riemannian curvature in the two-dimensional directions containing \(\dot\gamma\) and \(\nu\) is equal to \(k_1\).

Now using Lemma 2, for each field \(\nu\) one can construct a sequence of triangles \(\Delta_n(\nu)\) converging to \(\gamma\) and such that, for any \(n\), the perimeter of \(\Delta_n(\nu)\) is strictly less than \(2\pi/\sqrt{k_1}\). Hence, and from condition \((2^{**})\), it follows that over each triangle \(\Delta_n(\nu)\) one can stretch a cone \(K_n(\nu)\) (the cone \(K_n(\nu)\) is obtained as the set of shortest curves joining one of the vertices of \(\Delta_n(\nu)\) with the points of the opposite side). For the cones \(K_n(\nu)\) the following lemmas are true.

Lemma 3. The Gaussian curvature of \(K_n(\nu)\) at each point does not exceed \(k_1\).

Lemma 3 follows from Synge’s lemma \((^{10})\).

Lemma 4. The area of the cone \(K_n(\nu)\) does not exceed the area of the triangle \(\Delta_n^L(\nu)\), constructed on a sphere of radius \(1/\sqrt{k_1}\) with the same side lengths as the triangle \(\Delta_n(\nu)\).

Lemma 4 follows from a theorem of A. D. Aleksandrov \((^{11})\). From Lemmas 3 and 4 one obtains an upper estimate for the integral curvature of the cone \(K_n(\nu)\). On the other hand, from the Gauss–Bonnet theorem one can obtain a lower estimate for the integral curvature of \(K_n(\nu)\) in terms of the angles of \(\Delta_n(\nu)\). Comparing these estimates, one obtains that the Gaussian curvature of \(K_n(\nu)\) is everywhere almost equal to \(k_1\), and the area of \(K_n(\nu)\) is almost equal to \(2\pi/\sqrt{k_1}\). Passing now to the limit as \(n\to\infty\), we obtain that there exists a \(q\)-parameter family of surfaces \(F\), isometric to a two-dimensional hemisphere of radius \(1/\sqrt{k_1}\), whose boundary coincides with \(\gamma\).

Now it is no longer difficult to show that the totality of all surfaces of this family forms a \((q+1)\)-dimensional surface \(F^{q+1}\), isometric

\((q+1)\)-dimensional sphere of radius \(1/\sqrt{k_1}\). Finally, from \((2^{**})\) and the preceding it is not difficult to obtain that \(F^{q+1}\) is a totally geodesic surface in \(R_{k_1}^n(a)\).

The proofs of Theorems 2 and 3 are obtained by applying arguments analogous to those given above; only in the proof of Theorem 3 it is necessary to use certain topological considerations, in particular, the theorem of F. Browder \((^{12})\).

REFERENCES

1. A. V. Pogorelov, Matem. sborn., 18 (60), 181 (1964).
2. W. Klingenberg, Ann. Math., 69, 654 (1959).
3. W. Klingenberg, Comm. Math. Helv., 34, 47 (1961).
4. V. A. Toponogov, DAN, 154, No. 5, 1047 (1964).
5. W. Klingenberg, Comm. Math. Helv., 34, f. 1 (1960).
6. M. Morse, The Calc. of Variat. in the Large, 1934.
7. I. Tsukamoto, Tohoku Math. J., 18, No. 2, 138 (1966).
8. V. A. Toponogov, Sibirsk. matem. zhurn., 8, No. 5 (1967).
9. M. Berger, Ann. Scuola Norm. Sup. Pisa, Ser. III, 14, f. II (1960).
10. J. L. Synge, Proc. London Math. Soc., 25, 274 (1926).
11. A. D. Aleksandrov, Akad. wiss. Forsch. Math., No. 1, 33 (1957).
12. W. Browder, Bull. Am. Math. Soc., 68, No. 3, 202 (1962).