V. A. TOPONOGOV

AN ESTIMATE FOR THE LENGTH OF A CLOSED GEODESIC IN A COMPACT RIEMANNIAN SPACE OF POSITIVE CURVATURE

(Presented by Academician I. N. Vekua on July 22, 1963)

1. In this paper we shall consider an infinitely differentiable* complete simply connected Riemannian space \(R_{k_0}^{m}\), whose curvature at every point and in every two-dimensional direction does not exceed 1 and is not less than \(k_0\), where \(k_0\) is some number greater than zero.

The main result of the paper consists in the proof of the following theorem.

Theorem 1. The length of any closed geodesic in \(R_{k_0}^{m}\) is not less than \(2\pi\).

This theorem for \(m=2\) was proved by A. V. Pogorelov (see \((^1)\)). For even \(m=2n\) its proof was given by W. Klingenberg (see \((^{2,3})\)). The same author proved this theorem for \(m=2n+1\) and \(k_0>1/4\). Thus it remained to consider the case where \(m=2n+1\) and \(k_0\) is an arbitrary positive number. This case will be considered in the article.

By the usual arguments, using Morse’s theorem on the distance between conjugate points in \(R_{k_0}^{m}\) (see \((^{2,4})\)), one can show that Theorem 1 is equivalent to the following theorem:

Theorem 2. Every arc of a geodesic of the space \(R_{k_0}^{m}\) of length not greater than \(\pi\) is shortest.

2. Let two geodesics \(\gamma_0\) and \(\overline{\gamma}_0\) issue from the point \(A_0\), with endpoints at the points \(B_0\) and \(\overline{B}_0\) and lengths \(l_0\) and \(\overline{l}_0\); \(l_0<\pi\), \(\overline{l}_0<\pi\). Introduce along \(\gamma_0\) and \(\overline{\gamma}_0\) Fermi coordinate systems \(\{x^i\}\) and \(\{\overline{x}^i\}\) \((^5)\). Denote by \(A_k^i\) the transition matrix from the system \(\{\overline{x}^i\}\) to the system \(\{x^i\}\) at the point \(A_0\). Suppose that at the points \(B_0\) and \(\overline{B}_0\) vectors \(h\) and \(\overline{h}\) are given, whose coordinates are \(h^i\) and \(\overline{h}^{\,i}\), and at the point \(A_0\) a vector \(\lambda\) is given, whose coordinates in the system \(\{x^i\}\) are \(\lambda^i\), and in the system \(\{\overline{x}^i\}\) are \(\overline{\lambda}^{\,i}\). Let \(h_1=\lambda_1\), \(\overline{h}_1=\overline{\lambda}_1\). In the directions of the vectors \(\lambda\), \(h\), and \(\overline{h}\) draw geodesics and mark on them the points \(A_0(\delta)\), \(B_0(\delta)\), and \(\overline{B}_0(\delta)\) at distance \(\delta\) from the points \(A_0\), \(B_0\), and \(\overline{B}_0\). Join the point \(A_0(\delta)\) with \(B_0(\delta)\) and \(\overline{B}_0(\delta)\) by geodesics \(\gamma_0(\delta)\) and \(\overline{\gamma}_0(\delta)\).

Denote by \(\psi(\delta)\) the angle between \(\gamma_0(\delta)\) and \(\overline{\gamma}_0(\delta)\) at the point \(A_0(\delta)\). By \(\xi_j^i(x^1)\), \(\zeta_j^i(x^1)\) \(\bigl(\overline{\xi}_j^i(\overline{x}^1), \overline{\zeta}_j^i(\overline{x}^1)\bigr)\) denote the fundamental system of solutions of the Jacobi equations along \(\gamma_0\) \((\overline{\gamma}_0)\) with initial conditions

\[ \xi_j^i(0)=0,\quad \xi_j^{i\,\prime}(0)=\delta_j^i,\quad \zeta_j^i(0)=\delta_j^i,\quad \zeta_j^{i\,\prime}(0)=0. \]

Lemma 1.

\[ \left.\frac{d\psi}{d\delta}\right|_{\delta=0} = -\left(\sum_{i=2}^{m} A_1^i\beta^i + A_i^1\overline{\beta}^{\,i}\right)\big/ \sin \psi(0), \]

where \(\beta^i\) and \(\overline{\beta}^{\,i}\) satisfy the system of equations:

\[ \beta^i \xi_i^j(l_0)=h^j-\lambda^i \zeta_i^j(l_0), \]

\[ \overline{\beta}^{\,i}\overline{\xi}_i^j(\overline{l}_0) = \overline{h}^{\,j} - \overline{\lambda}^{\,i}\overline{\zeta}_i^j(\overline{l}_0), \qquad i,j=2,\ldots,m. \]

* The requirement of infinite differentiability is not essential; it suffices to require twice continuous differentiability of \(R_{k_0}^{m}\).

The proof of this lemma can be obtained by a direct calculation of the derivative \(\psi'(\delta)\), using the special properties of the metric tensor in the Fermi coordinate system.

Proof of Theorem 1. We shall briefly set out the idea of the proof of Theorem 1. Suppose that Theorem 1 is false. Then there exists a closed geodesic \(\gamma\) of length \(2l<2\pi\). We may assume that \(\gamma\) has the least length among all closed geodesics. Let \(l_0\) be such a number that \(3l_0<2l,\ 4l_0>2l\). Denote by \(\Gamma_\varepsilon\) the class of closed quadrilaterals defined by the following conditions: a quadrilateral \(m\in\Gamma_\varepsilon\) if: 1) the length of each side does not exceed \(l_0\); 2) the perimeter of \(m\) does not exceed \(2l+\varepsilon\); 3) there exists a family \(H_m(t)\) of closed piecewise smooth curves such that \(H_m(0)=\gamma;\ H_m(1)=m;\ l(H_m(t))<2l+\varepsilon\). Here \(\varepsilon>0\) is an arbitrary number, and \(l(H_m(t))\) is the length of the curve \(H_m(t)\).

a) We shall prove that in \(\Gamma_\varepsilon\) there exists a quadrilateral (possibly degenerate) whose perimeter does not exceed \(3l_0<2l\). Denote by \(\psi(m)\) the smallest angle of \(m\). Let
\[ \psi_0=\inf_{m\in\Gamma_\varepsilon}\psi(m). \]
Construct a sequence \(m_k\in\Gamma_\varepsilon\) such that
\[ \lim_{k\to\infty}\psi(m_k)=\psi_0. \]
Let \(m_0\) be a limiting quadrilateral for the sequence \(m_k\).

Two cases are possible:

I. \(m_0\) is a degenerate quadrilateral. Then, by virtue of condition 1) in the definition of \(\Gamma_\varepsilon\), the assertion of part a) is valid.

II. \(\psi(m_0)=\psi_0\). We shall prove that in this case \(\psi_0=0\). Thereby in this case also we shall prove the assertion of part a). Suppose that \(\psi_0\ne0\). Let \(A_0,A_1,A_2\), and \(A_3\) \((A_4=A_0)\) be the vertices of \(m_0\), and let \(\psi_0\) be the angle at the vertex \(A_0\). Denote the side \(A_iA_{i+1}\) by \(\gamma_i\), and by \(\tau_i^+,\tau_i^-\) the unit vectors at the points \(A_i\), tangent to \(\gamma_i\). Consider the set \(N\) of all unit vectors at the point \(A_0\). Transport each vector of the set \(N\) parallel along \(m_0\). By the simple connectedness of \(R_{k_0}^{2n+1}\), the resulting mapping \(f\) of the set \(N\) onto itself is homotopic to the identity; and since, moreover, \(N\) is homeomorphic to the sphere \(S^{2n}\) of dimension \(2n\), there exists a fixed point of the mapping \(f\) ((6), p. 604). In the language of parallel translation this means that at the point \(A_0\) there exists a vector \(\lambda_0\) which, under parallel translation along \(m_0\), passes into itself. Let \(\lambda_i\) be the vector at the point \(A_i\) obtained from \(\lambda_0\) by parallel translation along \(m_0\). Define at the point \(A_1\) \((A_3)\) the set \(N_1\) \((N_3)\) of unit vectors by the conditions: \(h\in N_1\) \((\bar h\in N_1)\), if: 1) \((h,-\tau_1^-)=(\lambda_0,\tau_0^+)\) \((\bar h,\tau_3^+)=(\lambda_0,-\tau_0^-)\); 2) \((h,\lambda_1)\ge 1-\varepsilon_1\) \(((\bar h,\lambda_3)\ge 1-\varepsilon_1)\), where \((a,b)\) is the scalar product of the vectors \(a\) and \(b\), and \(\varepsilon_1>0\) is an arbitrary number.

Take two vectors \(h_1\in N_1\) and \(h_3\in N_3\). At the point \(A_2\) find a vector \(h_2\) such that
\[ (h_1,\tau_1^+)=(h_2,-\tau_2^-),\qquad (h_3,-\tau_3^-)=(h_2,\tau_2^+). \]
In the directions of the vectors \(h_0=\lambda_0,h_1,h_2\), and \(h_3\), draw geodesics and take on them the points \(A_i(\delta)\) at distance \(\delta\) from the points \(A_i\) (\(\delta>0\), if \(A_i(\delta)\) lies from \(A_i\) in the direction of the vectors \(h_i\), and \(\delta<0\) in the opposite case). Joining the points \(A_i(\delta)\) and \(A_{i+1}(\delta)\) by shortest arcs, we obtain a quadrilateral \(m(\delta,h_i)\).

Introduce along \(\gamma_0\) \((\gamma_3)\) a Fermi coordinate system \(\{x^i\}\) \((\{\bar x^i\})\). Let \(h^i\) and \(\bar h^i\) be the coordinates of the vectors \(h_1\) and \(h_3\). Then, as follows from Lemma 1,
\[ \left.\frac{d\psi(m(\delta,h_i))}{d\delta}\right|_{\delta=0} = -\frac{\sum A_1^i\beta^i+A_1^i\bar\beta^i}{\sin\psi_0}, \tag{1} \]
where \(\beta^i\) and \(\bar\beta^i\) satisfy the system of equations
\[ \begin{aligned} \beta^j\xi_j^i&=h^i-\lambda^j\xi_j^i,\\ \bar\beta^j\bar\xi_j^i&=\bar h^i-\bar\lambda^j\bar\xi_j^i, \end{aligned} \qquad i,j=2,\ldots,2n. \tag{2} \]

If, for at least one \(i\), \((\tau_i^+, \tau_j^-, \lambda_i) \ne 0\), then, using (1) and (2), one can prove the existence of vectors \(h_1 \in N_1\) and \(h_3 \in N_3\) such that \(\psi'(0) \ne 0\). For definiteness let \(\psi'(0) < 0\). Then, for sufficiently small \(\delta\) \((\delta > 0)\), in the quadrilateral \(m(\delta, h_i)\) the angle at the vertex \(A_0(\delta)\) is less than \(\psi_0\). If \(\varepsilon_1\) is taken sufficiently small, then, with the aid of Synge’s lemma \((^7)\), one can prove that \(m(\delta, h_i) \in \Gamma_\varepsilon\).

The contradiction obtained in this case proves the assertions of part a). If, however, \((\tau_i^+, \tau_i^-, \lambda_i)=0\) for all \(i\), then one can prove that the plane \(\tau_0^+ \wedge \tau_0^-\), determined by the vectors \(\tau_0^+\) and \(\tau_0^-\), is carried into itself under parallel displacement along \(m_0\). But then the set of all unit vectors at the point \(A_0\) orthogonal to the plane \(\tau_0^+ \wedge \tau_0^-\) is also carried into itself under parallel displacement along \(m_0\). Using the latter fact and the one-connectedness of \(R_{k_0}^{2n+1}\), one can, as was done at the beginning of part a), prove the existence of a vector \(\mu_0\) orthogonal to the plane \(\tau_0^+ \wedge \tau_0^-\) and carried into itself under parallel displacement along \(m_0\). Thus this case is reduced to the preceding one.

b) In view of the arbitrariness of \(\varepsilon\), from the assertion of part a) one can derive the existence of such a family \(\overline H(t)\) of closed piecewise-smooth curves that \(\overline H(0)=\gamma\), \(\overline H(1)=\gamma_1\), \(l(\gamma_1)\le 3l_0<2l\), and \(l(\overline H(t))\le 2l\) for any \(t\). Apply to the family \(\overline H(t)\) the Morse deformation (see, for example, \((^2)\), p. 71). The family \(H(t)\) thereby obtained will again have the same properties as \(\overline H(t)\), and, in addition, if for some \(t\), \(l(H(t))=2l\), then the curve \(H(t)\) is a closed geodesic of length \(2l\).

Let \(t_0\) be the greatest value among all \(t\) for which \(l(H(t))=2l\). For each curve \(H(t)\), for \(t\ge t_0\), construct a surface \(F(t)\) of minimal area with boundary \(H(t)\). Since \(l(H(t))<2l\) for \(t>t_0\), the theorem on the isoperimetric inequality \((^9)\) can be applied to them. Hence it follows that the area of the surface \(F(t)\)

\[ S(t)<2\pi \quad \text{for } t>t_0 . \tag{3} \]

The surface \(F(t_0)\) is a minimal surface; therefore its relative curvature is nonpositive, and consequently its Gaussian curvature does not exceed the maximum of the Riemannian curvature of \(R_{k_0}^{2n+1}\), i.e. 1. Let us now apply the Gauss–Bonnet theorem to \(F(t_0)\). It follows from it that the total curvature of \(F(t_0)\) is equal to \(2\pi\), since \(H(t_0)\) is a closed geodesic. But, on the other hand, the total curvature of \(F(t_0)\) does not exceed \(S(t_0)\cdot 1\), which, by virtue of (3), is strictly less than \(2\pi\). The contradiction obtained proves Theorem 1, and together with it Theorem 2.

Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR

Received
12 VII 1963

CITED LITERATURE

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2. W. Klingenberg, Ann. Math., 69, 654 (1959).
3. W. Klingenberg, Comm. Math. Helv., 35, 47 (1961).
4. M. Morse, The Calculus of Variations in the Large, 1934.
5. É. Cartan, Geometry of Riemannian Spaces, Moscow–Leningrad, 1936.
6. P. S. Aleksandrov, Combinatorial Topology, 1947.
7. I. L. Synge, Proc. London Math. Soc., 25, 274 (1926).
8. G. Seifert, W. Threlfall, Variationsrechnung im Grossen, 1947.
9. A. D. Aleksandrov, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 38, 5 (1951).