MATHEMATICS

V. I. ARNOLD

SEVERAL REMARKS ON FLOWS OF LINEAR ELEMENTS AND FRAMES

(Presented by Academician A. N. Kolmogorov, 9 XII 1960)

Many problems of mechanics lead, as is well known, to geodesic flows (see \([1–4]\)). In the present note more general dynamical systems in the spaces of linear elements and frames of a Riemannian manifold are defined—isotropic flows. These include the flow associated with curves of constant geodesic curvature and with the motion of a charged point on a smooth surface in the presence of a magnetic field.

§ 1. Let \(M\) be an \(n\)-dimensional Riemannian manifold. A \(k\)-frame, or \(k\)-hedron, on \(M\) is a pair \(\omega=(x,\xi^k)\), where \(x\in M\) is the carrier, and \(\xi^k\) is an ordered set \((\xi_1,\ldots,\xi_k)\) of pairwise orthogonal unit vectors of the tangent space to \(M\) at the point \(x\). Frames with a common carrier form a homogeneous space \(\Xi_k\), and all frames on \(M\) form the space \(\Omega_k\). We define the volume element in the space \(\Omega_k\) by the formula \(d\Omega=dM\,d\Xi\), where \(dM\) is the volume element on \(M\), and \(d\Xi\) determines the invariant measure in the space \(\Xi_k\).

By a flow of \(k\)-hedra we shall mean a one-parameter group \(S^t\) of transformations of the space \(\Omega_k\): \(\omega\to S^t\omega\). The trajectories of the flow will be the lines \(\Gamma\) on \(M\) formed by the carriers \(x(t)\) of the elements \(S^t\omega\). By a tangent flow we shall mean such a flow of \(n\)-hedra for which the frame \(S^t\omega\) is the accompanying frame of the trajectory \(x(t)\). Denote by \(v\) the speed of motion of \(x(t)\) along \(\Gamma\), and by \(k_1,\ldots,k_{n-1}\) the curvatures of \(\Gamma\). Clearly, a tangent flow is determined by specifying the functions \(v(\omega)\), \(k_1(\omega),\ldots,k_{n-1}(\omega)\) on \(\Omega_n\).

Definition. An isotropic flow is a tangent flow for which the speed \(v\) is constant, while the curvatures \(k_1(x),\ldots,\ldots,k_{n-1}(x)\) depend only on the carrier \(x\) and do not depend on the directions of the vectors \(\xi_1,\ldots,\xi_n\) of the frame \(\omega\).

In particular, the geodesic flow is isotropic: \(v\equiv 1,\ k_i\equiv 0\).

§ 2. It turns out that an isotropic flow is a dynamical system with an integral invariant. As is known, the measure \(d\Omega\) is an invariant measure of the geodesic flow.

Theorem 1. The transformations \(S^t\) of any isotropic flow preserve the measure \(d\Omega\).

The proof is based on the fact that the infinitesimal transformation \(S^{dt}\) consists of an infinitesimal transformation of the geodesic flow and an infinitesimal rotation.

For \(n=2\), an isotropic flow is determined by the geodesic curvature \(k(x)\) of the trajectory passing through the point \(x\) of the surface \(M\). Such a flow is isomorphic to motion at fixed energy in a dynamical system with Lagrangian function \(L_2+L_1\), containing terms quadratic and linear in the velocities. In this case Theorem 1 follows from Liouville’s theorem. Eliminating the cyclic coordinate, one can easily

to investigate the course of curves of constant curvature on surfaces of revolution. These curves were studied by Minding \((^{5})\) and Darboux \((^{6})\).

§ 3. Also readily amenable to study are cyclic flows, in which \(k_1,\ldots,k_{n-1}\) are constant, on manifolds of constant curvature \(K\). Let us consider the Lobachevskii plane \((n=2,\ K=-1)\). A standard device makes it possible to pass to any surface of constant negative curvature \((^{2})\).

Cycles of curvature \(k\) on the Lobachevskii plane are divided into three classes: cycles proper \((k^2>1)\), horicycles \((k^2=1)\), and hypercycles \((k^2<1)\). In the Poincaré model these are, respectively, circles not intersecting the absolute, tangent to it, and intersecting the absolute at an angle \(\alpha\); \(\cos\alpha=k\).

Theorem 2. Every hypercyclic \((k^2+K<0)\) flow on a surface of constant negative curvature \(K\) is isomorphic to a geodesic flow.

The proof is based on the fact that a hypercycle of curvature \(k\) is an equidistant curve \(\Gamma_\tau\), i.e. it is separated from some geodesic \(\Gamma\) by a constant distance \(\tau\) (where \(k=-\sqrt{-K}\operatorname{th}\sqrt{-K}\tau\)). With the equidistants \(\Gamma_\tau\) on any surface one can associate a tangent flow isomorphic to the geodesic flow. On a surface of constant curvature the flow of equidistants \(\Gamma_\tau\) is cyclic.

§ 4. Analogous considerations in Lobachevskii space make it possible to prove the following:

Theorem 3. Every cyclic flow on an \(n\)-dimensional manifold of constant negative curvature \(-1\) belongs to one of the following three types:*

1. The flow is isomorphic to the generalized geodesic flow \((k_1=0)\).
2. The flow is isomorphic to the generalized horicyclic flow \((k_1=1,\ k_2=0)\).
3. The flow is isomorphic to one for which the carrier \(S^t\omega\) is fixed.

The usual geodesic flow belongs to flows of the first type (for it \(\nu=1,\ k_1=k_2=\ldots=k_{n-1}=0\)); all these flows are very similar to it (see Theorem 5). Among flows of the second type the ordinary horicyclic flow plays the same role \((\nu=k_1=1;\ k_2=\ldots=k_{n-1}=0)\). Flows of the third type are not ergodic; the ergodic components are tori of dimension \(r\); generally speaking, \(r=[n/2]\); on each torus the flow has a discrete spectrum with \(r\) generators.

The question of which type a given flow belongs to is decided as follows:

Theorem 4. In a \(2r\)-dimensional space of curvature \(-1\), a cyclic flow belongs to type 1, 2, or 3 according as \(k_1^2\) is less than, equal to, or greater than

\[ \chi^2 = 1+\frac{k_2^2}{k_3^2} +\frac{k_2^2k_4^2}{k_3^2k_5^2} +\cdots+ \frac{k_2^2k_4^2\cdots k_{2r-2}^2}{k_3^2k_5^2\cdots k_{2r-1}^2}. \]

In a \((2r+1)\)-dimensional space all flows with \(k_{2r}\ne0\) are of type 1, while for \(k_{2r}=0\) the flow is of type 1, 2, or 3 according as \(k_1^2\) is less than, equal to, or greater than \(\chi^2\).

§ 5. The methods of recent works of Ya. G. Sinai \((^{4,7})\), devoted to the ordinary geodesic flow, are applicable also to the generalized one.

Theorem 5. A flow \(S^t\) of type 1 on a compact manifold of constant negative curvature is a \(K\)-system \((^{4,7})\). The horispherical flow \((^{4,7})\) is conjugate to \(S^t\).

One can calculate the entropy \((^{8})\) of the flow \(S^t\). Denote by \(h(k_1,\ldots,k_{n-1})\) the entropy per unit time of the cyclic flow with velocity 1 and

* In each two-dimensional direction.

curvatures \(k_1,\ldots,k_{n-1}\) on a manifold of constant curvature \(-1\). Then

\[ h(k_1,\ldots,k_{n-1})=h(0)v, \]

where \(h(0)=h(0,\ldots,0)\) is the entropy of the geodesic flow, computed in \((^4)\), and \(v(k_1,k_2,\ldots,k_{n-1})\) is the speed of motion in the generalized geodesic flow isomorphic to \(S^t\).

In particular, for \(n=2\) we have

\[ h(k)=h(0)\sin\alpha=h(0)\sqrt{1-k^2}, \]

and for \(n=3\) the number \(v^2=x\) is the positive root of the equation

\[ x^2+(k_1^2+k_2^2-1)x+k_2^2=0. \]

In the case of a surface \((n=2)\), when \(k^2\to1\), then \(h\to0\). This suggests that the entropy of the horocyclic flow \(h(1)\) is zero. The latter was proved by B. M. Gurevich \((^9)\). Probably the entropy of any flow of type 2 is zero.

§ 6. The spectra of the geodesic and horocyclic flows were first found \((^3,^10)\) on the basis of an algebraic construction of I. M. Gelfand and S. V. Fomin; this construction depends on a certain group \(G\) and its subgroups: a compact subgroup \(K\), a discrete subgroup \(D\), and a one-parameter subgroup \(g^t\).

Theorem 6. If the group \(G\) is the group of motions of \(n\)-dimensional Lobachevsky space and \(K\) is the rotation group of an \((n-k)\)-dimensional Euclidean space, then the corresponding dynamical system is isomorphic to one of the cyclic flows of \(k\)-frames on a manifold of constant negative curvature; conversely, all such flows can be obtained in this way.

The proof is based on the connection between curves of constant curvature and one-parameter groups of motions; conjugate groups correspond to isomorphic systems.

§ 7. An interesting topological characteristic of a dynamical system is the group of rotation numbers \((^{11},^{12})\). I. M. Gelfand and I. I. Pyatetskii-Shapiro computed the rotation numbers of the geodesic and horocyclic flows on a surface of constant negative curvature \((^{13})\).

Theorem 7. If an isotropic flow on a Riemannian manifold distinct from the two-dimensional torus and from the Klein bottle is ergodic, then all rotation numbers are equal to zero and the flow has no nonconstant continuous eigenfunctions.

The proof is based on the fact that the trajectory of an ergodic isotropic flow passes equally often through each point in every direction.

Moscow State University
named after M. V. Lomonosov

Received
26 XI 1960

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