A flow in the space of bihedra of an $n$-dimensional Riemannian manifold $M^n$ (usually closed) for which the concept of a horocycle is defined; the horocycle flow describes the movement of the bihedra along the horocycles which they define.
The basic cases in which the concept of a horocycle is defined are those in which the curvature of the Riemannian metric is negative, and either $n=2$ or the curve is constant. With a bihedron, i.e. an orthonormal $2$-frame $(x,e_1,e_2)$ ($x\in M^n$; $e_1,e_2$ are mutually orthogonal unit tangent vectors at the point $x$) is associated the horocycle $h(x,e_1,e_2)$ through $x$ in the direction of $e_2$. It is situated on the horosphere $H(x,e_1)$ through $x$, the $(n-1)$-dimensional manifold orthogonal to the family of geodesic lines, asymptotic (in the positive direction) to the geodesic line which passes through $x$ in the direction of $e_1$. The direction on $h$ defined by $e_2$ is taken to be positive (in case $n=2$ this is the only role of $e_2$; $H$ and $h$ can have self-intersections; the simplest way to avoid the ambiguity which can arise from this is to carry out analogous constructions not in $M^n$, but in its universal covering manifold — when the curvature is constant, this is the ordinary $n$-dimensional Lobachevskii space — and to project the horocycle obtained there into $M^n$). Under the action of a horocycle flow, the bihedron $(x,e_1,e_2)$ during time $t$ passes to
where $x(t)$, when $t$ increases, moves at unit velocity along $h(x,e_1,e_2)$ in the positive direction, the unit vector $e_1(t)$ is orthogonal to $H(x,e_1)$ at the point $x(t)$ (the choice of one of the two possible directions for $e_1(t)$ is made by continuity) and $e_2(t)=dx(t)/dt$.
Horocycle flows were studied because they played an important role in the research of geodesic flows (cf. Geodesic flow) on manifolds of negative curvature . Now this role is played by certain foliations (cf. Foliation) arising from the theory of $Y$-systems (cf. $Y$-system), and horocycle flows became a research subject in its own right. The properties of a horocycle flow have been well established (see –, ). For various generalizations, see –.
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Horocycle flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Horocycle_flow&oldid=32463