Bounded geodesics and Hausdorff dimension

Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all (p, v)∈ SM, where p ∈M and v is a unit tangent vector at p, the geodesic through p in the direction of v is dense in M. A theorem of Dani [Dl] says that the set of all (p, v)∈SM for which the corresponding geodesic is bounded (namely those with compact closure in M) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

On a Generalization of the First Curvature of a Curve in a Hypersurface of a Riemannian Space

The unit tangent vector at a point of a curve in a hypersurface of a Riemannian space has two derived vectors along the curve, one with respect to the Riemannian space in which the hypersurface is imbedded and one with respect to the hypersurface itself. When the former vector is decomposed along the directions normal and tangent to the hypersurface, its tangential component, which is called the first curvature vector of the curve at the point in the hypersurface, is exactly the latter vector.

On the concept of gravitational force and Gauss's theorem in general relativity

Whittaker and Ruse have developed forms of Gauss's theorem in general relativity, their theorems connecting integrals of normal force taken over a closed 2-space V2 with integrals involving the distribution of matter taken over an open 3-space bounded by V2. The definition of force employed by them involves the introduction of a normal congruence (with unit tangent vector λi), the “force” relative to the congruence being the negative of the first curvature vector of the congruence (– δλi/δs). This appears at first sight a natural enough definition, because – δλi/δs at an event P represents the acceleration relative to the congruence of a free particle travelling along a geodesic tangent to the congruence at P. In order to give physical meaning to this definition of force it is necessary to specify the congruence λi physically, and it would seem most natural to choose the congruence of world-lines of flow of the medium. Supposing certain conditions satisfied by this congruence (cf. Ruse, loc. cit.), the theory of Ruse is applicable, and from this follows a form of Gauss's theorem.