"[...] the illustration of a space of constant positive measure of curvature by the familiar example of the sphere is somewhat misleading. Owing to the fact that on the sphere the geodesic lines (great circles) issuing from any point all meet again in another definite point, antipodal, so to speak, to the original point, the existence of such an antipodal point has sometimes been regarded as a necessary consequence of the assumption of a constant positive curvature. The projective theory of non-Euclidean space shows immediately that the existence of an antipodal point, though compatible with the nature of an elliptic space, is not necessary, but that two geodesic lines in such a space may intersect in one point if at all." (Felix Klein, "The Most Recent Researches in Non-Euclidian Geometry", [lecture] 1893)
"Imagine the forehead of a bull, with the protuberances from which the horns and ears start, and with the collars hollowed out between these protuberances; but elongate these horns and ears without limit so that they extend to infinity; then you will have one of the surfaces we wish to study. On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears. Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear. [...] If, therefore, a material point is thrown on the surface studied starting from a geometrically given position with a geometrically given velocity, mathematical deduction can determine the trajectory of this point and tell whether this path goes to infinity or not. But, for the physicist, this deduction is forever useless. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered."
"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947)
"Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur.." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)
"Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points." (Stephen Hawking, "A Brief History of Time", 1988)
"Be careful about infinity: we want an infinite number of geometrically distinct geodesics. As a kinematic motion, running twice along a periodic geodesic is different from running only once, but it is not geometrically distinct. For example when working on counting functions one should be careful to distinguish between the counting function for geometric periodic geodesics and that for parameterized ones. The question is difficult because the standard ways to prove existence of periodic geodesics consider them as motions." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"In general, when looking for the minimum of some quantity, the natural approach since the appearance of calculus is to look for minima among the cases where the first variation is zero. Then one studies the case at hand directly or computes the second variation, etc. We did that amply for geodesics and for the isoperimetric inequality. But systoles are not accessible to calculus, in some sense because of their nature, or perhaps because we lack the required tools." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Thanks to the uniqueness of solutions of ordinary differential equations, in particular for the geodesic equations, periodicity of a geodesic occurs precisely when the geodesic is a loop with the same initial and final velocity. Using the point of view from the unit tangent bundle, and the notion of geodesic flow, the periodic geodesics are precisely the periodic flow lines of the geodesic flow. Note that a periodic geodesic is permitted self-intersections. Those without self-intersections are called simple." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!"
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