On the Local Uniqueness of Geodesics

Abstract

are meaningful and define the geodesics associated with (1). Professor Wintner has raised the question of whether or not geodesics are uniquely determined (locally) by (2) and by a set of initial conditions (even though the coefficient functions FVjk are not subject to a uniqueness restriction, say of the Lipschitz type). It turns out that the answer to this question is of a very surprising nature. On the one hand, if it is only assumed, as above, that E, F_, G are of class CU, then the solutions of (2) are not in general uniquely determined by initial conditions. On the other hand, if (1) is the first fundamental form of a (2-dimensional) surface of class U2 in a 3-dimensional Euclidean space, then the solutions of (2) are unique. This last statement can be reworded in the following manner: