578 GEOMETRY 



a problem im Kleinen, while the discovery of periodic orbits and the 

 theory of the stability of the solar system are typical problems im 

 Grossen. 



The principal problems in this field of geometry are connected 

 with closed curves and surfaces. Of special importance are the inves- 

 tigations relating to the closed geodesic lines which can be drawn 

 on a given surface, since these are apt to lead to the invention of 

 methods applicable to the wider field of dynamics. Geodesies may 

 in fact be defined dynamically as trajectories of a particle constrained 

 to the surface and acted upon either by no force or by a force due to 

 a force function U whose first differential parameter is expressible 

 in terms of U. The few general theorems known in this connection 

 are due in the main to Hadamard (Journal de Mathematiques, 1897, 

 1898). Thus, on a closed surface whose curvature is everywhere 

 positive, a point describing a geodesic must cross any existing closed 

 geodesic an infinite number of times, so that, in particular, two 

 closed geodesies necessarily intersect. 1 On a surface of negative 

 curvature, under certain restrictions, there exist closed geodesies 

 of various topological types, as well as geodesies which approach 

 these asymptotically. 



As regards surfaces all of whose geodesies are closed, the investi- 

 gations have been confined entirely to the case of surfaces of revo- 

 lution, the method employed being that suggested by Darboux in 

 the Cours de Mecanique of Despeyrons. Last year Zoll 2 succeeded 

 in determining such a surface (beyond the obvious sphere) which 

 differs from the other known solutions in not having any singularities. 

 Analogous problems in connection with closed lines of curvature 

 and asymptotic lines will probably soon secure the consideration 

 they deserve. 



A problem of different type is the determination of applicability 

 criteria valid for entire surfaces. The ordinary conditions (in terms 

 of differential parameters) assert, for example, the applicability of 

 any surface of constant positive curvature upon a sphere; but the 

 bending is actually possible only for a sufficiently small portion of the 

 surface. A spherical surface as a whole cannot be applied on any 

 other surface, that is, cannot be bent without extension or tearing. 

 This result is analogous to the theorem known to Euclid, although 

 first proved by Cauchy, that a closed convex polyhedral surface is 

 necessarily rigid. Lagrange, Minding, and Jellet stated the result for 

 all closed convex surfaces, but the complete discussion is due to 

 H. Liebmann. 3 The theory of the deformation of concave surfaces 



1 In a paper read before the St. Louis meeting of the American Mathematical 

 Society, Poincare' stated reasons which make very probable the existence of at 

 least three closed geodesies on a surface of this kind. 



1 Math. Annakn, vol. LVII (1903). 



3 Gottingen Nachrichten, 1899; Math. Annalen, vols. LIII, LIV. 



