On a Special Class of Connected Surfaces. 



BV ARNOLD EMCH. 



Students of higher mathematics well know what an important place 

 the Riemann's Surfaces occupy in the Theory of Functions. The 

 interest in this branch of mathematics will doubtless be largely 

 increased in this country since the appearance of the excellent books 

 on this subject by Forsyth, and by Harkness and Morley, which will 

 certainly contribute to exacter logic, formality and intuition in 

 mathematics. 



My intention is to illustrate the utility of the theory of functions 

 in geometry. The reader, however, will understand that by this 

 illustration I do not mean a full development of the geometrical 

 features of the general theory of functions, or a treatment of certain 

 problems related to functions. A simple example out of many 

 possible ones may show that the study of the'theory of functions, 

 and here especially of Riemann's Surfaces, suggests new points of 

 view and ideas in branches of mathematics which do not seem to be 

 related in any way with the others; and that it is, therefore, of great 

 importance for mathematics in general. 



In regard to the general subject of Connected Surfaces and 

 Riemann's Surfaces, I refer to Chapter XIV and XV of the book of 

 Forsyth already mentioned. There we see that surfaces are at 

 present being considered in view of their use as a means of represent- 

 ing the value of a complex variable. Surfaces used for this purpose 

 may be classified according to their connectivity; and the question 

 now arises whether all surfaces of the same connectivity are equivalent 

 to one another, so that they can be transformed into one another. 



As long as continuity is maintained, geometrical transformation as 

 as well as physical deformation may affect the surface without de- 

 stroying the possibility of representing the values of a complex 

 variable on it, provided that certain conditions are satisfied. Hence 

 in the continuous deformation of a surface there may be stretching 

 and bending; but there must be no tearing and there must be no 

 joining. It is not necessary that the deformation of a surface, without 

 tears or joints, be actually possible; and it is sufficient that there 

 exists a point-to-point transformation between the surfaces in which 



(150) KAN. U.VIV. yUAK., VOL. III.. NO. i!. 1894. 



