3 
52 
Mathematics. “Continuous one-one transformations of surfaces in 
themselves”. (Bh communication ')). By Prof. L. B. J. BROUWER. 
In Cretie’s Journal, vol. 127, p. 186 Prof. P. Bourn has enun- 
ciated without proof the following theorem proved by me (as a 
particular case of a more general theorem) in vol. 71 of the Mathe- 
matische Annalen (compare there page 11+): 
“Werden die Punkte einer Kugeloberfliiche wieder tn Punkte der 
Kugeloberjliiche iibergefiihrt und gescheht diese Ueberführung durch 
stetige Bewegung, welche den Mittelpunkt nicht beriihrt, so kehrt 
mindestens ein Punkt in seine friihere Lage zuriick. Unter einer stetigen 
Bewegung ist hier eine Bewegung verstanden, bei welcher die recht- 
winktigen Koordinaten stetige Funktionen der Zeit und der Anfangs- 
werte sind.” 
Now I shall show here in the first place that the theorem enun- 
ciated and proved in the first communication on this subject ’), i. e. 
that each continuous one-one transformation with invariant indicatrix 
of a sphere in itself possesses at least one invariant point, may be 
considered as a particular case of the quoted theorem of Bout *). 
To that end I shall establish the following theorem : 
“Any continuous one-one transformation a with invariant mdicatria 
of a sphere in itself can be transformed by a continuous modification *) 
into identity” *). 
In order to prove this property we choose in the sphere two 
opposite points P, and P, determining a net of circles of longitude 
and latitude and passing by a@ into Q, and Q,. By means of a 
continuous series + of conform transformations of the sphere in 
itself we can transform Q, and Q, into P, and P,. Let c be an 
arbitrary circle of latitude, described in such a sense that 2, pos- 
sesses with respect to c the order ®) + 1, and c’ the image of c for 
at, then P, possesses also with respect to c’ the order + 1. 
1) Compare these Proceedings XI, p. 788; XII, p. 286; XIII, p. 767; XIV, 
p 300 (1909—1911). 
2) These Proceedings XI (1909), p. 797. 
5) This | indicated already shortly Mathem. Ann. 71 (1911), p. 325, footnote *). 
') Under a continuous modification of a univalent continuous transformation we 
understand in the following always the construction of 2 continuous series of uni- 
valent continuous transformations, 1.e. a series of transformations depending in 
such a manner on a parameter, that the position of an arbitrary point is a con- 
tinuous function of its initial position and the parameter. 
5) That this theorem wants a proof is shown by the fact that e.g. for a torus it 
does not hold. 
6) Compare e.g. J. Tannery, “Introduction à la théorie des fonctions d'une 
variable”, vol. II, p. 438. 
