106 
MATHEMATICS: S. LEFSCEETZ 
the real folds form equivalent p dimensional cycles, and the q ^ p dimen- 
sional ones which one of the folds contains are equivalent to some cycle, on 
any other. Moreover the real folds form homeomorphic manifolds. One of 
p 
them is defined by the relations Uf, = ^^^m^o^^m, where the (/)'s are real param- 
1 
eters varying from zero to one. It follows that any one of the folds corre- 
sponds point by point to the interior of a ^^-dimensional cube in the same 
manner as the points of a ring to those within a square. From this follows 
readily that the real folds are two sided manifolds of 7-th index of connec- 
tion equal to (pj) ; this being also the number of independent real cycles of Vp. 
We may remark finally that each real fold is transformed into itself by the 
two sets of birational transformations, (uh, + Ci^), {uh, — Uh-\- c^), where 
the (<:)'s are real constants. 
6. It is of interest to consider the types of real Jacobi varieties. — The 
Jacobi variety Vp of a curve of genus p, Cp, is an abelian variety of rank one 
belonging to the period matrix of p independent integrals of the first kind of 
Cp. Let Cp be real and its integrals of the first kind Vi,V2 . . . . Vp real in form 
with a real point 'Mo for lower limit of integration. We assume that Cp has 
q > 0 real branches. We may then take for Vp a representation (1), the (/)'s 
being real functions of the variables u, now defined by U/j 
^ J Mo 
the points Mi, M2, . . . Mp, being alone variable on Cp. In order that the 
point which corresponds to them on Vp be real it is necessary and sufficient 
that the set of the (M)'s coincide with that of their conjuguates, — a corollary 
of the fact that when a point of Vp moves on a real fold the increments of 
the (u)'s are real. 
Let Bi, B2, . . . Bg, be the real branches of C^. If we constrain 0:1 of 
the {M)^s to be on Bi, on B2, . . . ag on Bg, the corresponding point of 
Vp will be on a definite real fold Fi of the variety, provided of course that 
p — 'E ai he ^0 and even. Assume one of the (q;)'s, for example ai, to be 
greater than one. I say that the fold F2 corresponding to the integers 
ai — 2, CX2, . . . ag, coincides with Fi. Indeed a ready corollary from our 
previous discussion is that two distinct real folds of an abelian variety have 
no common points, while Fi and F2 intersect along the p — 1 dimensional 
manifold obtained when ai of the (MYs of which two coincide are on Bi, 
the p — ai remaining being disposed as previously. Hence ii q ^ p 1, Vp 
has as many real folds as there are independent solutions of the congruence 
— p = 0, modulo 2, that is 2^~\ li q > p -{- 1, Vp would have more 
than 2^ real folds. As this is impossible, q ^ p -\- 1 and we have here a 
new proof of Harnack's theorem that a curve of genus p has at most p + 1 
real branches. 
We see then that a Jacobi variety can have any number of real folds given 
by 2^ 5 being an integer at most equal to the genus p. 
