Page 82 - Curvature and Homology

64 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

2.5. Periods
General line integrals of the form

are often studied as functionals of the arc (or chain) C under the
conditions that the functions p = p(x, y) and q = q(x, y) are of class
k 2 1 in a plane region D and that C is allowed to vary in D. A
particularly important type of line integral has the characteristic property
that the integral depends only on its end points, that is if C and C'
have the same initial and terminal points

This is equivalent to the statement that

over any closed curve (or cycle) r. Now, a necessary and sufficient
condition that the line integral (2.5.1) be a function of the end-points
of C is that the differential p dx + q dy be an exact differential, or,
in the language of Chapter I that the linear differential form a = p dx +
q dy be an exact differential form. The most important consequence is
Cauchy's theorem for simply connected regions. If a & a holomorphic
dijgerential and D a simply connected region, then

If we put

then f is a linear functional (or cochain) and, in general
f (c') = f (c) + f (r) (2.5.6)

where I' is the cycle C' - C. The integral f(r) is called a period of the
form a. Hence (2.5.6) may be stated as follows: The values of the
line integral (2.5.1) along various chains with the same initial and
terminal points are equal to a given value of the integral plus a period.