Recall that a topological space is said to be finitely connected if its z-th homotopy
groups are finitely generated for all i (cf. [Bu, §29.6]). Let M be a finitely connected
2-dimensional smooth manifold with finitely connected (but not necessarily compact)
piecewise smooth boundary dM. Then, M has finitely many ends ej, where 1 j
k, (i.e. M has k boundary components in the sense of [AS, Chap. I, 36C]). If
we compactify M by adding at infinity points in 1-1 correspondence with the ejs, we
get a compact topological manifold with boundary M':=M u {e\, ... ,£*}. The
ends ejs are of two types according to whether ej belongs to the boundary dM' or to
the interior Int(M') of M\ In addition suppose that M is a complete Riemannian
manifold. A complete (or as we prefer to say a maximal) geodesic of M is said to be
distant if it is entirely contained into a sufficiently small neighborhood of some ej.
Assume that both the total curvature c(M) of M and the total geodesic curvature
K(M) of dM with respect to M are defined and that the sum K(M) + c(M) has
a meaning. Then the curvature at infinity K^iM) of M is defined to be fCo(M) :=
2%%(M) - K(M) - c(M), where %{M) is the Euler characteristic of M. By a
suitable generalization of Cohn-Vossen's theorem (see 3.4 below), one has JC»(M)
n%(dM), where %(dM) is the number of noncompact connected components of
dM. For each end ej there exists an arbitrarily small neighborhood Mj of ej such
that if ej e dM\ Mj := Mj - {ej} C M is a Riemannian half plane; if ej e
Int(M'), Mj = Mj - {ej} is a Riemannian half cylinder (see 3.4.2 and 3.4.3 for
definitions of Riemannian half planes and half cylinders). By the Gauss-Bonnet
theorem, each number K^Mj) is depending only on ej but independent of My, and
is called the curvature at ej. Applying Cohn-Vossen's theorem to each Mj, one has
f n if e; e dM '
\ 0 if ej e Int(M ')
Recieved by the editor July 16, 1991.
Previous Page Next Page