Introduction

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 '

Kjtfj)

J

\ 0 if ej e Int(M ')

Recieved by the editor July 16, 1991.

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