Total excess on length surfaces (Q5947413)

Let \( X \) be a connected complete length space, that is, \( (X, d) \) is a connected complete metric space such that for any points \( p, q \) in \( X \) there is a curve \( c \) from \( p \) to \( q \) of length \( d(p,q) \). Let \( \mathcal{U} \subset X \) be the set of all points having a neighborhood homeomorphic to a two-dimensional open disk and assume that \( X \) satisfies the following topological condition (T): \( \mathcal{P} = X \setminus \mathcal{U} \) is a discrete set of \( X \) and, for any \( p \in \mathcal{P} \), there exists a neighborhood \( U \) such that \( U \setminus \{ p \} \) consists of a finite number of two-dimensional annuli. The authors call such an \( X \) satisfying two additional technical conditions a good surface. For example, an Alexandrov space of curvature bounded from above satisfying condition (T) is a good surface. In a previous paper, the first author has introduced the concept of total excess for compact Alexandrov surfaces of curvature bounded from below [J. Math. Soc. Japan 50, 859-878 (1998; Zbl 0936.53044)], in a fashion similar to that of \textit{H. Busemann} for \(G\)-surfaces [The geometry of geodesics, Academic Press, New York, London (1955; Zbl 0112.37002)]. He also defined Gaussian curvature in terms of total excess. In the paper under review, the authors define the total excess for compact good surfaces and prove the generalized Gauss-Bonnet theorem for such surfaces. Also, the generalization of the total excess as well as of the Euler characteristic for noncompact good surfaces is considered. Following the method developed by the first author in a previous paper (see loc.cit.), it is shown that the Gaussian curvature of a good surface \( X \) with non-positive total excess can be defined almost everywhere on \( X \).