Surfaces with extreme value of curvature in Alexandrov spaces

In an Alexandrov space with curvature bound, we prove that a curvature takes the extreme value over some specially constructed surfaces if and only if each of the surfaces is totally geodesic and locally isometric to a surface with constant curvature.Introduction.An Alexandrov space is a locally compact complete length space (i.e., a space in which distance is measured by the infimum of lengths of curves) with curvature bounded either below or above in the distance comparison sense, that is, the Alexandrov-Toponogov comparison theorem holds for all small geodesic triangles.A complete Riemannian manifold with sectional curvature bounded either below or above is an Alexandrov space and in fact the difference lies in the differentiability.Until recently people discussed only C°°-Riemannian manifolds and forgot about other important aspects of metric spaces.It was the work of Gromov that ended this long sleeping period.Inspired by the idea developed by Gromov [11], [12], Alexandrov spaces got footlights, and it became known that they can be obtained as the so-called Gromov-Hausdorff limits (cf.[13], [15], [23]) of sequences of Riemannian manifolds belonging to a certain class determined by geometric quantities; curvature, diameter, and volume (cf.[17], [18], [24]).Since the notion of Alexandrov spaces is a generalization of Riemannian manifolds, is seems natural to consider the problem: To what extent can one extend results in Riemannian geometry to Alexandrov spaces?It is known that some well-known results in Riemannian geometry can be extended to finite Hausdorff dimensional Alexandrov spaces of curvature bounded below.For example, the Myers-Toponogov compactness theorem [6], the Diameter sphere theorem of Grove and Shiohama [19], [22], the fibration theorem of Yamaguchi [27], [28], and the Soul theorem of Cheeger and Gromoll [9], [22] can be generalized.It should also be mentioned that the isometry group of a finite Hausdorff dimensional Alexandrov space with lower curvature bounded is a Lie group [10].In this paper, we will show that for specially constructed surfaces Σ t (ί= 1, 2) in an Alexandrov space X with curvature bounded either below or above, the curvature 1991 Mathematics