D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders (Q6052380)

The theory of Riemannian symmetric spaces is important in many domains of mathematics and physics. In recent years, symmetric spaces have been studied from different viewpoints. Their classification is well-known. Symmetric spaces are characterized as Riemannian manifolds whose geodesic symmetries (that is, geodesic reflections with respect to all points) are globally defined isometries. These geodesic symmetries are also volume-preserving. \textit{J. E. D'Atri} and \textit{H. K. Nickerson}, [J. Differ. Geom. 3, 467--476 (1969; Zbl 0195.23604)] have studied Riemannian and pseudo-Riemannian manifolds all of whose (local) geodesic symmetries are volume-preserving (up to sign) or equivalently, which are divergence-preserving. Following Vanhecke, such spaces are called D'Atri spaces. This paper deals with this kind of spaces, that is, D'Atri spaces. More exactly, the authors try to characterize 3-dimensional Riemannian manifolds, in which the total scalar curvature of tubular surfaces of small radii about regular curves, or only about geodesic segments depends only on the length of the central curve and the radius of the tube. One of the main result of the paper is contained in Theorem 4.1, which gives conditions for a 3-dimensional Riemannian manifold to be a D'Atri space. In Theorem 4.2, the authors prove that if the manifold M is complete and the sectional curvature of M is bounded, then M is a D'Atri space. This is an important result because manifolds of bounded curvature are widely investigated in Riemannian geometry.