A concept of the mass center of a system of material points in the constant curvature spaces
The concept of centroid or mass center of a system of material points in \(n\)-dimensional Euclidean space fulfils the following axioms: 1. The centroid of a one-point-system is this point. 2. The centroid of the centroids of two systems of material points is the centroid of their union. 3. Multiplication of all material points of a system with the same mass factor results in multiplying their centroid with this factor. 4. The centroid is invariant under Euclidean motions. 5. The centroid- mapping is continuous with respect to the natural topology of material point systems. Generalizing these axioms to constant-curvature manifolds, it is shown that on constant-curvature manifolds there exists exactly one concept of centroid. The axioms remain meaningful on smooth manifolds with a transitively operating Lie group of motions. It is further shown that if this group is compact the manifold is a sphere. The non-compact case is open.
scientific article; zbMATH DE number 205385