The Riemannian center of mass was constructed in [GrKa] (1973). In [GKR1, GKR2, Gr, Ka, BuKa] (1974-1981 it was successfully applied with more refined estimates. Probably in 1990 someone renamed it without justification into karcher mean and references to the older papers were omitted by those using the new name. As a consequence newcomers started to reprove results from the above papers. -Here I explain the older history.The Euclidean center of mass is an affine notion. I will use discrete mass points rather than mass distributions, for simpler wording. Let p i ∈ R n be points of weight m i with m i = 1 (for convenience). The center is defined, without reference to a metric, as:This formulation does not carry over to Riemannian manifolds. A small change improves the situation. We define a vector field on R n by averaging the vectors from an arbitrary point x ∈ R n to the mass points p i :This vector field points from every point x to the center, C(p i ) = x + V (x), and it generalizes to a Riemannian manifold M :

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