Does invariance over all geodesic balls or circles of the same radius of integrals of the form $\int_B K^a dM$ or $\int_{\partial B} \kappa^a ds$ characterize constant curvature surfaces? These questions were left open in the previous post, and I answer them, and more, in today’s post.

This post began as an overlong Math Stack Exchange question/answer. A moderator suggested it was too long for MSE, and to try to pare the answer down and save this work on a separate personal page. It may be more pleasant to read this work by first exporting it to pdf, using your browser’s Print feature.