We investigate generalizations of many theorems of Riemannian geometry to Riemannian orbifolds. Basic definitions and many examples are given. It is shown that Riemannian orbifolds inherit a natural stratified length space structure. A version of Toponogov's triangle comparison theorem for Riemannian orbifolds is proven. A structure theorem for minimizing curves shows that such curves cannot pass through the singular set. A generalization of the Bishop relative volume comparison theorem is presented. The maximal diameter theorem of Cheng is generalized. A finiteness result and convergence result is proven for good Riemannian orbifolds, and the existence of a closed geodesic is shown for non-simply connected Riemannian orbifolds.



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URL: https://digitalcommons.calpoly.edu/math_fac/111