Appendix

The literature for the geometric properties of Riemannian Manifolds is immense and hence we cannothopetosurveythemhere;foranappetizer,wereferthereadertoBuragoetal.[93]andLee A few existing works focus on optimizing geodesically convex functions over Riemannian manifold with subgradient methods [83, 105, 106]. Inthecaseofnonlineargeometry,theliterature has been devoted on two different orthogonal axes:a) the existence of saddle point for min-max objectivebi-functions andb)the design ofalgorithms for the computation ofsuch points. Similar with the existence results, initially the developed methods referred to the computation of singularities in monotone variational operators typically in hyperbolic Hadamard manifolds with negativecurvature[124]. This formulation finds a wide range of real-world applications, e.g., non-negative principle component analysis, weighted max-cut and so on.