# Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations

arXiv: https://arxiv.org/abs/2605.02279 · HTML: https://mlsanity.com/paper/2605.02279/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 115 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 11 of 11)

- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Eigenvalue and Generalized Eigenvalue Problems: Tutorial](https://mlsanity.com/paper/1903.11240/) (2019)
- [KKT Conditions, First-Order and Second-Order Optimization, and Distributed Optimization: Tutorial and Survey](https://mlsanity.com/paper/2110.01858/) (2021)
- [Spectral, Probabilistic, and Deep Metric Learning: Tutorial and Survey](https://mlsanity.com/paper/2201.09267/) (2022)
- [Optimization on manifolds: A symplectic approach](https://mlsanity.com/paper/2107.11231/) (2021)
- [MixEst: An Estimation Toolbox for Mixture Models](https://mlsanity.com/paper/1507.06065/) (2015)
- [Riemannian stochastic variance reduced gradient on Grassmann manifold](https://mlsanity.com/paper/1605.07367/) (2016)
- [A Finsler Geometrical Programming for the Nonlinear Complementarity Problem of Traffic Equilibrium](https://mlsanity.com/paper/2109.01256/) (2021)
- [On Manifold Hypothesis: Hypersurface Submanifold Embedding Using Osculating Hyperspheres](https://mlsanity.com/paper/2202.01619/) (2022)
- [Accurate and fast matrix factorization for low-rank learning](https://mlsanity.com/paper/2104.10785/) (2021)

## Abstract (arXiv)

Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development. This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form. Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We further specialize these constructions to important matrix manifolds, including the Stiefel, Grassmann, and SPD (Symmetric Positive Definite) manifolds, providing explicit formulas widely used in optimization and geometric machine learning. This monograph develops a unified and implementation-oriented treatment of Riemannian geometry for optimization on manifolds. Its main contribution is the systematic organization and detailed derivation of classical geometric constructions in forms directly usable for algorithm design and numerical implementation. By connecting coordinate-level differential geometry with matrix-manifold formulas, the monograph bridges the gap between abstract theory and practical computation, and provides a reference for researchers and practitioners working in Riemannian optimization and related fields.