This session aims at providing a summary of the main concepts of Riemannian geometry of particular relevance to robotics. Our goal is not to give an in-depth introduction to Riemannian geometry, but rather to provide the theoretical background required to make use of Riemannian geometry in robotics. Some of the topics that we will cover are: The notion of curvature, smooth manifolds, tangent spaces, Riemannian metrics, geodesics, exponential and logarithmic maps, and parallel transport operations.

Despite recent progress in applying geometric deep learning to robotics, leveraging geometric symmetry in robot policy learning remains challenging in practice. Many existing methods can improve sample efficiency and generalization, but their complexity often makes them difficult to adopt widely across different robotic systems and tasks. In this tutorial, I will provide a practical overview of how geometric symmetry can be incorporated into robotic policies. I will discuss several complementary approaches, including equivariant neural networks, canonicalization, and symmetry-based data augmentation, and highlight their strengths, limitations, and practical trade-offs. Through these examples, the tutorial aims to make symmetry-aware robot learning more accessible and to help researchers develop policies that generalize more effectively from limited data.

This lecture explores the emerging synthesis of Geometric Algebra, Riemannian Geometry, and Quantum Computing as a unified mathematical and computational framework for advanced robotics modeling, learning, and control. We present how geometric algebra provides a coordinate-free language for representing spatial transformations, rigid body kinematics, dynamics, and sensorimotor interactions, enabling compact and expressive formulations for robotic perception and control. Building on this foundation, Riemannian geometry offers powerful tools for describing the nonlinear manifold structures underlying robot configuration spaces, optimization on curved spaces, geometric learning, and trajectory generation with intrinsic guarantees of stability and efficiency.

The lecture further examines how these geometric methods intersect with emerging quantum computational paradigms, highlighting the potential of quantum algorithms to accelerate optimization, enhance high-dimensional learning, and support scalable solutions to complex control and planning problems. Particular emphasis is placed on the integration of geometric priors into machine learning architectures, manifold-based representations for robot intelligence, and quantum-enhanced methods for adaptive and data-driven control.

By bridging classical geometry, modern machine learning, and quantum information science, this lecture aims to provide a coherent perspective on next-generation mathematical tools for robotics. Applications ranging from autonomous systems and embodied intelligence to geometric deep learning and quantum robotics are discussed, alongside open challenges and future research directions at the intersection of these rapidly evolving fields.