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.

Locomotion is the process by which an animal or robot uses internal changes in shape (the relative positions of its body components) to push against its environment and produce changes in its position (translations and rotations relative to the world reference frame). A key challenge in locomotion is identifying shape changes that avoid range-of-motion limits on the system’s joints while efficiently producing net displacements over distances significantly greater than the system’s body length. The solutions to this problem often take the form of gaits – cyclic shape changes – that maximize the ratio of net displacement to effort.

In this tutorial, I will discuss the use of Lie group theory and Riemannian geometry in understanding the process of locomotion and the structure of optimal gaits. Lie group theory provides a natural framework for discussing relative motions, and Riemannian geometry provides both a means of describing the physics underlying shape/position coupling and the costs of changing shape while subject to these physics.

Specific topics I will cover include:

• System kinematics with Lie groups

• Sub-Riemannian constraints between shape and position motions

• Lie brackets and curvature for understanding net position change over shape cycles

• The importance of coordinate choice when studying locomotion

• Geometric characterization of motion costs

• Applications of these principles to non-ideal systems and in machine learning

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.