The optimization algorithms from Chapters 4 and 6 require only rather simple tools from Riemannian geometry, all covered in Chapters 3 and 5 for embedded submanifolds then generalized in Chapter 8. This chapter provides additional geometric tools to gain deeper insight and help develop more sophisticated algorithms. It opens with the Riemannian distance then discusses exponential maps as retractions which generate geodesics. This is paired with a careful discussion of what it means to invert the exponential map. Then, the chapter defines parallel transport to compare tangent vectors in different tangent spaces. Later, the chapter defines transporters which can been seen as a relaxed type of parallel transport. Before that, we take a deep dive into the notion of Lipschitz continuity for gradients and Hessians on Riemannian manifolds, aiming to connect these concepts with the Lipschitz-type regularity assumptions we required to analyze gradient descent and trust regions. The chapter closes with a discussion of how to approximate Riemannian Hessians with finite differences of gradients via transporters, and with an introduction to the differentiation of tensor fields of all orders.