The work is an exploration to geometry that transcends traditional approaches, placing Felix Klein's transformative Erlangen Program firmly at its heart. It aims to reveal the essence of geometry through the unifying power of transformation groups and their actions on space. Klein's genius was to define each geometry by the group of transformations that preserve its fundamental properties. This book makes this powerful conceptual framework accessible and compelling. The reader is guided through this perspective, demonstrating how the structure and symmetries of each geometric realm – Euclidean, Similarity, Affine, Spherical, Projective, Inversive, Hyperbolic – are fundamentally defined and understood by the transformations that leave them invariant.

This book is a precise and clear mathematical presentation suitable for advanced undergraduates and graduate students, but also for mathematical educators as well as self-learners. It develops a sophisticated understanding of geometry through the intrinsic symmetries defined by transformation groups. Geometries is an invitation to experience the elegance, power, and breathtaking unity of geometry through the transformative lens pioneered by Klein. The reader is invites to master the language of symmetry and transformation groups, conquer challenging exercises with confidence, and unlock the interconnected universe of geometric spaces, where the abstract becomes tangible, and the transformation is the key.