The article "The Geometries of 3-Manifolds" by Peter Scott discusses the role of geometry in the theory of 3-manifolds, building on the work of Thurston. The author introduces the concept of "nice" metrics, particularly those of constant curvature, which provide new insights into the properties of 3-manifolds. These metrics allow observers to see the same picture regardless of their position or direction. The article also covers non-constant curvature metrics and their classification by Thurston. Scott begins by reviewing the geometry of 2-dimensional manifolds, including the Euclidean plane, the unit sphere, and the hyperbolic plane. He explains the discrete isometry groups of these spaces and their quotient surfaces, emphasizing the importance of fundamental regions and singular points such as cone points, reflector lines, and corner reflectors. The article then delves into the geometry of 3-manifolds, focusing on the eight 3-dimensional geometries identified by Thurston. These geometries are categorized based on their topological and geometric properties. Scott discusses the classification of 3-manifolds admitting these geometries, particularly those with constant curvature. He also explores the relationship between geometric and topological properties of 3-manifolds. Scott highlights the significance of Seifert fiber spaces, which are 3-manifolds foliated by circles, and their connection to various geometries. He provides a detailed description of these spaces and their role in understanding the geometry of 3-manifolds. Finally, Scott touches on the Geometrisation Conjecture, which posits that any compact, orientable 3-manifold can be decomposed into simpler pieces that admit geometric structures. He notes that while this conjecture remains unproven, significant progress has been made, and it is closely related to Thurston's work. Overall, the article provides a comprehensive overview of the geometric and topological aspects of 3-manifolds, emphasizing the importance of geometry in understanding their properties.The article "The Geometries of 3-Manifolds" by Peter Scott discusses the role of geometry in the theory of 3-manifolds, building on the work of Thurston. The author introduces the concept of "nice" metrics, particularly those of constant curvature, which provide new insights into the properties of 3-manifolds. These metrics allow observers to see the same picture regardless of their position or direction. The article also covers non-constant curvature metrics and their classification by Thurston. Scott begins by reviewing the geometry of 2-dimensional manifolds, including the Euclidean plane, the unit sphere, and the hyperbolic plane. He explains the discrete isometry groups of these spaces and their quotient surfaces, emphasizing the importance of fundamental regions and singular points such as cone points, reflector lines, and corner reflectors. The article then delves into the geometry of 3-manifolds, focusing on the eight 3-dimensional geometries identified by Thurston. These geometries are categorized based on their topological and geometric properties. Scott discusses the classification of 3-manifolds admitting these geometries, particularly those with constant curvature. He also explores the relationship between geometric and topological properties of 3-manifolds. Scott highlights the significance of Seifert fiber spaces, which are 3-manifolds foliated by circles, and their connection to various geometries. He provides a detailed description of these spaces and their role in understanding the geometry of 3-manifolds. Finally, Scott touches on the Geometrisation Conjecture, which posits that any compact, orientable 3-manifold can be decomposed into simpler pieces that admit geometric structures. He notes that while this conjecture remains unproven, significant progress has been made, and it is closely related to Thurston's work. Overall, the article provides a comprehensive overview of the geometric and topological aspects of 3-manifolds, emphasizing the importance of geometry in understanding their properties.