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3.3. Geometry and Topology (GTO)

This area covers the study of shape, space, and continuity, ranging from classical Euclidean geometry to the modern theory of manifolds, algebraic topology, and symplectic geometry, with deep connections to physics and data science.

Knowledge Area (KA)	Core Tier1	Core Tier2
3.3.1 Euclidean and Projective Geometry	1	1
3.3.2 Differential Geometry of Curves and Surfaces	1	1

3.3.1. GTO/Euclidean and Projective Geometry  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Axiomatic Euclidean geometry, isometries, symmetry groups, and the projective plane.
Topics:
Core

• Euclidean geometry: axioms, congruence, similarity, and classical theorems
• Isometries of the plane and space: reflections, rotations, translations, and glide reflections
• Symmetry groups: frieze groups, wallpaper groups, and the classification of finite isometry groups
• Projective plane: homogeneous coordinates, duality, and cross-ratio
• Conic sections in projective geometry: classification and projective transformations

Learning Outcomes:
Core:

1. Describe the axiomatic foundation of Euclidean geometry and identify the role of the parallel postulate [Familiarity]
2. Classify isometries of the plane and determine the symmetry group of a given geometric figure [Usage]
3. Apply projective coordinates and duality to prove classical theorems (Pappus, Desargues) [Assessment]

3.3.2. GTO/Differential Geometry of Curves and Surfaces  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Local and global theory of curves and surfaces in \(\mathbb{R}^3\): curvature, Gauss map, and the Gauss-Bonnet theorem.
Topics:
Core

• Space curves: curvature, torsion, Frenet-Serret frame, and the fundamental theorem of curves
• Regular surfaces in \(\mathbb{R}^3\): parametrizations, tangent planes, and first fundamental form
• Second fundamental form, principal curvatures, Gaussian and mean curvature, and the Gauss map
• Geodesics on surfaces and the Gauss-Bonnet theorem
• Gauss's Theorema Egregium: intrinsic nature of Gaussian curvature

Learning Outcomes:
Core:

1. Compute the curvature and torsion of a space curve from its parametric equations [Familiarity]
2. Calculate the Gaussian and mean curvature of parametric surfaces using the first and second fundamental forms [Usage]
3. Apply the Gauss-Bonnet theorem to relate the curvature of a surface to its topological Euler characteristic [Assessment]