Learning objectives and outcomes:
The course aims at introducing and developing the fundamental concepts
of differential and projective geometry of curves and surfaces,
in a rigorous yet concrete manner, based on examples, in order to
enhance the students' geometric intuition, abstraction and analytical skills, also in view of future applications.
GEOMETRY: Syllabus (tentative)
1. Review of techniques in mathematical analysis.
Elements of topology (in metric spaces).
2. Differential geometry of curves and surfaces
2.1 Curves - Differential geometry of plane and space curves
Arc length, curvature, curvature radius and osculating circle. Evolutes ed evolvents. Torsion, Frénet-Serret formulae. Fundamental Theorem (curvature and torsion characterize a curve up to rigid motion).
Examples: lines, conics (elementary theory) and other classical curves,
helices, Bézier curves.
2.2. Surfaces - Generalities. First and second fundamental forms, Meusnier's theorem, principal curvatures, principal lines. Euler's theorem.
Gaussian curvature. Covariant derivative e its geometric interpretation (Levi-Civita), parallel transport .
Gauss' Theorema Egregium. Fundamental theorem of surface theory: compatibility equations (Gauss-Codazzi-Mainardi, sketch).
Geodesics and their intrinsic and extrinsic properties.
Mechanical interpretation (brief digression on the Euler-Lagrange
equations). Geodesics on surfaces of revolution. Clairaut's theorem. Digression on elliptic integrals and applications to cartography.
Geodesic circles, Gauss' lemma,
intrinsic characterization of curvature (formula of Bertrand and Puiseux). Gauss' formula for geodesic triangles.
The Gauss-Bonnet theorem. The exponential map.
Normal and polar coordinates.
Examples: Planes, quadrics, (in particular, sphere and ellipsoid of revolution) ruled and developable surfaces, minimal surfaces,
pseudosphere, tori, the Klein bottle, the Moebius band, Bézier surfaces...
Conformal mappings. Examples of cartographic projections.
*3. The concepts of (abstract) differentiable and Riemannian manifold
Topological classification of surfaces (sketch).
*4. Projective geometry of curves and surfaces.
4.1 Review of basic projective geometry. Further developments in
4.2 Quadrics (projective approach).
Polarity. Metric properties.
NOTES: 1. Lecture notes will be made available (a revised version of the existing ones)
2. The program is tentative and subject to change: some topics may be deferred to future courses.
Written exam at the end of the course, followed by an oral test.