Geometry 4

A.Y. 2018/2019
Overall hours
Learning objectives
The aim of the course is to present an introduction to topological and differentiable manifolds.
Expected learning outcomes
Knowledge of some elementary properties of differential varieties.
Course syllabus and organization

Single session

Lesson period
Second semester
Geometria 4 (prima parte)
Course syllabus
Topological manifolds and topological manifolds with boundary. Topological surfaces.
Differentiable manifolds; examples (n-spheres, tori, projective spaces). Differentiable mappings. Rank theorem. Submanifolds.
Local theory of curves in R3. Curvature and torsion. Frenet Serret formulas.
Local theory of surfaces in R3. First fondamental form. Examples. Arc lengh of a curve on a surface. Gauss map. Second fondamental form. Normal curvature of a curve on a surface. Principal curvatures. Gaussian curvature, average curvature. Elliptic, parabolic and hyperbolic points.
Tangent spaces. Differential map.
Geometria 4 mod/02
Course syllabus
Partition of Unity.
Vector bundles and sections. Tangent and cotangent bundles.
Differential forms. Exterior differentiation. Volume form and orientation. Integration on manifolds. Stokes' theorem.
Geometria 4 (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 22 hours
Lessons: 36 hours
Professor: Bertolini Marina
Geometria 4 mod/02
MAT/03 - GEOMETRY - University credits: 3
Practicals: 22 hours
Lessons: 9 hours
Professor: Gori Anna