Course Details for A.Y. 2015/2016
Name:
Geometria III / Geometria III
Basic information
Credits:
: Bachelor Degree in Mathematics 9 CFU (b)
Degree(s):
Bachelor Degree in Mathematics 2^{nd} anno curriculum Generale Compulsory
Language:
Italian
Course Objectives
We foresee that the student will be able to deal with topology basic notions, that are necessary
during the B.D. in mathematics.
Moreover, after having learned a bunch of notions about curves and surfaces from
the intrinsic and extrinsic viewpoint, the student will be able to solve problems about
these topics.
Course Content
 Topology
1. Topology on a set. Continuous functions. Base of a topology. Metric space. Neghborhoods. Closure of a set and boundary. Interior of a set.2. Subspaces of a topological space. Product spaces (topological and metric) Product topology and continuous application. Product of any family of topological spaces (notions).3. Hausdorff spaces and continuous applications. Limits and semicontinuity.Separations. Topological spaces with numerable base.4. Connected spaces and continuous applications. Products of connected spaces. Connected components. Locally connected spaces (notions).5. Compact spaces and continuous applications. Product of compact spaces. Tychonoff theorem.
 Differential geometry
1. Parametrized curves, regular curves and arc length. Jordan theorem (statement).Arcwise connection (definition). Frenet frame. Existence and uniqueness theorem for curves, given curvature and torsion. Local canonical form.2. Regular surfaces. Graphs, inverse images of regular values. Surfaces invariant by rotation. Differentiable maps between regular surfaces. Differential of a differentiable map. Tangent vector to a surface. The set of tangent vectors to a surface coincide with the image of R^2by the differential of a parametrization.3. Normal vector to a surfaceFirst fundamental form. Length of a curve on a surface, angles between two curves on a surface. Gauss map, differential of the Gauss map. Normal curvature, normal section. Second fundamental form. Principal curvatures, OlindeRodrigues theorem. Gauss and mean curvature. Points of a surface: elliptic, parabolic, hyperbolic, planar. 4. Gauss map in local coordinates. Dupin indicatrix. Asymptotic directions, Conjugate directions. Asymtpotic curves equations.Asymtotic curves for catenoids and helicoids. 5. Minimal surfaces: definition and characterization as critical points of the area functional. Isothermal parameters. Coordinates functions of a minimal surface are harmonic with respect to isothermal parameters. Equation of a minimal graph. Scherk's surface.6. Isometry and local isometry between surfaces. Two surfaces are locally isometric if they have parametrizations with equal coefficients of the first fundamental form.Conformal map and locally conformal map. Statement: two regular surfaces are always locally conformal. Christoffel symbols, Gauss equation and CodazziMainardi equations. Egregium theorem.Fundamental theorem of local theory of surfaces (without proof).7. Tangent vector fields. Differentiability of a tangent vector field. Covariant derivative. Parallel vector fields.Vector fields along a curve. Parallel transport.Parallel transport is an isometry. Geodesic curvature. Algebraic value of the covariant derivative. Differential equations of the geodesics.8. Triangulation of a surface. Some notions about classification of compact surfaces. Genus of a surface.Gauss Bonnet theorem: local and global. Application of Gauss Bonnet theorem.Jacobi theorem.9. Differentiable vector field on a surface. HopfPoincare theorem and application.10. Exponential map on a surface and theorems.Geodesic coordinates and normal neighborhood. Minding theorem. Length of a geodesic circle. Computaton of the Gauss curvature in terms of the length of geodesic circles. Minimization properties of the geodesics. Rigidity of the sphere. Some generalization: Hopf and Alexandrov's theorem (only some notions)
Learning Outcomes (Dublin Descriptors)
On successful completion of this course, the student should

The student should have deep knowledge of the theory of curve and surfaces
immersed in R3 and good knowledge of the basic notions of intrinsic geometry of surfaces.
Moreover the student should acquire the basic notion of general topology.

The student should be able to solve problems about the theory of curves and surfaces immersed in R3 and some problem about intrinsic geometry of surfaces
Moreover the student should be able to recognize when the acquired notions of general topology are necessary to the comprehension of other
topics.

The student should be able to understand problems of curves and surfaces theory and topology and recognize the best method to solve them.

The student should be able to explain the statements and the proof s of the theorems about curves, surfaces and topology

The student should have acquired the ability of reading and understanding more advanced intrinsic theory of surfaces and topology.
Prerequisites and Learning Activities
first year courses of B.D. in mathematics
Assessment Methods and Criteria
The exam is as follows: first a written exam and,
if it is good, the student is allowed to participate to an oral exam.
Textbooks
 M. Abate, F. Tovena, Curve e Super , Springer.
 M. P. Do Carmo, Differential Geometry of Curves and Surfaces , Prentice Hall.
 V. Checcucci, A. Tognoli, E. Vesentini, Lezioni di Topologia Generale , Feltrinelli.
Course page updates
This course page is available (with possible updates) also for the following academic years:
To read the current information on this course, if it is still available, go to the university course catalogue .
Course information last updated on: 18 marzo 2015, 17:38