Module page
Istituzioni di Geometria Superiore
| academic year: | 2013/2014 |
| instructor: | Antonio Maschietti |
| degree course: | Mathematics for applications (magistrale) |
| type of training activity: | affine e integrativa |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/03 Geometria |
| teaching language: | italiano |
| period: | I sem (01/10/2013 - 18/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject: Euclideanspaces. Differentiantion, partialderivatives, inverse functiontheorem. Topologicalmanifolds. Topologicalproperties of manifolds. Smoothstructures. Examples of smoothmanifolds. Liegroups. Euclideansubmanifolds. Smoothfunctions and maps. The rank of a smoothmap.Partitions of unity. Generalities on curves. Tangentvectors to an euclideansubmanifold. Twodifferentdefinitionsoftangentspace to a smoothmanifold. Submersions, immersions and embeddings. Immersedmanifolds in euclideanspaces. Example: immersedcurves and surfacesin athreedimensionaleuclideanspace. Sard'stheorem. The tangent bundle. Vectorfields on manifolds. Flows. Liebrackets. The cotangent bundle. The differential of a function. Pullbacks. Line integrals. Conservative vectorfields. Tensors and tensorsfields on manifolds. Symmetrictensors. Riemannianmetrics. Example: metricproperties of surfaces: fundamentalforms, distance, angles, areas, curvatures, gaussian curvature; Gauss theorem. Integration on manifolds: Stokes'stheorem.Global properties: homology and cohomology, Betti numbers, Poincaré lemma, duality. Symplecticstructures and Hamiltoniansystems.
Suggested reading:
1) A. Maschietti, Introduzione alle varietà differenziabili. Lecture notes.
Furtherredings:
2) E. Sernesi, Geometria II. Bollati Boringhieri, 1994
Type of course: standard
Prerequisites: ALGEBRA LINEARE, CALCOLO 1 e 2, GEOMETRIA ANALITICA
Knowledge and understanding: Knowledge and understanding of basicconcepts of differentialgeometry: smoothmanifold, smoothmap, tangent bundle, vectorfield, flow of a field, tensorfield, differentialform, Liegroup;Riemanniangeometry; Integration onmanifolds; Local theory of curves: Frenetformulas; localtheory of surfaces: metricproperties, gaussian curvature, Gauss theorem.
Skills and attributes: Be able to calculate the Frenetapparatus of a curve defined by parametric or cartesianequations, the gaussian curvature, principalcurvatures and normalsections of a surfacedefined by parametric or cartesianequations. Be able to determinelocalcoordinates on a smoothmanifold and write down transitionmaps. Be able to study a smoothmap, usinglocalcoordinates, to ind the tangentspace to a manifold, to calculate the flow of a vectorfield. Be ableto operate with tensors, to calculateintegralson manifolds and recognizeexactforms, to calculatecohomolgygroups.
Personal study: the percentage of personal study required by this course is the 65% of the total.