Geometry 3

TOPOLOGY:

· Toplogical spaces: first definitions, examples, order relation among topolgies, basis.

Continuos functions: dfinitions and characterization; homeomorphisms; open and closed function

· Metric spaces: metric topology; basis for the topology induced by a metric; continuity in metric spaces

Induced topology: definitions and characterization; examples

Product topology: definition, basis, characterizations of continuos map in a product space

Quotient topology: definition and characterizations; saturated open subsets; universal property of the quotient; quotient by an equivalence relation. Examples of explicit constructions

Topological properties:
i. T1, T2, T3, T4 and their realtions: examples and counterexamples;
ii Connectedness: definition and characterization. connectedness in R and R^n and for product and quotient toplogical spaces
iii Path connectdness: definition, characterization, relation with connectnedess. Examples in R^n.
iv Compactness; definitions and first examples: Compactness; in R^n and in T2 spaces; Alexandrov's compattification

Homotopy between path and between topological spaces: definitions, the set of closed paths; homotpy among topological spaces and equivalence relation: examples and constructions of homotopies

Fundamental group: prelimnary results on groups; definitions of the fundamental group; simply connected spaces; Seifer van Kampen theorem. Exaples and explicit calculations on fundamental groups


CURVES AND SURFACES

Introduction to the curves in the planes and Frenet's equation
Diefences between the plane and the space: curves in the space. Vector product
Curvature, torsion and Frenet triple for curves in the space
Definition of Elementary surfaces. Definition of tangent space and affine tangent plane. Geometric interpretation of the tangent plane. Graph of a C^\infty function.
Definition of regular surface: examples. Surface as differential variety of dimension 2

First fundamental form: Rotation surfaces and ruled surfaces

Differentiable maps between surfaces and Gauss map.

Weingarten operator, normal nurvature. Meusnier theorem
Gauss curvatures in local coordinates
Local and global isometries. Gauss Egregium Theorem