Introduction to differential geometry

Mathematics BSc
Course description
2013.

Eötvös Loránd University
Faculty of Science
Institute of Mathematics

Borromean Rings

All courses
Core courses
- Algebra, number theory
- Analysis
- Geometry
  - Geometry 1
- Discrete mathematics
  - Discrete mathematics 1
  - Discrete mathematics 2
- Elementary mathematics
  - Elementary mathematics 1
- Informatics
  - Prog. fundamentals
Pure mathematics
- Algebra, number theory
- Analysis
  - Analysis 3
  - Analysis 4
  - Applied analysis
    - Numerical analysis
    - Appl. anal. comp.
  - Differential equations
    - Differential equations
    - Partial diff. equations
  - Topology
    - Introduction
    - Algebraic topology
  - Complex analysis
  - Functional analysis
  - Function series
- Geometry
- Operations research
  - Operations research 1
  - Operations research 2
- Probability theory
- Informatics
- Computer science
- Foundations of math.
  - Set theory
  - Mathematical logic
Applied mathematics
- Analysis
  - Analysis 3
  - Analysis 4
  - Analysis 5
  - Applied analysis
  - Differential equations
    - Differential equations
    - Partial diff. equations
  - Complex analysis
  - Functional analysis
- Geometry
- Operations research
  - Operations research 1
  - Operations research 2
- Probability theory
- Applied modules
- Informatics
- Algorithms
- Algebra
- Elementary mathematics
  - Elementary mathematics 2
  - Elementary mathematics 3
- Foundations of math.
BSc: general information
Studies

Introduction to differential geometry

Hours lect+pc	Credits lect+pc	Assessment	Specialization	Course code lect/pc	Semester	Status
2 + 2	2 + 3	exam + term grade	pure math.	mm1c1dg5m mm1c2dg5m	5	compulsory

Course coordinator

László Verhóczki
Department of Geometry
Institute of Mathematics

Strong	Weak	Prerequisites
Practice class
		Strong: Geometry1L (mm1c1ge2)
		Strong: Algebra2L (mm1c1al2)
		Strong: Analysis3L-p (mm1c1an3m)
Lecture
Lecture		Weak: practice class

Remarks

Syllabus designed by:

László Verhóczki, Department of Geometry, Institute of Mathematics.

Prerequisites

Analytical geometry, linear algebra, multivariable calculus

Literature

Manfredo P. do Carmo: Differential geometry of curves and surfaces. Prentice-Hall, New-Jersey, 1976.
B. Csikós: Differential Geometry. Lecture notes.

Syllabus

Parameterized smooth curves in Rⁿ. Moving Frenet frame of a curve of general type. Curvature functions, Frenet formulae. Osculating circle to a curve, evolute, involute, parallel curves. Umlaufsatz, Fenchel's theorem, Fáry-Milnor theorem.
Parameterizations of a smooth hypersurface in Rⁿ. Tangent space. First fundamental form. Area (volume). Normal curvature. Meusnier’s theorem. Second fundamental form, Weingarten map. Gaussian curvature. Moving Gauss frame, Christoffel symbols. Gauss and Codazzi-Mainardi equations. Fundamental theorem of hypersurfaces. Theorema Egregium. Geodesics.