Math 4019 Metric Differential Geometry

Three credits. Three hours of lecture per week. MATH 4031 ( Linear Algebra), MATH 3153.

  1. Unit Speed Reparametrization
  2. Winding Number and Signed Curvature
  3. Regular Surfaces I
  4. Regular Surfaces II
  5. Regular Surfaces III
  6. Jacobian of the Differential
  7. Some Regular Surfaces
  8. Surfaces of Revolution
  9. The Second Fundamental Form
  10. Gauss’s Theorema Egregium
  11. Minimal Surfaces
  12. Hyperbolic Paraboloid Exercise
  13. Self Adjointness
  14. Umbilics
  15. Isometries
  16. Connection Coefficients

Tangent, principal normal and binormal of a curve. Osculating plane. Curvature and torsion. The Frenet formulas. Special curves. Regular surfaces. The first and second fundamental forms. Total and mean curvature. The Fundamental theorem of surfaces. Minimal and ruled surfaces. Conformal mappings and isometries. Geodesics.

Course Objectives

Differential geometry is the study of curves, surfaces and their higher dimensional analogues by means of the calculus. It is one of the oldest and richest branches of mathematics, and remains central to modern pure mathematics as well as to much of theoretical physics. It provides a basis for understanding and solving a variety of problems in a wide range of areas including partial differential equations, geometric topology classical and continuum mechanics, elasticity theory, engineering, and general relativity.

The aim of this course is to give an elementary account of the geometry of curves and surfaces in two and three dimensions to students who have completed standard courses in multivariable calculus and linear algebra. This concrete introduction to the area will provide a good basis for the study of manifolds in general in a later more abstract course.

After this course the student is expected to:

  1. appreciate how the calculus can make precise, intuitive ideas of curvature and twisting, and calculate the simplest invariants of curves and surfaces.
  2. appreciate the use of linear algebra in a context where vector spaces without preferred bases arise naturally.
  3. be able to quote the definitions and results relating to each part of the syllabus, and to reproduce the proofs of some key results.
  4. understand the distinction between local and global properties in geometry.
  5. understand the distinction between intrinsic and extrinsic properties of surfaces.
  6. be aware that the Euler number, a topological quantity, can be related to the total curvature of a closed surface.

Syllabus

  1. Parametrized curves. (9 hours)
    1. The idea of a curve.
    2. Curves in two and three dimensions and examples.
    3. Reparametrization of nonsingular curves.
    4. Curvature and torsion. The Frenet formulae.
    5. Isoperimetric inequality.
    6. Fundamental theorem of curves: every curve in Euclidean 3-space is determined up to a rigid motion by its curvature and torsion functions.
  2. Surfaces in 3-space (12 hours)
    1. Regular surfaces.
    2. Coordinate patches.
    3. Inverse images of regular values.
    4. Curves in a surface.
    5. Tangent space.
    6. First fundamental form.
    7. Orientation and the definition of area.
    8. Computation in coordinate patches of area and surface integrals of functions.
  3. The second fundamental form. (12 hours)
    1. The Gauss map and associated quadratic form.
    2. The Gauss map in local coordinates.
    3. The local behavior of a surface. Elliptic, parabolic hyperbolic and planar points.
    4. Normal curvature in a prescribed direction and Euler’s formula.
    5. Principal curvatures and principal directions.
    6. Gaussian curvature K and mean curvature H.
    7. Gaussian curvature equals the Jacobian of the Gauss map.
    8. Ruled and minimal surfaces.
  4. The Intrinsic Geometry of surfaces. (12 hours)
    1. Isometries and conformal maps.
    2. Christoffel symbols.
    3. Gauss Theorema Egregium: Gaussian curvature is invariant under local isometries.
    4. Curves on a surface.
    5. The covariant derivative.
    6. Parallel transport and geodesics.
    7. Geodesic curvature.
    8. Geodesics of surfaces of revolution.
    9. Orientation and the Euler characteristic.
    10. Classification theorem for closed surfaces (the proof will be ommitted).
    11. The Gauss-Bonnet theorem and applications.

Text

  • M.P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

Bibliography

  1. B. O’Neill, Elementary Differential Geometry, Academic Press, 1997.
  2. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vols I, II, Publish or Perish Inc., 1975.
  3. D.J. Struik, Lectures on Classical Differential Geometry, Dover 1988.