MA 410 - Differential Geometry

MA 410 - Differential Geometry

This course is an introduction to the geometry of curves and surfaces. Topics include tangent vectors, tangent bundles, directional derivatives, differential forms, Euclidean geometry and calculus on surfaces, Gaussian curvatures, Riemannian geometry, and geodesics.

Prerequisites:    MA 227

Course Objectives

This one-semester course introduces to student major ideas of differential geometry and its applications to physics. Upon completion of this course students will have knowledge of the geometry of curves and surfaces, understand how calculus, topology and linear algebra contribute to studying geometrical objects, will be able to solve typical problems associated with this theory, and will be able to use standard software to visualize curves and surfaces and to perform standard calculations.

Learning Outcomes

  1. Vector and Euclidean spaces: Understand the following concepts: finite dimensional vector space over the field of real numbers, n-dimensional Euclidean space, norm, scalar and vector products, orthogonally. Demonstrate ability to operate with scalar and vector products in Euclidean spaces. Be able to produce norm induced by a scalar product and metric induced by a norm.
  2. Linear transformations: Understand the following concepts: linear mapping and linear form in finite dimensional vector spaces, rank of a linear mapping, and isomorphism.
  3. Topology: Be familiar with the following concepts of topological spaces, open and closed sets, injective, subjective, bijective, open, closed and continuous functions, homeomorphism and diffeomorphism.
  4. Curves in n-dimensional Euclidean space: Understand the definition of the regular curve, vector field along the curve. Know how to derive the formula for the length of a curve. Be able to operate with parametric and nonparametric forms of a curve, parameterize a curve by arc length.
  5. Frenet frame: Know the definitions of tangent, normal and binormal vector, curvature, curvature radius, and torsion. Be able to solve standard problems concerning motion of particles in space.
  6. Computer visualization of curves: Be able to use Mathematics to visualize plane and space curves with different parametrizations.
  7. Surfaces in n-dimensional Euclidean space: Be familiar with the following concepts: tangent vectors to n-dimensional Euclidean space, k-dimensional surface, local chart and atlas of a surface, smooth k-dimensional surface, surface parametrization, oriented Euclidean space, orienting atlas of a surface, orientable and non-orientable surface, boundary of a surface and its orientation.
  8. Area of a surface in Euclidean space:  Be familiar with the formula for an area of a k-dimensional smooth surface. For any given smooth surface in three-dimensional space be able to calculate the area for a given surface and to visualize the surface using appropriate software.
  9. Elements of multilinear algebra: Understand the concepts of bilinear mapping, alternating multilinear mapping, tensor product of vector spaces, external product of vector spaces.
  10. Differential forms: Be familiar with the following concepts: real-valued differential p-form, differential form defined on a smooth surface, exterior differentiation of a differential from, work form of a vector field F, flux form of a vector field V in a domain D. Be able to write coordinate expressions of differential forms and transform it under a mapping.
  11. Integral of differential form: Know the definitions of the integral of a k-form w over a given chart of a smooth k-dimensional surface and of the integral of a form over an oriented surface. Be able to apply this definition to find the work of a field and a flux across the surface. Understand that the area of a surface can be considered as the integral of a form, be able to write the volume element of an oriented smooth surface in Cartesian coordinates and apply it to determining the mass of a surface whose density is known.
  12. Fundamental theorems: Be familiar with Green’s Theorem, the Gauss-Ostrogradskii formula, and Stokes’ theorem and the following operators of vector analysis: grad, curl, div. and nabla.
  13. The Gauss-Bonnet theorem for surfaces: Be familiar with this theorem and its applications to vector fields on surfaces.

 

Textbooks

Do Carmo, Manfredo, Differential Geometry of Curves and Surfaces, Prentice Hall.

O'Neill, Barrett, Elementary Differential Geometry

Gray, Alfred, Elsa Abbena, and Simon Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica.

Zorich, Vladimir, Mathematical Analysis II  (for topics in differential forms).

Dugundji, James, Topology.

Zamansky, M., Linear Algebra and Analysis.