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Differential Geometry: A Brief Overview

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Geometry Definitions;

The vast subject of differential geometry (also called Euclidean differential geometry) is the study of the geometry of curves and surfaces, including manifolds–higher dimensional surfaces. The “classical” version uses differentials (small changes) and infinitesimals; the “modern” approach does not [1]. The field was first initiated by German mathematician Carl Friedrich Gauss (1777-1855).

Differential geometry has many applications in physics, including solid mechanics, computer tomography, general relativity, and quantum field theory.

Problems in Differential Geometry

The typical problems approached in differential geometry are [2]:

  • Analyzing the relationship between the shortest path on a surface and the concept of a straight line,
  • How to measure a curve’s curvature within a surface versus within an encompassing space,
  • How to measure a surface’s curvature,
  • Measuring the the shortest path within a surface between two points on the surface.

As a concrete example, Gauss used calculus to show that a cylinder’s intrinsic curvature (within its surface) is the same as that of a plane; This can be visualized by slicing a cylinder along its axis and flattening it. On the other hand, a sphere cannot be flattened to create a plane [3].

differential geometry

Gauss showed that a cylinder (left) can be flattened to form a plane (right).


Prerequisites for Learning Differential Geometry

Prerequisites for learning about differential geometry typically include multivariable calculus and linear algebra. Techniques used in the subject include:

  • Complex analysis (which centers around complex numbers and explores the functions and concepts associated with them),
  • Dynamical systems (where function describes a point’s time dependence in a geometrical space),
  • Ergodic theory (statistical properties of deterministic dynamical systems),
  • Harmonic analysis (representation of functions and signals as the superposition of waves),
  • Lie groups (groups that are differentiable manifolds),
  • Non-linear analysis (the nonlinear relations that hold between applied forces and displacements),
  • Operator theory (the study of linear operators on function spaces),
  • Ordinary and partial differential equations,
  • Topology (where an object’s properties are preserved under continuous deformations).

Although differential geometry draws it’s name from its use of ideas and techniques from differential calculus, it is possible to grasp the basics without calculus. For example, Hilbert & Cohn-Vosen’s introductory text Geometry and the Imagination[4] uses no calculus, drawing only on algebraic and geometric techniques instead. A typical course in differential geometry runs one semester but two or three semesters may be needed for a “comprehensive introduction”[5]; It is a prerequisite for Riemannian geometry.

References

[1] P. Petersen. Classical Differential Geometry. Retrieved March 5, 2021 from: https://www.math.ucla.edu/~petersen/DGnotes.pdf
[2] Differential Geometry. Retrieved March 5, 2021 from: http://pi.math.cornell.edu/~dwh/papers/EB-DG/EB-DG-web.htm
[3] N. Faulkner, W.L. Hosch (Eds.) Geometry.
[4] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, New York: Chelsea, 1990.
[5] Rutgers School of Arts & Sciences. 01:640:432 – Introduction to Differential Geometry. Retrieved March 5, 2021 from: https://www.math.rutgers.edu/academics/undergraduate/courses/978-01-640-432-introduction-to-differential-geometry

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Stephanie Glen. "Differential Geometry: A Brief Overview" From GeometryHowTo.com: Geometry for the rest of us! https://www.geometryhowto.com/differential-geometry-a-brief-overview/
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