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].

## 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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