Differential Geometry - Applications

Applications

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

• In physics, three uses will be mentioned:
  • Differential geometry is the language in which Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of space-time. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
  • Differential forms are used in the study of electromagnetism.
  • Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
• In economics, differential geometry has applications to the field of econometrics.
• Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry.
• In engineering, differential geometry can be applied to solve problems in digital signal processing.
• In probability, statistics, and information theory, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry, particularly via the Fisher information metric.
• In structural geology, differential geometry is used to analyze and describe geologic structures.
• In computer vision, differential geometry is used to analyze shapes.
• In image processing, differential geometry is used to process and analyse data on non-flat surfaces.
• Grigori Perelman's proof of the Poincaré conjecture using the techniques of Ricci flows demonstrated the power of the differential-geometric approach to questions in topology and it highlighted the important role played by its analytic methods.