**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 PoincarĂ© 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.

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