Differential Geometry - 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.

Read more about this topic:  Differential Geometry

Other articles related to "applications, application":

Building On OpenStep
... The idea was to use OpenStep code as a basis for network-wide applications running across different platforms, as opposed to using CORBA or some other system ... Unlike OpenStep, which defined an operating system that applications would run in, under PDO the libraries were compiled into the application itself, creating a stand-alone "native ... supporting code (Objective-C and the libraries), PDO applications were nevertheless considerably smaller than similar CORBA solutions, typically about ...
Biotechnology - Applications - Agriculture - Improved Taste, Texture or Appearance of Food
... The first genetically modified food product was a tomato which was transformed to delay its ripening ... Researchers in Indonesia, Malaysia, Thailand, Philippines and Vietnam are currently working on delayed-ripening papaya in collaboration with the University of Nottingham and Zeneca ...
X Servers - Limitations and Criticism - Network
... manner similar to GNU Screen in relation to terminals), and other applications and toolkits provide related facilities ... interface (mouse, keyboard, monitor) of a running application to be switched from one location to another without stopping and restarting the application ... This can be important in some applications, such as process monitoring and control ...
Photoresist Categories - Applications
... Microelectronics This application, mainly applied to silicon wafers/silicon integrated circuits is the most developed of the technologies and the most specialized in the field ...
Transistor - Comparison With Vacuum Tubes - Advantages
... to replace their vacuum tube predecessors in most applications are Small size and minimal weight, allowing the development of miniaturized electronic devices ... voltages, making transistors suitable for small, battery-powered applications ... for cathode heaters required after power application ...