Differential geometry is a discipline of mathematics which is used to study the geometric problems by suing the techniques of differential and integral calculus including linear and multi-linear algebra. Although geometry is an old branch of mathematics but differential geometry is a new concept in comparison of it and has been developed in eighteenth and nineteenth century which reached to its peak in twentieth century with the help of other disciplines of mathematics. It is similar to differential topology and covers overall topics of 3D geometry including plane, spaces and curves. Major branches of differential geometry are:
Pseudo-Riemannian geometry: is used to generalize the Riemannian geometry in the case of metric tensor does not need positive-definite. It’s a special case of Lorentzian manifold is the base of Einstein’s general relativity theory of geometry.
Finsler geometry: It is used to study the objects by using Finsler manifold (a differential manifold with a Finsler metric)
Symplectic geometry: It is the study of symplectic manifold by using symplectic manifolds.
Complex differential geometry: It is also known as Complex and Kahler geometry and studies the complex manifold with the help of complex structure.
CR geometry is used to study the intrinsic geometry of boundaries of domains in a complex manifold.
Differential topology: is another discipline of differential geometry which studies the global geometric invariants without a symplectic or metric form. Mostly it is used to study in astronomy to measure the space (sky) and to describe the space-time, positions of satellites, astronomical objects such as planets, asteroids, etc.
Applications of differential geometry: Differential geometry is extensively used in various branches of engineering, physics, economics, geometric modeling, computer-aided geometric design, probability, statistics, information technology, structural geology, image processing and economics.