Differential geometry

In mathematics, differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and multilinear algebra, to study problems in geometry. 

Shapes, like triangles, immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultra parallel lines are examined using differential geometry. image: wikipedia

The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. 

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. 

Differential geometry is closely related to differential topology, and to the geometric aspects of the theory of differential equations. 

The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field.

source: http://en.wikipedia.org/w/index.php?title=Differential_geometry&oldid=604228191


Commentary:

The main application of this is used in physics, in particular the mathematics of general relativity, which describes how massive objects bend space.

Within in this field one of the most important types of geometry is called Riemannian geometry.