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'''Differential geometry''' is a [[geometry]] where differences between [[point]]s are measured using a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].  

'''Differential geometry''' is a branch of mathematics which makes use of techniques of [[analysis]] to study geometric problems.  Initially, geometers primarily sought to understand the geometry of curves and surfaces in 3-dimensional [[Euclidean space]], and many important early results in the subject are due to [[Carl Friedrich Gauss|Gauss]].  Other early pioneers included [[Bernhard Riemann]] and Tullio Levi-Civita.

 

The primary objects of study in differential geometry are ''smooth'' and ''Riemannian'' [[manifold|manifolds]].  A typical example of such an object is a smooth surface in R^3, for example, the unit sphere.  Typical questions that a differential geometer might ask about a manifold include:

 

* Given two points, what is the shortest path between those two points staying on the manifold?  These length-minimizing paths are termed ''geodesics''.  On the sphere, the geodesics are [[great circle|great circles]].

* Is there a natural notion of distance on the manifold?  In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!

* How should one [[derivative|differentiate]] a function?  What does it mean to take the derivative of a function on the sphere?  This question leads to the development of objects called ''connections''.

 

Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].  

[[Category:Geometry]]

[[Category:Geometry]]