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'''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.

'''Differential geometry''' is a branch of mathematics which makes use of techniques of [[analysis]], particularly [[calculus]], 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:

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: