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'''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.
'''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]]s. 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]].
* 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]]s.
* Can the manifold be "flattened out"? Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing. How can we tell if it is possible to flatten a given surface? This question leads to the study of ''curvature''.
* Can the manifold be "flattened out"? Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing. How can we tell if it is possible to flatten a given surface? This question leads to the study of ''curvature''.
* 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!
* 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!