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Differential geometry

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
Differential geometry is a [mathematical] discipline that uses the methods of [differential] and [integral] [calculus] , as well as [linear] and [multilinear algebra] , to study problems in [geometry] . The theory of plane and space [curves] and of [surfaces] in the three-dimensional [Euclidean space] formed the basis for its initial development in the eighteenth and nineteenth century. Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on [differentiable manifold] s. It is closely related to [differential topology] , and to the geometric aspects of the theory of [differential equation] s. [Grigori Perelman] 's proof of the [Poincaré conjecture] using the techniques of [Ricci flow] demonstrated the power of the differential-geometric approach to questions in [topology] and highlighted the important role played by the analytic methods. [Differential geometry of surfaces] already captures many of the key ideas and techniques characteristic of the field.

Branches of differential geometry
Riemannian geometry

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