Differential geometry of surfaces - Revision history

Karl Jones: /* External links */

2016-04-22T21:38:14Z

‎External links

Karl Jones: /* Description */

2016-02-11T14:34:53Z

‎Description

Karl Jones at 14:29, 11 February 2016

2016-02-11T14:29:13Z

Karl Jones: Created page with "In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian..."

2016-02-11T14:28:18Z

Created page with "In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian..."

New page

In [[mathematics]], the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

== Description o==

Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.

One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane.

These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections.

On the other hand extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied.

This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.

== See also ==

* [[Differential geometry]]
* [[Geometry]]
* [[Mathematics]]

== External links ==

* [https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces Differential geometry of surfaces] @ Wikipedia

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	* [https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces Differential geometry of surfaces] @ Wikipedia		* [https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces Differential geometry of surfaces] @ Wikipedia
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	One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by [[Carl Friedrich Gauss]] (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.		One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by [[Carl Friedrich Gauss]] (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
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	Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves.		Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves.

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-In [[mathematics]], the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
+In [[mathematics]], the '''differential geometry of surfaces''' deals with the [[differential geometry]] of smooth surfaces with various additional structures, most often, a Riemannian metric.
-== Description o==
+== Description ==
 Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.
-One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
+One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by [[Carl Friedrich Gauss]] (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
-Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane.
+Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves.
 These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections.