en>AReu: /* Chain maps and tensor product */ - Typo in formula corrected

2014-06-12T16:02:24Z

Chain maps and tensor product: - Typo in formula corrected

en>Mark viking: /* de Rham cohomology */ remove self-redirect

2014-02-03T23:44:45Z

de Rham cohomology: remove self-redirect

← Older revision		Revision as of 01:44, 4 February 2014
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	~~My name~~: ~~Leslie FitzGibbon~~<br>~~Age~~: 29<br>~~Country~~: ~~Germany~~<br>~~Town~~: ~~Stuttgart Lederberg~~ <br>~~Post code~~: ~~70619~~<br>~~Street~~: ~~Ollenhauer Str~~. 23<br><br>~~Look~~ at ~~my web page~~; [http://~~safedietplansforwomen~~.com/~~bmi~~-~~chart bmi chart~~]		[[File:Gauss-Bonnet theorem.svg\|thumb\|300px\|An example of complex region where Gauss-Bonnet theorem can apply. Shows the sign of geodesic curvature.]]
			The '''Gauss–Bonnet theorem''' or '''Gauss–Bonnet formula''' in [[differential geometry]] is an important statement about [[surface]]s which connects their geometry (in the sense of [[curvature]]) to their topology (in the sense of the [[Euler characteristic]]). It is named after [[Carl Friedrich Gauss]] who was aware of a version of the theorem but never published it, and [[Pierre Ossian Bonnet]] who published a special case in 1848.

			== Statement of the theorem ==
			Suppose <math>M</math> is a [[Compact space\|compact]] two-dimensional [[Riemannian manifold]] with boundary <math>\partial M</math>. Let <math>K</math> be the [[Gaussian curvature]] of <math>M</math>, and let <math>k_g</math> be the [[geodesic curvature]] of <math>\partial M</math>. Then
			:<math>\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \, </math>
			where ''dA'' is the [[volume element\|element of area]] of the surface, and ''ds'' is the line element along the boundary of ''M''. Here, <math>\chi(M)</math> is the [[Euler characteristic]] of <math>M</math>.

			If the boundary <math>\partial M</math> is [[piecewise smooth]], then we interpret the integral <math>\int_{\partial M}k_g\;ds</math> as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the [[angle]]s by which the smooth portions turn at the corners of the boundary.

			==Interpretation and significance==
			The theorem applies in particular to compact surfaces without boundary, in which case the integral
			:<math>\int_{\partial M}k_g\;ds</math>

			can be omitted. It states that the total Gaussian curvature of such a closed surface is equal to 2π times the Euler characteristic of the surface. Note that for [[orientable manifold\|orientable]] compact surfaces without boundary, the Euler characteristic equals <math>2-2g</math>, where <math>g</math> is the [[genus (mathematics)\|genus]] of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and <math>g</math> counts the number of handles.

			If one bends and deforms the surface <math>M</math>, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will. The theorem states, somewhat surprisingly, that the
			total integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its total curvature is 4π (the Euler characteristic of a sphere being 2), no matter how big or deep the dent.

			Compactness of the surface is of crucial importance. Consider for instance the [[unit disc\|open unit disc]], a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss–Bonnet formula does not work. It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value 2π.

			As an application, a [[torus]] has Euler characteristic 0, so its total curvature must also be zero. If the torus carries the ordinary Riemannian metric from its embedding in '''R'''<sup>3</sup>, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting in total curvature 0. It is not possible to specify a Riemannian metric on the torus with everywhere positive or everywhere negative Gaussian curvature.

			The theorem also has interesting consequences for triangles. Suppose ''M'' is some 2-dimensional Riemannian manifold (not necessarily compact), and we specify a "triangle" on ''M'' formed by three [[geodesic]]s. Then we can apply Gauss–Bonnet to the surface ''T'' formed by the inside of that triangle and the piecewise boundary given by the triangle itself. The geodesic curvature of geodesics being zero, and the Euler characteristic of ''T'' being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle. Since the turning angle at a corner is equal to π minus the interior angle, we can rephrase this as follows:
			:The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle.
			In the case of the plane (where the Gaussian curvature is 0 and geodesics are straight lines), we recover the familiar formula for the sum of angles in an ordinary triangle. On the standard sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is always bigger than π.

			== Special cases ==
			A number of earlier results in spherical geometry and hyperbolic geometry over the preceding centuries were subsumed as special cases of Gauss–Bonnet.

			=== Triangles ===
			In [[spherical trigonometry]] and [[hyperbolic trigonometry]], the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse) amount by which its exterior angles fail to add up to 360°.

			The area of a [[spherical triangle]] is proportional to its excess, by [[Girard's theorem]] – the amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°.

			The area of a [[hyperbolic triangle]], conversely is proportional to its ''defect,'' as established by [[Johann Heinrich Lambert]].

			=== Polyhedra ===
			{{main\|Descartes' theorem on total angular defect}}
			[[Descartes' theorem on total angular defect]] of a [[polyhedron]] is the polyhedral analog:
			it states that the sum of the defect at all the vertices of a polyhedron which is [[homeomorphic]] to the sphere is 4π. More generally, if the polyhedron has [[Euler characteristic]] <math>\chi=2-2g</math> (where ''g'' is the genus, meaning "number of holes"), then the sum of the defect is <math>2\pi \chi.</math>
			This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices).

			Thinking of curvature as a [[Measure (mathematics)\|measure]], rather than as a function, Descartes' theorem is Gauss–Bonnet where the curvature is a [[discrete measure]], and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem.

			==Combinatorial analog==
			There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following one. Let <math>M</math> be a finite 2-dimensional [[pseudo-manifold]]. Let <math>\chi(v)</math> denote the number of triangles containing the vertex <math>v</math>. Then
			:<math> \sum_{v\in{\mathrm{int}}{M}}(6-\chi(v))+\sum_{v\in\partial M}(3-\chi(v))=6\chi(M),\ </math>
			where the first sum ranges over the vertices in the interior of <math>M</math>, the second sum is over the boundary vertices, and <math>\chi(M)</math> is the Euler characteristic of <math>M</math>.

			Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons. For polygons of n vertices, we must replace 3 and 6 in the formula above with n/(n-2) and 2n/(n-2), respectively.
			For example, for [[quadrilateral]]s we must replace 3 and 6 in the formula above with 2 and 4, respectively. More specifically, if <math>M</math> is a closed 2-dimensional [[digital manifold]], the genus turns out <ref>Chen L and Rong Y, Linear Time Recognition Algorithms for Topological Invariants in 3D, arXiv:0804.1982, ICPR 2008</ref>
			:<math> g = 1+(M_{5} +2 M_{6}-M_{3})/8, \ </math>
			where <math>M_{i}</math> indicates the number of surface-points each of which has <math>i</math> adjacent points on the surface.

			==Generalizations==
			Generalizations of the Gauss–Bonnet theorem to ''n''-dimensional Riemannian manifolds were found in the 1940s, by [[Carl B. Allendoerfer\|Allendoerfer]], [[André Weil\|Weil]], and [[Shiing-Shen Chern\|Chern]]; see [[generalized Gauss–Bonnet theorem]] and [[Chern–Weil homomorphism]]. The [[Riemann–Roch theorem]] can also be seen as a generalization of Gauss–Bonnet.

			An extremely far-reaching generalization of all the above-mentioned theorems is the [[Atiyah–Singer index theorem]].

			A generalization to 2-manifolds that need not be compact is [[Cohn-Vossen's inequality]].

			==References==
			{{reflist}}

			==External links==
			* {{springer\|title=Gauss-Bonnet theorem\|id=p/g043410}}
			*[http://mathworld.wolfram.com/Gauss-BonnetFormula.html Gauss–Bonnet Theorem] at Wolfram Mathworld

			{{DEFAULTSORT:Gauss-Bonnet theorem}}
			[[Category:Theorems in differential geometry]]
			[[Category:Riemann surfaces]]

en>Stausifr: Disambiguated: kernel (mathematics) → kernel (set theory)

2012-05-27T06:25:24Z

Disambiguated: kernel (mathematics) → kernel (set theory)

New page

My name: Leslie FitzGibbon Age: 29 Country: Germany Town: Stuttgart Lederberg Post code: 70619 Street: Ollenhauer Str. 23 Look at my web page; [http://safedietplansforwomen.com/bmi-chart bmi chart]

Chain complex - Revision history

en>AReu: /* Chain maps and tensor product */ - Typo in formula corrected

en>Mark viking: /* de Rham cohomology */ remove self-redirect

en>Stausifr: Disambiguated: kernel (mathematics) → kernel (set theory)