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	The Gauss-Bonnet Theorem and Its Generalization
		The Gauss-Bonnet Theorem in 3D space says that the integral of the Gaussian curvature over a closed smooth surface is equal to 2π times the Euler characteristic of the surface.
		Note that the right-hand side of the formula fits into the scheme of the Gauss-Bonnet Theorem in that the Euler characteristic of a plane polygon is just 1 because for a polygon there is 1 face and the number of edges and vertices are equal.
		The theorem may be used to determine the effect of non-smooth singularities such as conical points or ridges.
	www.applet-magic.com /gaussbonet.htm (0 words)

	Descartes 7
		The Gauss-Bonnet Theorem states that for any smooth surface S the integral of the Gaussian curvature is equal to 2
		Moreover, negative curvature would not have seemed a natural concept, since at the beginning of his career (he was twenty-four in 1620) he was reluctant to consider negative numbers at all.
		Finally, for Descartes the distinction between a vertex and the measure of the (planar or solid) angle at that vertex was not explicit; the lack of this distinction probably kept him from the combinatorial version of his theorem that Euler derived.
	e-math.ams.org /featurecolumn/archive/descartes7.html (356 words)

	Generalized Gauss-Bonnet theorem (Site not responding. Last check: )
		In mathematics, the generalized-Gauss-Bonnet theorem presents Euler characteristic of closed Riemannian manifold as an integral of a certain polynomial from its curvature.
		It is a direct generalization of the Gauss-Bonnet theorem to general even dimension.
		As with the Gauss-Bonnet theorem, there are generalizations when M is a manifold with boundary.
	www.xasa.com /wiki/en/wikipedia/g/ge/generalized_gauss_bonnet_theorem.html (139 words)

	Gauss-Bonnet theorem
		The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).
		Suppose M is a compact two-dimensional orientable Riemannian manifold with boundary ∂M.
		The theorem requires, somewhat surprisingly, that the total integral of all curvatures will remain the same.
	www.ebroadcast.com.au /lookup/encyclopedia/ga/Gauss-Bonnet_theorem.html (120 words)

	Springer Online Reference Works (Site not responding. Last check: )
		The Gauss–Bonnet theorem was known to C.F. Gauss [1]; it was published by O.
		Bonnet [2] in a special form (for surfaces homeomorphic to a disc).
		Other generalizations of the theorem are connected with integral representations of characteristic classes by parameters of the Riemannian metric [4], [6], [7].
	eom.springer.de /G/g043410.htm (272 words)

	Bonnet biography
		From 1868 Bonnet assisted Chasles at the École Polytechnique, and three years later he became a director of studies there.
		In 1878 Bonnet succeeded Le Verrier to the chair at the Sorbonne, then in 1883 he succeeded Liouville as a member of the Bureau des Longitudes.
		A formula for the line integral of the geodesic curvature along a closed curve is known as the Gauss-Bonnet theorem.
	www-gap.dcs.st-and.ac.uk /~history/Biographies/Bonnet.html (341 words)

	Bonnet
		Bonnet was elected to the Academy of Sciences in 1862 to replace
		Bonnet made major contributions introducing the notion of geodesic curvature.
		Bonnet's work used a special coordinate system an a surface such as isothermic and tangential coordinates.
	www.educ.fc.ul.pt /icm/icm2003/icm14/Bonnet.htm (310 words)

	Bonnet (print-only) (Site not responding. Last check: )
		Bonnet was elected to the Academy of Sciences in 1862 to replace Biot.
		In 1878 Bonnet succeeded Le Verrier to the chair at the Sorbonne, then in 1883 he succeeded Liouville as a member of the Bureau des Longitudes.
		Bonnet's work used a special coordinate system an a surface such as isothermic and tangential coordinates.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Bonnet.html (349 words)

	Bonnet biography
		From 1868 Bonnet assisted Chasles at the École Polytechnique, and three years later he became a director of studies there.
		In 1878 Bonnet succeeded Le Verrier to the chair at the Sorbonne, then in 1883 he succeeded Liouville as a member of the Bureau des Longitudes.
		A formula for the line integral of the geodesic curvature along a closed curve is known as the Gauss-Bonnet theorem.
	www-history.mcs.st-and.ac.uk /history/Biographies/Bonnet.html (341 words)

	[No title] (Site not responding. Last check: )
		In 1848 O. Bonnet extended this formula to smooth closed curves on the surface by a limiting argument which is like the extension from polygons to curves men- tioned above.
		It is very convenient to have a name for important theorems and the main point is that people should know approximately what theorem is meant by the name rather than who gets the credit.
		It is clear that at that time Hopf did not know that the Gauss-Bonnet theorem held for all dimensions and thus was a generalization of 180 degree theorem.
	www.math.purdue.edu /research/atopology/Gottlieb/bonnet.txt (3637 words)

	Gauss–Bonnet theorem - Wikipedia, the free encyclopedia
		The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces 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.
		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.
	en.wikipedia.org /wiki/Gauss-Bonnet_theorem (774 words)

	Omega Art - Gaussian curvature and the Gauss-Bonnet theorem
		The examples given are rather simple (cones and pyramids aren’t exactly difficult surfaces) and merely hint at the general underlying mathematics.
		Mathematically this tends to get rather complex very quickly, although the general idea remains valid: arrows perpendicular to a surface that travel along a closed path correspond to a piece of the unit sphere.
		The Gauss-Bonnet theorem is an amazing and non-trivial piece of mathematics, which we won’t describe in detail here.
	www.omega-art.com /math/gauss.html (0 words)

	Gaussian curvature Summary
		A Gauss map is a function describing the distance to a sphere from an orientable surface in Euclidean space.
		Gauss's 1828 excellent theorem or Theorema egregium states that the Gaussian curvature depends only on the first fundamental form (metric tensor) and its derivatives and not on the second fundamental form.
		A corollary of this theorem is that the Gaussian curvature is invariant under isometric deformations of the surface.
	www.bookrags.com /Gaussian_curvature (1037 words)

	[No title] (Site not responding. Last check: )
		A new proof of the Gauss-Bonnet theorem based on its reduction to a theorem about dual cones is given.
		Two different proofs of the dual cones theorem are given, one based on "moving fronts" and another on a heuristic mechanical analogy.
		As a corollary of this approach we obtain the observation that the total geodesic curvature of a curve on a surface is an invariant of the Gauss map, and that dual curves have reciprocal geodesic curvatures.
	www.math.psu.edu /levi/bike.html (179 words)

	Gauss (Site not responding. Last check: )
		He was a child prodigy, as so many others and one of his achievements of which he was most proud was to have discovered, at age nineteen, that the regular 17 sided polygon was constructible with ruler and compass.
		By then Gauss had already received his doctorate; in his doctoral dissertation he gave six different proofs of what is known as the fundamental theorem of algebra.
		Gauss' great love was always number theory; he once said "Mathematics is the queen of science, and number theory is the queen of mathematics," but there is no area of mathematics or physics in which he did not make fundamental contributions.
	www.math.fau.edu /schonbek/Modern_Analysis/calcmath15.html (320 words)

	Gauss-Bonnet Surface Description
		The Gauss-Bonnet theorem is used to describe the total curvature of a closed surface.
		A closed surface is one that does not have any boundaries, such as a sphere or a cube.
		Moreover, using the Gauss-Bonnet formula it is possible to calculate the total curvature of any genus of shape (as long as it is a closed surface) just by counting the number of holes in that shape.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/AV0405/LINDSELL/Gauss-Bonnet.html (806 words)

	Intute: Science, Engineering and Technology - Search results
		Gaussian curvature and the Gauss-Bonnet theorem : an introduction for mathematically inclined non-mathematicians
		It covers: the work of Carl Friedrich Gauss, Alexander von Humboldt and Wilhelm Weber; explorations and surveys to study the Earth's magnetic field; and Michael Faraday's work on lines of force (field lines) and the disk dynamo.
		In addition, important concepts in sampling required for the understanding of the sampling theorem and the problem of aliasing are addressed.
	www.intute.ac.uk /sciences/cgi-bin/search.pl?term1=Gauss-Jordan (1716 words)

	Introduction
		In particular, by the Hodge theorem the dimension of the kernel of
		Even more generally, the Atiyah-Singer index theorem, dating from the early 1960s, shows that the index of any elliptic first order geometric operator D is given by such an integral, even though the index need not have an obvious topological interpretation.
		This proof assumes the existence of an integral kernel, the heat kernel, for heat flow for the Laplacians on forms; the construction of the heat kernel is in Chapter 3.
	math.bu.edu /INDIVIDUAL/sr/webbook/node2.html (1894 words)

	Synge's Theorem
		It seems Sygne's Theorem relates the differential concept of curvatrue to the globle concept of genus...
		I don't know about his "theorem", but here is an account of Synge the man.
		Gauss-Bonnet is a beautiful theorem and maybe its relevance to stringy business has been explained already (Lethe could help us on that) and who knows, maybe Synge's is too.
	www.physicsforums.com /showthread.php?t=9114 (1749 words)

	math lessons - Gauss-Bonnet theorem
		The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).
		The theorem applies in particular if the manifold does not have a boundary, in which case the integral
		The theorem requires, somewhat surprisingly, that the total integral of all curvatures will remain the same.
	www.mathdaily.com /lessons/Gauss-Bonnet_theorem (159 words)

	Math 4530-1 spring 2005 lecture notes page
		apr1.pdf 4.4 Ros' proof of Alexandrov's sphere theorem for compact constant mean curvature surfaces.
		mar21.pdf 3.4-3.5 Liebmann's amazing theorem that compact surfaces with constant Gauss curvature must be spheres.
		feb11.pdf linalg: matrix of a linear transformation, self-adjoint operators, the spectral theorem; background we apply to the shape operator.
	www.math.utah.edu /~korevaar/4530spring05/4530lectures.html (399 words)

	Edinburgh Mathematics Programme
		Principal curvatures (as eigenvalues of II relative to I), Gauss and mean curvature.
		Statement of general Stokes' theorem and its relation to vector calculus.
		The syllabus takes a very particular route through this material so as to arrive at the Gauss-Bonnet theorem, which is usually not covered at this level.
	www.maths.ed.ac.uk /~derek/Syll/DGe.html (426 words)

	Googlism : what is gauss-bonnet theorem
		gauss-bonnet theorem is presented as a geometric version of green's theorem or the divergence theorem
		gauss-bonnet theorem is one of the most important theorems in differential geometry
		gauss-bonnet theorem is so interesting that various authors could not resist including parts of its history in their textbooks
	www.googlism.com /what_is/g/gauss-bonnet_theorem (158 words)

	Gauss-Bonnet Formula -- from Wolfram MathWorld
		So if you distort the surface and change the curvature at any location, regardless of how you do it, the same total curvature is maintained.
		Another way of looking at the Gauss-Bonnet theorem for surfaces in three-space is that the Gauss map of the surface has map degree given by half the Euler characteristic of the surface
		This makes the Gauss-Bonnet theorem a simple consequence of the Poincaré-Hopf index theorem, which is a nice way of looking at things if you're a topologist, but not so nice for a differential geometer.
	mathworld.wolfram.com /Gauss-BonnetFormula.html (411 words)

	Euler's Theorem as the Path towards Geometry by Emil Saucan in the Nexus Network Journal vol. 7 no. 1 (Spring 2005)
		Moreover, faces as homeomorphic images of the disk and the topological concept of map (as opposed to that of a mere graph embedding) come under study naturally.
		Trees are the simplest truly interesting graphs and they provide us with a third Proof of Euler's Theorem: Von Staudt's proof [11] based upon spanning trees.
		Paradoxically, the benefit of this proof is that it does not extend to surfaces of positive genus -- thus allowing yet a different insight into the topology of surfaces.
	www.cirm.univ-mrs.fr /EMIS/journals/NNJ/Saucan.html (1902 words)

	INI : Abstracts : GMR : The Gauss-Bonnet Theorem for Poincare-Einstein metrics (Site not responding. Last check: )
		A useful tool in the study of the AdS/CFT correspondence is the renormalized volume.
		For four dimensional Poincare-Einstein manifolds, a theorem of Mike Anderson relates the renormalized volume and the Euler characteristic of the underlying manifold with boundary.
		We will discuss the extension of the renormalization procedure to curvature integrals on these manifolds and the proof of the corresponding Gauss-Bonnet theorem in all even dimensions.
	www.newton.cam.ac.uk /programs/GMR/albin.html (77 words)

	Transactions of the American Mathematical Society
		Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on
		J. Brüning, The signature theorem for manifolds with metric horns, Journées: ``Équations aux Dérivées Partielles'' (Saint-Jean-de-Monts, 1996), Exposé II, École Polytech., Palaiseau, 1996.
		Stokes theorem and Hodge theory for singular algebraic varieties, to appear in Math.
	www.ams.org /tran/2003-355-04/S0002-9947-02-03168-9/home.html (0 words)

	Generalizing the Gauss-Bonnet Theorem Gauss-Bonnet Theorem
		Question: According to the Gauss-Bonnet theorem, if X is a compact, even- dim hypersurface in R^(k+1), then integral of K over X = Vol(S^k^)*Chi(X)/2.
		However, all of the proofs I have seen of this theorem assert that X must be a k-manifold, i.e.
		Is it possible to generalize the Gauss Bonnet theorem to non-regular hypersurfaces?
	www.newton.dep.anl.gov /newton/askasci/1993/math/MATH012.HTM (0 words)

	m138b (Site not responding. Last check: )
		Gauss' Theorema Egregium, parallel transport and covariant differentiation, geodesics, exponential sprays, the Gauss-Bonnet Theorem and its applications, models for hyperbolic geometry.
		Completeness and the Hopf-Rinow Theorem, first and second variations of arc length, Bonnet's Theorem for surfaces with positive curvature, Hadamard's Theorem for surfaces with negative curvature.
		This outline leaves substantial time for additional topics to be chosen by the instructor.
	math.ucr.edu /home/UndergradInfo/pages/m138b (80 words)

	Math Club Schedule of Events
		In this talk we explain these concepts by focusing in at its core, using the Foucault pendulum on the physics side and the Gauss-Bonnet theorem on the mathematics side.
		From the mathematics perspective, we will explore basic spherical geometry and prove the Gauss-Bonnet theorem for the round sphere by elementary means.
		From the physics perspective, we give a geometric explanation of the Foucault pendulum that establishes it as a prototype for a geometric phase.
	www.nd.edu /%7Emathclub/colloquia.html (991 words)

	Introduction to Differential Geometry 1 -- from Mathematica Information Center
		We study the Gauss map, Gauss, mean and principal curvatures for surfaces in space, and Gauss curvature for abstractly defined surfaces.
		We also study "extremal" objects, such as distance and energy minimizing curves on surfaces, and area minimizing surfaces.
		Examples of some of the graphics used in teaching Differential Geometric topics in the course with Mathematica are assembled on the "Differential Geometry Images" page which is linked to our homepage http://www.math.uiowa.edu/~seaman/.
	library.wolfram.com /infocenter/Courseware/271 (1044 words)

	Generalized Gauss-Bonnet Theorem
		using classic spherical trigonometry which is a special case of GBT (the Gauss-Bonnet Theorem) but we still have no closed forms for A
		In our terminology the exterior angle at a vertex of a polyhedron is the content of the set of tangents there.
		The sum of the exterior angles of a convex polyhedron in R
	www.cap-lore.com /MathPhys/Simplex/GenGaussBonnet.html (0 words)

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