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	Generalized Gauss-Bonnet theorem - Wikipedia, the free encyclopedia
		In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed Riemannian manifold as an integral of a certain polynomial derived 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.
	en.wikipedia.org /wiki/Generalized_Gauss-Bonnet_theorem (156 words)

	Gauss map -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		The Gauss map can be defined the same way for (additional info and facts about hypersurface) hypersurfaces in, this way we get a map from a hypersurface to the unit sphere.
		Finally, the notion of Gauss map can be generalized to an oriented submanifold of dimension in an oriented ambient (additional info and facts about Riemannian manifold) Riemannian manifold of dimension.
		In that case, the Gauss map then goes from to the set of tangent -planes in the tangent bundle.
	www.absoluteastronomy.com /encyclopedia/G/Ga/Gauss_map.htm (327 words)

	The Gauss-Bonnet Formula for Curves
		The Gauss-Bonnet Formula for curves states that the integral of the curvature around a closed curve in a plane plus the sum of the turning angles at corner points is equal to 2π.
		The proof of this theorem in the standard sources is tedious and involved.
		A curvature function κ(s) could be defined as the generalized* function which is the derivative of α(s) but this is usually not what is done.
	www.applet-magic.com /gaussbf.htm (362 words)

	Pfaffian - Wikipedia, the free encyclopedia
		The Pfaffian is an invariant polynomial of a skew-symmetric matrix (Note that it is not invariant under a general change of basis but rather under a proper orthogonal transformation).
		In particular, it can be used to define the Euler class of a Riemannian manifold which is used in the generalized Gauss-Bonnet theorem.
		The term Pfaffian was introduced by Arthur Cayley, who used the term in 1852: "The permutants of this class (from their connection with the researches of Pfaff on differential equations) I shall term Pfaffians." The term honors German mathematician Johann Friedrich Pfaff.
	www.wikipedia.org /wiki/Pfaffian (319 words)

	Gauss–Bonnet theorem - Wikipedia, the free encyclopedia
		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 requires, somewhat surprisingly, that the total integral of all curvatures will remain the same.
		A generalization to n dimensions was found in the 1940s, by Allendoerfer, Weil, and Chern.
	en.wikipedia.org /wiki/Gauss-Bonnet_theorem (180 words)

	Euler characteristic (Site not responding. Last check: 2007-11-05)
		In general, the Euler characteristic is a topological invariant, i.e., any two polyhedra that are homeomorphic to each other have the same Euler characteristic.
		One can therefore extend the definition to more general surfaces than polyhedra, and speak of the Euler characteristic of, for example, a torus, which would be the Euler characteristic of any polyhedron homeomorphic to a torus.
		A discrete analog of this result is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry).
	www.sciencedaily.com /encyclopedia/euler_characteristic (514 words)

	Gauss-Bonnet theorem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		Suppose is a (A small cosmetics case with a mirror; to be carried in a woman's purse) compact two-dimensional (additional info and facts about orientable) orientable (additional info and facts about Riemannian manifold) Riemannian manifold with boundary.
		The theorem applies in particular if the manifold does not have a boundary, in which case the integral can be omitted.
		A generalization to dimensions was found in the 1940s, by Allendoerfer, (United States mathematician (born in France) (1906-1998)) Weil, and (additional info and facts about Chern) Chern.
	www.absoluteastronomy.com /encyclopedia/g/ga/gauss-bonnet_theorem1.htm (174 words)

	[No title]
		The definition of a g* roup in general* is just an abstraction, where the functions become undefined elements and composition is the undefined operation which satisfies the group laws of as* so- ciativity and existence of identity and inverse, these laws being the relations * that equivalences satisfy.
		This magnificent theorem is easier to apply than the coincidence theo- rem and so the Lefschetz number and fixed point index are met more frequently in interesting situations than the coincidence number and coincidence indices.
		Gauss showed for a triangle whose sides areRgeodesics on a surface M in three- space that the sum of the angles equals ss + M KdM, where K is the Gaussian curvature of the surface.
	hopf.math.purdue.edu /Gottlieb/unity.txt (8065 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-11-05)
		Albert Einstein's generalized theory of gravitation was an attempt to ascertain a universal law of gravitation and the electromagnetic force as a unified field theory.
		General Hospital is the longest-running daytime soap opera on the American ABC television network, and is also the longest-running soap opera produced in Hollywood.
		In militaries, a general order is a published directive, originated by a commander, and binding upon all personnel under his command, the purpose of which is to enforce a policy or procedure unique to his unit's situation which is not otherwise addressed in applicable service regulations, military l..
	kamelya.info /browse.php?title=G/GE/GEN (10826 words)

	[No title]
		Just as angle is the length, or 1-volume, of a region of the unit circle in the one dimension case, we can think of the area, or 2-volume, of a region on the unit sphere in three space, denoted by $S^2$, as a representation of angle in the two dimensional case.
		So we have greatly generalized the original 180 degree theorem theorem about the triangle for polygons by the theorem that the total curvature of a simple closed curve is $2 \pi$.
		Since $\chi (M) = 0$ for closed odd-dimensional manifolds, the theorem as stated by Hopf did not seem to generalize to the odd dimensional case, and in particular did not generalize the 180 degree theorem, which as we saw {\it is} generalized by the Gauss-Bonnet Formula.
	www.math.purdue.edu /~gottlieb/Papers/bonn2.tex (6385 words)

	[No title]
		But from the mathematical point of view the exterior angles are supe- rior because they can be defined in more generality than for polygons and more importantly they are the form in which the angular information is presented in a far reaching theorem.
		The Nineteenth Century The Gauss-Bonnet Theorem is such an interesting result that various authors could not resist including parts of its history in their textbooks.
		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.
	hopf.math.purdue.edu /Gottlieb/bonnet.txt (3637 words)

	iqexpand.com (Site not responding. Last check: 2007-11-05)
		For closed Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature--see the Gauss-Bonnet theorem for two-dimensional case and generalized Gauss-Bonnet theorem for general case.
		A discrete analog of the Gauss-Bonnet theorem is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry).
		The concept of Euler characteristic of a bounded finite poset is another generalization, important in combinatorics.
	euler_characteristic.iqexpand.com (1012 words)

	Research (Site not responding. Last check: 2007-11-05)
		We use Clifford algebras to generalize the Cauchy Integral Formula to compact n-dimensional manifolds.
		In order to generalize the Gauss-Bonnet theorem to non-compact surfaces an analog of integration must be developed.
		By the uniformization theorem we may as well consider hyperbolic space modulo a discrete subgroup of isometries and we discuss the case of hyperbolic space in detail.
	www.math.unb.ca /~thj/Academic/Abstracts.html (429 words)

	Curriculum Vitae
		Generalized symplectic geometries and the index of families of elliptic problems, Memoirs A.M.S.
		A nice byproduct of this work is an ``adiabatic proof'' of a very general result concerning the cobordism invariance of the index of families.
		On a theorem of Henri Cartan concerning the equivariant cohomology, math.DG/0005068, An.
	www.nd.edu /~lnicolae/vita.html (3209 words)

	Speakers CurGeo 03
		It is an interesting problem, both from a mathematical and a physical point of view, to classify VA's which are generated by a Virasoro element L, a space g of even primary fields of conformal weight 1 (currents) and a space U of odd primary fields of conformal weight 3/2.
		We prove that natural operators defined on the spaces of general linear connections on vector bundles, on the spaces of linear symmetric connections on base manifolds and on certain tensor bundles can be factorized through the curvature tensors of linear and classical connections, the tensor fields and their covariant differentials with respect to both connections.
		The relation between generalized Jacobi morphisms and generalized Bianchi identities for field theories is analyzed.
	diffiety.org /conf/curgeo03/participants.htm (1905 words)

	Euler characteristic (Site not responding. Last check: 2007-11-05)
		One can therefore extend the definition to moregeneral surfaces than polyhedra, and speak of the Euler characteristic of, for example, a torus, which would be the Euler characteristic of any polyhedron homeomorphic to a torus.
		For Riemannian manifolds, the Euler characteristic canalso be found by integrating the curvature, see the Gauss-Bonnet theoremfor two-dimensional cas and generalized Gauss-Bonnet theorem for general case.
		A discrete analog of this result is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron;see defect (geometry).
	www.therfcc.org /euler-characteristic-35910.html (443 words)

	gauss bonnet theorem
		In mathematics, 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.
		A generalisation to n dimensions was found in the 1940s, by Allendoerfer, Weil, and Chern.
	www.fact-library.com /gauss_bonnet_theorem.html (159 words)

	Binghamton University, Mathematical Sciences, Research Interests
		In this case the theorem was uncovered through exploration with the computer algebra package Magma, which is well worth checking out.
		For example, the Euler characteristic of a surface is a topological invariant based its usual definition in terms of a triangulation of the surface.
		However, it may also be considered geometric in view of the Gauss-Bonnet theorem or spectral in view of the Hodge theorem.
	www.math.binghamton.edu /dept/server/research.html (1474 words)

	Generalized Gauss-Bonnet theorem - Encyclopedia, History, Geography and Biography
		Generalized Gauss-Bonnet theorem - Encyclopedia, History, Geography and Biography
		Let M be a compact Riemannian manifold of dimension 2n and $\Omega$ be the curvature form of the Levi-Civita connection.
		Generalized Gauss-Bonnet theorem, Further generalizations and See also:.
	www.arikah.net /encyclopedia/Chern-Gauss-Bonnet_theorem (191 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 /people/sr/webbook/node2.html (1894 words)

	Generalized Gauss-Bonnet Theorem (Site not responding. Last check: 2007-11-05)
		There is a generalized Gauss-Bonnet theorem, from 1943, which seems not to be online but to which there a few online references.
		Here is a physics paper with references that gives a brief introduction to the theorem, but beware the error in their formula for ω
		Exterior trihedral angles are depicted nicely towards the bottom of the above theorem reference.
	www.cap-lore.com /MathPhys/Simplex/GenGaussBonnet.html (230 words)

	CIRP Annals - 1991 (Site not responding. Last check: 2007-11-05)
		After presenting a brief summary of the present tendencies relative to the design, preparation and management of the fixtures, the plant and its components are described in their general structure, posing particular focus on the geometric and functional features of the modular elements, designed in order to facilitate the automatic operations of manipulation and assembly.
		Three fundamental results are obtained; the first two relate the required shears to the fiber curvatures (normal and geodesic), and the third, called the Gauss-Bonnet theorem, relates the shears to the Gaussian curvature of the part and the fiber orientation.
		The punch was made of rubber instead of steel and a strong counter force was generated by a rod type spring or by a hydraulic pressure unit.
	www.cirp.net /publications/1991.html (8843 words)

	Theorem of Euler (Site not responding. Last check: 2007-11-05)
		A Parametrized Index Theorem for the Algebraic K-Theory Euler Class, by William...
		Henri LIFCHITZ's Generalization of Euler-Lagrange theorem and new primality test...
		E.W. Dijkstra Archive: A (new?) proof of a theorem of Euler's on partitions (EWD...
	www.scienceoxygen.com /math/309.html (144 words)

	Topics: G
		In general: A function that allows determination of some quantities as coefficients in a series expansion.
		Hist: In art, it has generally been considered to be the most pleasing to the eye, and has been used in works from the Pyramids to paintings by Rembrandt.
		Present in the shapes of hurricanes, spiral galaxies, and some biological structures such as the chambered nautilus, it describes a logarithmic spiral.
	www.phy.olemiss.edu /~luca/Topics/g.html (1712 words)

	Master of science in mathematics (Site not responding. Last check: 2007-11-05)
		It examines generation of measures: measures on algebras of sets, the extension of measures, outer measure, the Caratheodory and Hahn extension theorems, the Lebesgue Stieltjes measures and the Riesz representation theorem for a bounded positive linear functional on C ([ a,b]).
		The course discusses the general solution of Riccati's equation; when one particular integral is known, when two particular integrals are known, when three particular integrals are known.
		Spectrum of a linear operator, study of compact operators in B(H), where H is a Hilbert space, spectral theorem for compact normal and self-adjoint operators and Fredholm alternative in Hilbert space.
	www.cuea.edu /faculties/science/masters/mscm.htm (4803 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Gauss-Bonnet theorem (Was Re: Yet another radians question) Date: 6 May 1999 16:07:33 GMT Newsgroups: sci.physics,sci.astro,sci.math,alt.math.moderated Keywords: piecewise-linear versions of Gauss-Bonnet curvature theorem In article
		Well, you'll have deviations from "flatness" at almost all points, in general, so it isn't "summing" which you'll have to do but rather integration (unless you stay in the polyhedral -- piecewise-linear -- category).
		Quoting page V.387: "In 1943 Allendoerfer and Weil proved a generalization of the Gauss-Bonnet formula for a polyhedral piece of a Riemannian manifold imbedded in Euclidean space; using this, they were able to obtain a proof of the general Gauss-Bonnet Theorem for [real-analytic] manifolds, by means of a triangulation." Ref.: Trans.
	www.math.niu.edu /~rusin/papers/known-math/99/gauss_bonnet (353 words)

	Graduate Course Descriptions (Site not responding. Last check: 2007-11-05)
		General theorems of partial differentiation; implicit function theorems; vector calculus in 3-space; line and surface integrals, classical integral theorems.
		Mathematical treatment of random number generation and application of these tools to mathematical topics in Monte Carlo method, limit theorems and stochastic processes for the purpose of gaining mathematical insight.
		The residue theorem, the argument principle, harmonic functions and the Dirichlet problem, analytic continuation and the monodromy theorem, entire and meromorphic functions, the Weierstrass product representation and the Mittag-Leffler partial fraction representation, special functions, conformal mapping and the Picard theorem.
	www.math.ncsu.edu /grad/brochure/courses.html (2183 words)

	CMS Winter 2002 Meeting
		Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, no general method has been proposed which is practical for any but very small groups.
		On a compact Riemannian manifold M with a boundary D we consider the problem of the reconstruction of the Riemannian metric g if are known the lengths of geodesics with endpoints on the boundary D of M.
		This is done by showing, firstly, that the space generated by the representations mentioned above is spanned by residues of Eisenstein series associated to an unramified automorphic characters of a maximal split torus of G.
	www.cms.math.ca /Events/winter02/abs/CP (1608 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		As described on page 464 of my article, Hopf asked for an intrinsic proof and generalization of the even dimensional case of his Satz VI, the topological Gauss-Bonnet theorem.
		This is clearly stated in Theorem I on page 101 of C.B. Allendoerfer and Andre Weil, Amer.
		The Nash embedding theorem which states that every Riemannian manifold can be found as a submanifold of Euclidean space was proved in the 1950's.
	www.math.purdue.edu /~gottlieb/Papers/AW (279 words)

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