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Topic: Fundamental theorem of Riemannian geometry

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	Geometry - Search View - MSN Encarta
		The Bolyai-Lobachevsky geometry, often called hyperbolic non-Euclidean geometry, describes the geometry of a plane consisting only of the points on the inside of a circle in which all possible straight lines are chords of the circle.
		Similarly Riemannian, or elliptic non-Euclidean geometry, is the geometry of the surface of a sphere in which all straight lines are great circles.
		His most famous proof, a theorem that bears his name, states that the square of the longest side of a right triangle is equal to the sum of the squares of the other two sides.
	encarta.msn.com /text_761569706__1/Geometry.html (5322 words)

	PlanetMath: first fundamental form
		The first fundamental form is related to the area form as follows.
		Theorem 1 (Theorema Egregium) The Gaussian curvature of a surface is unchanged under isometries (because it only depends on the first fundamental form).
		This theorem is not obvious, since the usual definitions of the Gaussian curvature are not invariant (they depend on the particular embedding of the surface in
	planetmath.org /encyclopedia/FirstFundamentalForm.html (746 words)

	Geometry
		The phrase "discrete geometry," which at one time stood mainly for the areas of packing, covering, and tiling, has gradually grown to include in addition such areas as combinatorial geometry, convex polytopes, and arrangements of points, lines, planes, circles, and other geometric objects in the plane and in higher dimensions.
		Similarly, "computational geometry," which referred not long ago to simply the design and analysis of geometric algorithms, has in recent years broadened its scope, and now means the study of geometric problems from a computational point of view, including also computational convexity, computational topology, and questions involving the combinatorial complexity of arrangements and polyhedra.
		The first Theorem was formulated as a consequence of the second one (it is a form of the strong Maximum Principle for parabolic equations).
	www.wordtrade.com /science/mathematics/geometry.htm (6586 words)

	toponogov
		In the fundamental works of A.D.Aleksandrov, synthetic methods are again used because objects under study are not smooth enough for applications of the classical analysis methods.
		This in-depth theorem is the basis of modern investigations of the relations between curvature properties, geodesics behaviour, and the topological structure of Riemannian spaces.
		Visually Riemannian space may be characterized in such a way that in a small neighbourhood of its arbitrary point the geometry of a space does not differ from the usual Euclidean geometry, and the difference is less when the taken neighbourhood is smaller.
	math.haifa.ac.il /ROVENSKI/toponogov_e.html (1521 words)

	The Mathematics of Fermat's Last Theorem
		Theorem B is even harder still, and it is the theorem of which Andrew Wiles first claimed a proof in 1993, thus proving FLT as well.
		Theorem B certainly seems, to one unfamiliar with the territory, to be quite technical and abstruse.
		Theorem B and more general forms of the Taniyama-Shimura Conjecture can be viewed in yet another way to affirm that there is a very significant relationship between modular functions and elliptic curves.
	cgd.best.vwh.net /home/flt/fltmain.htm (2364 words)

	Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
		The key idea in Finsler geometry is to consider the projectivized tangent bundle PTM (i.e., the bundle of line elements) of the manifold M. The main reason is that all geometric quantities constructed from F are homogeneous of degree zero in y and thus naturally live on PTM, even though F itself does not.
		The fundamental tensor $g_{ij}$ is defined as the y-Hessian $(\frac{1}{2} \, F^2)_{y^i y^j}$.
		As is well known, Riemannian geometry can be handled, elegantly and efficiently, by tensor analysis on M. Its handicap with Finsler geometry arises from the fact that the latter needs more than one space, for instance PTM in addition to M, on which tensor analysis does not fit well.
	www.math.iupui.edu /~zshen/Finsler/history/chern.html (2855 words)

	Geometry / Topology at Michigan State University
		The goal is to understand gravitational radiation in terms of geometric analysis (in particular, to understand the Peeling Theorem without recourse to conformal compactifications, and to understand the relation between ADM and Bondi mass).
		Students should be familiar with the objects of Riemannian geometry: manifolds, vector and tensor fields, differential forms, metrics and curvature, etc. and have a solid knowledge of linear algebra.
		This course starts with introduction to classical algebraic geometry: We begin by introducing affine and projective varieties, study their basic properties and eventually concentrate on the study of algebraic curves.
	www.mth.msu.edu /Related/gt/Courses.htm (743 words)

	Geometry from a Differentiable Viewpoint
		Differential geometry is a subject of basic importance for all mathematicians, regardless of their special interests, and it also furnishes essential ideas and tools to physicists and engineers.
		The modern subject turns on problems that have emerged from the new foundations that are far removed from the ancient roots of geometry, and when we teach the new and cut off the past, students are left to find their own way to a meaning of "geometry" in differential geometry.
		In a very lively manner the spherical and hyperbolic geometries, the classical theory of curves and surfaces and a great part of Riemannian geometry are presented, as well as some applications (the tautochrone and accurate clock of Huygens, map projections and mathematical cartography, Lorentz manifolds as space-time models).
	math.vassar.edu /faculty/McCleary/Geom.page (966 words)

	Differential Geometry at Notre Dame
		The striking feature of modern Differential Geometry is its breadth, which touches so much of mathematics and theoretical physics, and the wide array of techniques it uses from areas as diverse as ordinary and partial differential equations, complex and harmonic analysis, operator theory, topology, ergodic theory, Lie groups, non-linear analysis and dynamical systems.
		We have proved this for compact Riemannian spaces with positively pinched curvature and in another direction established that if two compact surfaces of negative curvature and finite area have the same length data for marked closed geodesics then the two surfaces must be isometric.
		Much of the progress in Riemannian geometry that took place over the last decades has been made via the use of deep analytic techniques on non-compact manifolds.
	www.nd.edu /~jcao/dg.html (822 words)

	Ma 157a, Introduction to Riemannian Geometry
		This is not a textbook which carefully covers foundations of the field, but an 800 page attempt to survey all of modern Riemannian geometry.
		It is a great place to see what Riemannian geometry is all about, and also to get further intuition about basic concepts (there are several hundred figures and innumerable examples).
		There is another point of view one can take on Riemannian geometry which deemphasizes the role of differentiability and focuses on more intrinsically metric-space notions.
	www.its.caltech.edu /~dunfield/classes/2006/157a/index.html (812 words)

	Courses in the Department of Mathematics
		Hartogs’ Theorem), a deeper study of Riemann surfaces, the uniformization theorem, the Dirichlet problem in higher dimensions, differential equations in a complex domain and the Riemann-Hilbert problem, Hardy spaces.
		Inverse and implicit function theorems, transversality, Sard’s theorem and the Whitney embedding theorem.
		Riemannian metrics, connections and curvature on vector bundles, the Levi-Civita connection, and the multiple interpretations of curvature.
	catalogs.uchicago.edu /divisions/math-courses.html (2661 words)

	2006-2007 Course Register
		Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory. The instructor may elect to cover other topics such as Morse Theory, h-cobordism theorem, characteristic classes, cobordism theories.
		Fundamentals of topological groups. Haar measure. Representations of compact groups. Peter-Weyl theorem. Pontrjagin duality and structure theory of locally compact abelian groups.
		Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.
	www.upenn.edu /registrar/register/math.html (4476 words)

	Differential Geometry, Volume 4
		Jörgens theorem; surfaces of constant curvature 0; surfaces of constant
		The author has pulled together the main body of "classical differential geometry" that forms the background and origins of the state of the theory today.
		His unusually extensive chapter on Riemannian submanifolds goes beyond being a good exposition of readily available material, and performs a scholarly service.
	www.mathpop.com /bookhtms/dg4.htm (426 words)

	Course Outline for Math 241AB
		First, there was the theory of minimal surfaces in Riemannian manifolds, which in the hands of Meeks, Schoen and Yau led to advances in three-dimensional topology and a resolution of the positive mass conjecture from general relativity in 1979.
		An application is the sphere theorem of Micallef and Moore (1988): A compact simply connected manifold with positive isotropic curvature is homeomorphic to a sphere.
		This theorem extended the earlier sphere theorems of Berger, Klingenberg and Toponogov, and showed that a new type of curvature, isotropic curvature, is relevant to the study of minimal surfaces.
	www.math.ucsb.edu /~moore/s241.html (1009 words)

	[No title]
		The Cheeger-Gromoll splitting theorem states that a complete Riemannian manifold with nonnegative Ricci curvature is isometric to the product of some Euclidean space and a manifold N where N has no lines, a line being defined as a smooth curve that minimizes distance in both directions.
		Namely, we show that if a manifold $M$ with almost nilpotent fundamental group admits a metric with $-a^2\le sec(M)\le -1$, then the infimum of $a^2$ over all such metrics is equal to square of the nilpotency class of the fundamental group of $M$.
		Abstract: One of the most important subfields of Riemannian Geometry is the study of the Laplace spectrum of a compact Riemannian manifold.
	www.math.ucsb.edu /~wei/dgs03.html (1413 words)

	Higher Dimensional Geometry
		Using time as the fourth dimension, four-dimensional beings would see humans as an infinite series of static forms that represent all motions of life moving through time as seen all at once; a section of such a worldsheet is seen in the image at right.
		Relativity, a description of gravity, was one of the first theories to simplify the laws of nature in higher dimensions and was based on Riemannian geometry.
		Moreover, space's curvature in higher dimensions is describable with Riemannian geometry and is responsible for the forces felt by three-dimensional beings.
	library.thinkquest.org /27930/geometry.htm?tqskip1=1&tqtime=0512 (3058 words)

	Undergraduate Courses
		It covers topics such as: the unique factorization theorem for ideals in rings of algebraic integers, integral bases, the discriminant, the different, ramification, the finiteness theorem for ideal-class groups, Dirichlet's theorem on groups of units of rings of algebraic integers etc. The prerequisites are Algebra 401-402 or an equivalent one year course in algebra.
		The fundamental group of a knot or a link complement will be the central algebraic focus, and spanning surfaces will be the main geometric tool.
		Theory of curves and surfaces in Euclidean space: Frenet equations, fundamental forms, curvatures of a surface, theorems of Gauss and Mainardi-Codazzi, curves on a surface; introduction to tensor analysis and Riemannian geometry; theorema egregium; elementary global theorems.
	mathnt.mat.jhu.edu /new/undergrad/courses_v.htm (1763 words)

	Saccheri's Solution to Euclid's BLEMISH
		Non-Euclidean geometry is one of the marvels of mathematics and even more marvelous is how it gradually evolved through a process of eliminating flaws in logical reasoning.
		As he proceeded through the theorem, he treats the three cases O, E, A one after another hoping to reduce cases O and A to the Euclid case.
		Those credited with the discovery of non-Euclidean geometry such as Riemann, Lobachevsky, Bolyai and Gauss all built on the work of the first geometer to treat the matter seriously and rigorously, Jerome Saccheri, who in uncovering the consequences of the acute angle hypothesis, directed subsequent geometers to a new geometry.
	www.faculty.fairfield.edu /jmac/sj/sacflaw/sacflaw.htm (3512 words)

	Mathematics and Computer Science Courses \| College of the Holy Cross (Site not responding. Last check: 2007-10-12)
		A first course in the differential geometry of curves and surfaces for students who have completed Mathematics 241 and a semester course in linear algebra.
		The major theorems will be studied along with their proofs and the computer will be used as a research tool to do experiments which motivate and illustrate the theory.
		Topics include the fundamentals of two and three dimensional graphics such as clipping, windowing, and coordinate transformations (e.g., positioning of objects and camera), raster graphics techniques such as line drawing and filling algorithms, hidden surface removal, shading, color, curves and surfaces and animation.
	www.holycross.edu /academics/math/courses (2685 words)

	Geometry, euclidian geometry, elements of geometry (Site not responding. Last check: 2007-10-12)
		a Riemannian metric, the fundamental theorem of Riemannian geometry, Levi-Civita connection geometry..
		Projective geometry is a beautiful subject which has some remarkable applications beyond those in standard textbooks geometry.
		The earliest recorded beginnings of geometry may be traced to Ancient Egypt geometry.
	www.educationalend.com /geometry.html (259 words)

	when he was 3 years old he corrected his father (Site not responding. Last check: 2007-10-12)
		Some mathematicians consider the German mathematician Gauss to be the greatest of all time, and almost all consider him to be one of the three greatest, along with Archimedes and Newton; in contrast, he is hardly known to the general public.
		When he was asked to supervise the geodetic survey of Hanover, he solved the computational problems that arose by founding the study of the intrinsic differential geometry of curved surfaces, which was to be fundamental to Riemannian geometry and Einstein's theory of relativity.
		He was the first to give a proof of the fundamental theorem of algebra that every polynomial with real or complex coefficients has at least one root; he gave four different proofs during his life.
	www.ee.umd.edu /~andrekff/gauss.htm (613 words)

	Description of Research - Non-immersion problems in Riemannian geometry
		In [7] we proved, using the local theory developed by Cartan and the growth of the fundamental group, that a hyperbolic manifold with non-abelian fundamental group (most of them have this property) cannot be immersed in the (2
		n+1, with negative Ricci curvature, as follows: a) If n=3 then the infimum of the length of the second fundamental form of M is zero (this is the three dimensional version of a conjecture of Milnor for surfaces).
		The argument uses the structure theorem for Euclidean hypersurfaces of non-negative curvature (due to H. Wu and others).
	www.nd.edu /~fxavier/Research/Riemannian.htm (235 words)

	Fundamental Theory Group - Research in Elementary Particles and Fields
		One of the great triumphs of twentieth century physics was the elucidation of the structure of matter and the forces that govern it.
		Its interrelation with mathematical ideas from modern geometry and topology have broadened our horizons and led to physics beyond the Standard Model.
		Following the general ideas of Connes, Professor Wali (along with Nguyen Ai Viet) has developed a formalism that follows closely the Riemannian geometric approach to construct action functionals on a two sheeted space-time that can be looked upon as discretized version of Kaluza-Klein theory.
	physics.syr.edu /research/fundamental_theory/particles.html (2216 words)

	Lecture Notes on Differential Geometry (Site not responding. Last check: 2007-10-12)
		Regular values, proof of fundamental theorem of algebra, Smooth manifolds with boundary, Sard's theorem, and proof of Brouwer's fixed point theorem.
		The notion of distance on a Riemannian manifold and proof of the equivalence of the metric topology of a Riemannian manifold with its original topology.
		Riemannian connections, brackets, proof of the fundamental theorem of Riemannian geometry, induced connection on Riemannian submanifolds, reparameterizations and speed of geodesics, geodesics of the Poincare's upper half plane.
	www.math.gatech.edu /~ghomi/LectureNotes/index.html (408 words)

	Math 402: Differential Geometry*, Spring 2004
		Differential geometry is the study of geometric figures using the methods of calculus.
		Especially important is the notion of a Riemannian metric which provides concepts of length, angle, volume, and many notions of curvature.
		In a general Riemannian manifold, the notion of a line is replaced by a geodesic which is a curve locally minimizing length or energy.
	math.rice.edu /~hardt/402S04 (774 words)

	Ricci Flow, 3-Manifolds and Geometry — Program
		Hamilton's 3-manifolds with positive Ricci curvature theorem: background and basic techniques used in its proof - linearization of the Ricci tensor, short time existence, basic evolution equations, maximum principles, curvature pinching estimates, convergence criteria.
		Geometrization (3 lectures) : The eight basic 3-dimensional geometries, prime decomposition of 3-manifolds, incompressible tori, Thurston's geometrization conjecture on 3-manifolds, graph manifolds.
		Fundamental results in differential geometry which are used in Perelman's work:
	www.claymath.org /programs/summer_school/2005/program.php (535 words)

	Amazon.com: Riemannian Geometry: Books: Manfredo P. Do Carmo,Francis Flaherty (Site not responding. Last check: 2007-10-12)
		Most books about Riemannian Geometry can be quite difficult to read and understand if you are not lucky to have a good teacher that explains you the contents in a more intuitive way.
		Exposition of key concepts of RG (affine connection, riemannian connection,geodesics, parallelism and sectional curvature,...) are well motivated and concisely explained with numerous motivating and not so difficult execises.
		I have gone through many books about riemannian geometry, only to find that most of them are playing magic in front of me. When it comes to curvature and variation of energy (arc length), most of the book are just playing around with the notations without drawing any geometric insight.
	www.amazon.com /Riemannian-Geometry-Manfredo-Do-Carmo/dp/0817634908 (1860 words)

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