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Topic: Hadamard Cartan theorem

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	Search Results for theorem*
		Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'.
		The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years.
		It is nevertheless certain that the theorem on the sum of the three angles of the triangle should be considered one of those fundamental truths that are impossible to contest and that are an enduring example of mathematical certitude.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=theorem*&CONTEXT=1 (16791 words)

	RAP
		This theorem strongly resembles the Cartan-Hadamard Theorem of
		Riemannian geometry because it uses local curvature conditions to draw powerful conclusions about the global geometry and topology of a space.
		April 24, Bogdan Petrenko, UIUC; Cartan-Hadamard Theorem and Alexandrov patchwork
	www.math.uiuc.edu /~kapovich/rap1.html (279 words)

	Search Results for theorem
		His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis.
		The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper by Jacobi (1836).
		Mercer's theorem about the uniform convergence of eigenfunction expansions for kernels of operators appears in his 1909 paper Functions of positive and negative types and their connection with the theory of integral equations published in the Philosophical Transactions.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=theorem&CONTEXT=1 (15689 words)

	Amazon.fr : Lectures on Space of Nonpositive Curvature: Livres: Werner Ballmann (Site not responding. Last check: 2007-11-01)
		In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces.
		In the smooth case, this is known and is the content of the Rank Rigidity theorem.
		An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book.
	www.amazon.fr /exec/obidos/ASIN/3764352426 (494 words)

	Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
		Here the remarkable fact is that the same formula holds, with the sectional curvature in the Riemannian case replaced by the flag curvature in the general Finsler case.
		As a consequence all the classical theorems, such as the Hadamard-Cartan theorem on manifolds of nonpositive curvature, the Bonnet-Myers theorem, the Synge theorem, the first comparison theorem of Rauch as well as the Bishop-Gromov volume comparison theorem, extend to the Finsler setting.
		The dimension of the space of all harmonic forms of degree $p$ is the $p$-th Betti number of the manifold.
	www.math.iupui.edu /~zshen/Finsler/history/chern.html (2855 words)

	CBMS BRIDSON ABSTRACTS
		The word problem as measured by Dehn functions; connection to isoperimetric properties of manifolds via the Filling Theorem; the isoperimetric spectrum; the clear demarkation of hyperbolic groups and non-positively curved groups (again).
		The contrasting theorems of Bridson in the combable case.
		Theorems establishing the equivalence of hyperbolic and atoroidal properties in various contexts (3-manifolds, analytic manifolds, new results in 2003--2004).
	www.albany.edu /~ted/abstracts.html (712 words)

	CBMS BRIDSON ABSTRACTS
		The word problem as measured by Dehn functions; connection to isoperimetric properties of manifolds via the Filling Theorem; the isoperimetric spectrum; the clear demarkation of hyperbolic groups and non-positively curved groups (again).
		The contrasting theorems of Bridson in the combable case.
		Theorems establishing the equivalence of hyperbolic and atoroidal properties in various contexts (3-manifolds, analytic manifolds, new results in 2003--2004).
	www.math.albany.edu /~ted/abstracts.html (712 words)

	EULER Record Details
		Finsler versions of the Hopf-Rinow, Cartan-Hadamard and Bonnet theorems are proved.\par The second chapter is devoted to the geometry of complex Finsler metrics.
		Finally, the Cartan connection and the Chern- Finsler connection are compared.\par In the third chapter there are brought together, in a unified form, the results and applications that motivated the work of the authors.
		Finally, a characterization of strongly convex circular domains is given in terms of differential geometric properties of the Kobayashi metric.\par I recommend the book to everyone who appreciates the beauty of both real and complex Finsler geometry or who uses differential geometry for global approaches of certain problems of real and complex analysis.
	www.emis.de /projects/EULER/detail?ide=1994abatfinsmetrglob&matchno=54&matchtotal=190&q=subseries (485 words)

	Mathematics 217B: Differential Geometry
		Class 11 (Monday, February 6): The Hopf-Rinow theorem; extremal properties of geodesics in Lorentzian manifolds
		Class 19 (Monday, February 27): The Cartan-Hadamard theorem; properties of the fundamental group of a manifold with non-positive sectional curvature
		Class 25 (Monday, March 13): Toponogov's theorem; an estimate for the volume of a tubular neighborhood of a closed geodesic
	math.stanford.edu /~brendle/math217b.html (731 words)

	Mathematics (Site not responding. Last check: 2007-11-01)
		Eigenvalues, eigenvectors, diagonizable matrices, the Cayley- Hamilton theorem.
		Properties of integers, equivalence relations, groups, sub-groups, cyclic groups, normal sub-groups, Lagrange's theorem, quotient groups, the homomorphisms theorems, rings and fields: definition and examples, polynomial rings, the Euclidean algorithm and the g.c.m., zero divisors, integral domains, ideals, quotient rings, and the homomorphism theorem, unique factorization in rings of polynomials over a field.
		Lebesgue measure, measurable functions, integrable functions and convergence theorems, the relation between the Riemann and Lebesgue integrals, monotone functions and functions of bounded variation, differentiation of monotone functions, absolutely continuous functions, the Riemann-Stieltjes integral, the theorems of Fubini and Tonelli.
	www.math.technion.ac.il /department/courses/sub010.html (3670 words)

	Springer Online Reference Works
		The study of the topology of Riemannian manifolds is usually preceded by the study of their metric properties.
		There are also comparison theorems for volumes of tubular neighbourhoods of submanifolds (see [13], [16]).
		Extremal theorems are not always connected with estimates of the curvature.
	eom.springer.de /R/r082160.htm (1742 words)

	Atlas: Developability and nonpositive curvature by Andre Haefliger (Site not responding. Last check: 2007-11-01)
		Theorem Any groupoid of local isometries which is connected, Hausdorff, complete and of non-positive curvature is developable.
		This means that such a groupoid is equivalent to the groupoid associated to an action of a group G on a complete simply-connected metric space [X\tilde] of non-positive curvature.
		The proof is an adaptation of the proof of the Cartan-Hadamard theorem given by S. Alexander and R.L. Bishop.
	atlas-conferences.com /cgi-bin/abstract/cabb-31 (281 words)

	DC MetaData for Preprint Nr. 2055
		Abstract: In this paper we study Banach--Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point.
		In this context we generalize the classical theorem of Cartan--Hadamard, saying that the exponential function is a covering map.
		Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.
	wwwbib.mathematik.tu-darmstadt.de /Preprints/shadows/pp2055.html (125 words)

	: John M. Lee,J. M. Lee (Site not responding. Last check: 2007-11-01)
		The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation.
		The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem.
		This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way.
	www.amazon.fr /exec/obidos/ASIN/0387983228 (355 words)

	《微分流形和黎曼流行（影印版）》图书详细资料信息 / China-Pub (Site not responding. Last check: 2007-11-01)
		Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms.
		I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem.
		I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions.
	www.china-pub.com /computers/common/info.asp?id=19796 (268 words)

	Mathematics > Geometry
		This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups.
		Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology.
		It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.
	www.alltextbook.com /h_Books/491546_7.html (2364 words)

	Graduate Course Descriptions (Site not responding. Last check: 2007-11-01)
		Cartan: Elementary theory of analytic functions of one or several complex variables, Dover.
		Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.
		These notions will be explained by examining simple concrete examples of dynamical systems such as translations and automorphisms of tori, expanding maps of the interval, Markov chains, etc. Fundamental theorems of ergodic theory such as the Poincare recurrence theorem, and the Birkhoff ergodic theorem will be presented.
	www.math.toronto.edu /graduate/courses/descriptions.html (3907 words)

	2007-2008 Faculty of Arts and Science Calendar (Site not responding. Last check: 2007-11-01)
		The basic concepts of calculus: limits and continuity, the mean value and inverse function theorems, the integral, the fundamental theorem, elementary transcendental functions, Taylor’s theorem, sequence and series, uniform convergence and power series.
		Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers.
		Cauchy’s theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.
	www.artsandscience.utoronto.ca /ofr/calendar/crs_mat.htm (2671 words)

	Faculty Foster (Site not responding. Last check: 2007-11-01)
		Differentiable manifolds; Tangent space, vector fields; Frobenious theorem; relation between Lie sub-algebras and Lie subgroups; Cartan's theorem on closed subgroups; one parameter subgroups; Exponential maps; Adjoint representation; Homogenous spaces; Compact Lie groups; symmetric spaces.
		Normal families and applications, Riemann mapping theorem, Conformal mapping of a sequence of domains; Modular function, Hyperbolic metric, Elementary theory of univalent functions, Lowner's theory, Dirichlet problem, Green's function and conformal mapping; transfinite diameter and capacity; Symmetrization, Extremal length and prime ends.
		Rotation theorems, Radii of star-likeness and convexity, principle of subordination, p-valent, mean p-valent and circumferentially mean p-valent functions.
	www.iitk.ac.in /math/files/doct_courses.html (2535 words)

	New Zealand Mathematical Societu Newsletter Number 84, April 2002 (Site not responding. Last check: 2007-11-01)
		It is well-known that this theorem does not extend to the class of all matroids.
		Berwald and Cartan generalised different aspects of this connection to Finsler geometry and obtained two different connections which possessed the second property but not the first, while Chern's connection (which is identical with Rund's) has the first property but not the second.
		We are introduced to the Finsler versions of well-known theorems from Riemannian geometry and finally we study the Riemannian manifold as a special case of the Finsler theory.
	ifs.massey.ac.nz /mathnews/Nzms84/news84a.shtml (16991 words)

	Read This: Math Talks for Undergraduates
		The motivation is Cauchy's residue theorem from complex analysis, but Lang "is basically concerned with real analysis in the plane." I found this the least exciting of the talks.
		Then, as applications and examples, he discusses the Weierstrass approximation theorem, Fourier series, harmonic functions, Poisson families and solutions of the heat equation.
		We start with Apollonius's theorem (though Lang doesn't call it that), define a Bruhat-Tits space, prove Serre's theorem that a subset has a circumcentre, leading to the Bruhat-Tits fixed point theorem.
	www.maa.org /reviews/langtalks.html (1068 words)

	m568summary
		estimates, geodesic triangles on Hadamard manifolds (law of cosines, angle inequalities).
		Existence of closed geodesics in free homotopy classes (Cartan's theorem).
		Volume of geodesic balls/spheres in terms of Jacobi fields on Hadamard manifolds.
	www.math.utk.edu /~freire/m568summary.html (197 words)

	New Zealand Mathematical Societu Newsletter Number 83, December 2001
		David won the prize with a short speech giving a proof of the topological theorem that there must be at least one point in the ocean at which the mean tidal range is zero.
		These gems include Pythagoras' theorem, the golden rectangle and tilings with congruent tiles or just a few polygonal shapes but arranged in a pattern which is scalable (similar to a subdivision of itself into smaller tiles of the same shapes) but never repeats, featuring the results of Berger, Penrose and Conway.
		When a pupil asked "What is Pythagoras' theorem?" he replied "The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides".
	ifs.massey.ac.nz /mathnews/NZMS83/news83.htm (17750 words)

	Abbeys Bookshop - Fundamentals of DIfferential Geometry
		Introduces the concepts in differential topology, differential geometry, and differential equations, and some of the basic theorems in the three areas.
		Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds", the focus changed from the general theory of manifolds to general differential geometry.
		The Theorem of Frobenius * II: Metrics, Covariant Derivatives and Riemannian Geometry: Metrics.
	www.abbeys.com.au /items/29/75/16 (146 words)

	Bao,
		comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature.
		methods, fixed point theorems, and connections with differential equations.
		theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution
	www.yurinsha.com /322/p18.htm (1257 words)

	Summer School in Low Dimensions - Home page (Budapest, August 3 - 14, 1998)
		Neumann will start with a new short proof of the toral decompositon theorem, then survey the current state of Thurston's geometrization conjecture, which says that the components of the toral decompositon of a three-manifold should be geometric.
		One of the basic results in this area is a generalization of the Cartan--Hadamard Theorem: if a nonpositively curved space is complete and simply connected then it is contractible.
		It follows that if a nonpositively curved space is compact, then it is an Eilenberg--MacLane space of type K(G,1) where G denotes its fundamental group.
	www.math-inst.hu /~carlos/diffabs.html (417 words)

	Mathematics
		Fostering the ability to understand theorems and their purpose by studying selected groups of theorems in contexts that are new to the students and not part of the regular introductory courses.
		Complex functions and their differentiability, complex integrals, power series, the Cauchy theorem and formulas, residues and applications to evaluating integrals, conformal mappings and applications.
		Asymptotic analysis, initial-value problems, multiscale analysis and the averaging theorem, homogenization theory, boundary- and transition-layer problems, matched asymptotic expansions, relaxation oscillations and propagating transition layers.
	www.acs.utah.edu /GenCatalog/1038/crsdesc/math.html (4875 words)

	A Cartan-Hadamard Theorem for Banach-Finsler Manifolds (ResearchIndex) (Site not responding. Last check: 2007-11-01)
		In this paper we study Banach{Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive dierential in every point.
		In this context we generalize the classical theorem of Cartan{ Hadamard, saying that the exponential function is a covering map.
		We apply this to symmetric spaces and thus obtain criteria for Banach{Lie groups with an involution to have a polar decomposition.
	citeseer.ist.psu.edu /377343.html (293 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		In general, geometry determines some properties about the (global) shape of the space.
		For instance, if a space has a non-positively curved geometry, after "unwrapping" the space we obtain another space that, unlike a doughnut, can be shrunk within itself to a single point (Cartan-Hadamard theorem).
		This research project is oriented towards studying such relations on spaces with non-positive geometry: how the geometry (local data) determines the global shape (topology) and also how topology could determine the geometry.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9505136.txt (272 words)

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