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Topic: Curvature of Riemannian manifolds

Related Topics

Weyl curvature

Ricci curvature tensor

Einstein manifold

Riemannian manifold

Ricci flow

Intrinsic metric

Laplacian

General relativity

Sphere

Euclidean geometry

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	Curvature - Wikipedia, the free encyclopedia
		For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
		Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
		Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection.
	en.wikipedia.org /wiki/Curvature (1191 words)

	Curvature (Site not responding. Last check: 2007-11-02)
		The of curvature at points on physical curves be measured in diopters (alternative spelling: dioptre); a diopter is per meter.
		curvature is closely related to the first of surface area in particular a minimal surface like a soap film has mean curvature zero and soap bubble has constant mean curvature.
		Unlike Gauss the mean curvature depends on the embedding instance a cylinder and a plane are locally isometric but the mean curvature of a is zero while that of a cylinder nonzero.
	www.freeglossary.com /Curvature (877 words)

	Research Papers of P. Gilkey
		[8] The spectral geometry of a Riemannian manifold, J.Diff.Geo.
		[60] The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary.
		[127] Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture.
	darkwing.uoregon.edu /~gilkey/respap.html (3586 words)

	Curvature (Site not responding. Last check: 2007-11-02)
		right For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
		Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
		In cosmology, the concept of "curvature of space" is considered, which is the curvature of corresponding pseudo-Riemannian manifolds, see Curvature of Riemannian manifolds.
	curvature.kiwiki.homeip.net (874 words)

	Curvature of Riemannian manifolds - Wikipedia, the free encyclopedia
		In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point.
		It is the Gauss curvature of the σ-section at p; here σ-section is a locally-defined piece of surface which has the plane σ as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of σ under the exponential map at p.
		For a manifold of constant curvature, the Weyl tensor is zero.
	en.wikipedia.org /wiki/Curvature_of_Riemannian_manifolds (962 words)

	toponogov
		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.
		Riemannian spaces are defined usually by some sufficiently complicated constructions, using concepts of the analysis.
		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)

	Amazon.ca: Riemannian Manifolds : An Introduction to Curvature: Books: John M. Lee (Site not responding. Last check: 2007-11-02)
		It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds.
		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.
	www.amazon.ca /Riemannian-Manifolds-Introduction-John-Lee/dp/0387983228 (458 words)

	Craig Sutton Colloquium Abstract (Site not responding. Last check: 2007-11-02)
		Under this scheme geometers tend to study either Riemannian manifolds of nonpositive sectional curvature or Riemannian manifolds of nonnegative sectional curvature.
		In particular, the theory of manifolds of negative sectional curvature is quite rich and examples are plentiful.
		Specifically, there are very few examples of manifolds of positive sectional curvature in the literature and there are few known topological obstructions to having positive sectional curvature.
	www.haverford.edu /math/colloquium/sutton-abs.html (200 words)

	Sectional curvature - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.isi.jhu.edu) (Site not responding. Last check: 2007-11-02)
		In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.
		Riemannian manifolds with constant sectional curvature are the most simple.
		The model manifolds for the three geometries are hyperbolic space, Euclidean space and a unit sphere.
	en.wikipedia.org.cob-web.org:8888 /wiki/Sectional_curvature (220 words)

	CMS/CAIMS Summer 2004 Meeting
		The spectral geometry of the Riemann curvature tensor
		The Blaschke conjecture aims to classify the Riemannian manifolds whose cut loci are metric spheres-in other words, spaces in which light rays shooting out of a given point must all focus at the same time.
		A curvature invariant of order n is a scalar obtained by contraction from a polynomial in the Riemann tensor and its covariant derivatives up to the order n.
	www.cms.math.ca /Events/summer04/abs/ait.html (1338 words)

	Alibris: Curvature
		The Curvature of Spacetime: Newton, Einstein, and Gravitation
		This work is devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow.
		The Curvature Scale Space technique was selected as a contour shape descriptor for MPEG-7 after substantial and comprehensive testing, which demonstrated the superior performance of the CSS-based descriptor.
	www.alibris.com /search/books/subject/Curvature (495 words)

	Torsion in almost Kaehler geometry (Site not responding. Last check: 2007-11-02)
		A classification of almost Kaehler (or even more general almost Hermitian manifolds) was proposed by A. Gray in [?] by considering three classes of AK manifolds whose Riemannian curvature operator has a certain degree of ressemblence with that of a Kaehler manifolds.
		They are defined as those AK - manifolds whose torsion tensor is of special algebraic type with respect to the Kaehler nullity of the almost Kaehler structure.
		As an alternative definition, these manifolds have two complementary foliations, one given by the Kaehler nullity and the other one given by its Riemannian orthogonal.
	www-irm.mathematik.hu-berlin.de /~nagy/torsionAKresz.html (933 words)

	Geometry from a Differentiable Viewpoint
		The concepts of arc length, curvature, and torsion are introduced and it is shown that these quantities form a complete set of invariants in the sense that any two curves for which these quantities are the same differ at most by a rigid motion.
		Integrals of the Gaussian curvature are computed in Chapter 12 leading to the Gauss-Bonnet theorem and its global consequences.
		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)

	projekte
		The aim of the Special Semester is to bring together highly qualified researchers working on different aspects of the geometry of manifolds with indefinite metrics.
		Whereas in Riemannian geometry in the last 30 years an essential progress was made in the investigation and classification of different classes of Riemannian manifolds (e.g.
		manifolds with an addi-tional geometric structures, manifolds with conditions on curvature, homogeneous Riemannian spaces), similar results for pseudo-Riemannian manifolds are rare and many problems are still open.
	www.mathematik.hu-berlin.de /~baum/ESI2005.html (369 words)

	[No title]
		Research InterestsDifferential Geometry: pseudo-Riemannian manifolds, the Riemann curvature tensor, and the covariant derivatives of the Riemann curvature tensor.
		“Curvature homogeneous signature (2,2) manifolds,” Proceedings of the 9th International Conference “Differential Geometry and its Applications.” Joint with P. Gilkey and S. Nikcevic.
		University of Oregon Topology/Geometry Seminar “A study of algebraic invariants of curvature tensors,” November 30, 2004.
	noether.uoregon.edu /~dunn/CV.doc (1024 words)

	Robert C. McOwen's List of Publications (Site not responding. Last check: 2007-11-02)
		Remarks on singular elliptic theory for complete Riemannian manifolds (with H.O. Cordes), Pacific J. The behavior of the Laplacian on weighted Sobolev spaces, Comm.
		Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds (with P. Aviles), J.
		Prescribed curvature and singularites of conformal metrics on Riemann surfaces, J.
	www.math.neu.edu /~mcowen/McOwen_publications.html (338 words)

	Read This: A Panoramic View of Riemannian Geometry
		In his latest book A Panoramic View of Riemannian Geometry, Marcel Berger does a remarkable job giving an in-depth survey ranging over almost the full spectrum of Riemannian geometry, furnishing the reader with some of the most exciting and elegant topics from classical to modern, and from local to global Riemannian geometry.
		The author does not follow the traditional definition-theorem-proof approach; instead, he only motivates and presents, without detailed proofs, the best possible results in many areas known to date, thereby providing interested readers with a valuable source and efficient means for learning about the latest advances.
		However, this book should not be, as cautioned by the author, used as a handbook or primer of Riemannian geometry; rather, with careful selection according to individual's taste, it can be a great reference which may enlarge the breadth of one's knowledge and enhance one's research.
	www.maa.org /reviews/riemannpanorama.html (583 words)

	Andrejewski Vorlesung: Jeff Cheeger (Site not responding. Last check: 2007-11-02)
		At a given point, the curvature of the manifold measures the asymptotic rate of approach to Euclidean geometry as the scale parameter goes to zero.
		As explained in our first lecture, the bad examples of Riemannian manifolds of bounded sectional curvature, say K\le 1, are precisely those which are very collapsed.
		Remarkably, it turns out that all (sufficiently collapsed) such spaces must exhibit a high degree of (generalized) circular symmetry; in the present instance, this symmetry is given by rotation about the axis of the wire.
	www.uni-math.gwdg.de /andrej/cheeger.html (328 words)

	Cornell Math - MATH 757, Fall 2000
		Comparison Geometry studies how metric invariants of a space, especially curvature, determine its topology.
		While the focus of the course will be Riemannian manifolds, sectional curvature and Ricci curvature, we will also explore how the notion of curvature can be extended to more general metric spaces.
		Prerequisites: An introduction to Riemannian geometry, 651 and a willingness to accept results from 662 as needed.
	www.math.cornell.edu /Courses/GradCourses/FA00/757.html (95 words)

	S. Alexander Publications (Site not responding. Last check: 2007-11-02)
		(with R. Bishop), Spines and topology of thin Riemannian manifolds with boundary (pdf), (ps), Trans.
		______, The Fary-Milnor theorem in Hadamard manifolds, Proc.
		______, Geometric curvature bounds in Riemannian manifolds with boundary, Trans.
	www.math.uiuc.edu /~sba/sbapubs.html (307 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		We want to apply analytic methods to problems in geometry and general relativity.
		(N), and F a symmetric monotone curvature function defined on an open convex cone in
		We consider Lorentzian manifolds N having a big crunch singularity and want to analyze, if a smooth extension of N through the singularity to a new spacetime
	www.math.uni-heidelberg.de /studinfo/gerhardt/dfg (204 words)

	Amazon.com: Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics): Books: John M. Lee (Site not responding. Last check: 2007-11-02)
		Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) by John M. Lee
		I mean, I sorta remember stuff like algebra and geometry and triangles and proofs and things like that, and all that math stuff helped me get through the first four chapters.
		After that, the book picked up pace and finished really strong with comparisons of manifolds on both positive and negative curvatures.
	www.amazon.com /Riemannian-Manifolds-Introduction-Curvature-Mathematics/dp/0387983228 (1373 words)

	Cornell Math - MATH 652, Fall 2003
		The prerequisites are advanced calculus, linear algebra, and point set topology.
		The topics covered will include: topological manifolds, differentiable manifolds, immersions and embeddings, tangent bundles, fibre bundles, vector fields and dynamical systems, Frobenius' theorem, Lie groups, differential forms, integration on manifolds, Stokes theorem, connections, Riemannian manifolds, geodesics, curvature.
		The last four topics are also covered quite thoroughly in MATH 662 and so they will probably be discussed to a lesser extent than the other topics.
	www.math.cornell.edu /Courses/GradCourses/FA03/652.html (118 words)

	I.G. Nikolaev: Publication List
		Space of directions at a point of a space of curvature not greater than K. (Russian) Sibirsk.
		Parallel translation of vectors in spaces with curvature that is bilaterally bounded in the sense of A. Aleksandrov.
		Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. Aleksandrov.
	www.math.uiuc.edu /~inik/papers.htm (824 words)

	Table of contents for Library of Congress control number 97165734 (Site not responding. Last check: 2007-11-02)
		A genealogy of noncompact manifolds of nonnegative curvature: history and logic R. Greene 5.
		Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers G. Perelman 9.
		Example of a complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and with nonunique asymptotic cone G. Perelman 11.
	www.loc.gov /catdir/toc/cam024/97165734.html (189 words)

	Member Page
		Mean curvature flow singularities for mean convex surfaces
		The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality
		Convexity estimates for mean curvature flow and singularities of mean convex surfaces
	www.aei.mpg.de /english/php-Skripte/quMembPage/index.php?personKey=huisken (107 words)

	Courses: Math Department, WCAS, Northwestern University
		Manifolds, connections, Riemannian manifolds, curvature, second variation of arc-length, selected other topics.
		Introduction to basic differential topology, manifolds, fixed point theorems, Brouwer degrees, fundamental groups and covering spaces.
		Principal types of processes (such as stationary, Markov, Gaussian) and their general theory; detailed study of some processes such as the Poisson and Brownian motion processes; selected topics such as sums of independent random variables, waiting times, branching processes, information theory.
	www.math.northwestern.edu /courses/GradCourses.html (693 words)

	Petersburg Department of Steklov Institute (Fontanka, 27)
		Wilking: Group actions on manifolds of positive sectional curvature
		Schwachhoefer: Cohomogeneity one manifolds and lower curvature bounds
		Rodionov and V. Slavskii: One-dimensional sectional curvature of Riemannian manifolds
	www.pdmi.ras.ru /EIMI/2002/geo2/18.html (86 words)

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