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Topic: Sectional curvature

Related Topics

Ricci curvature tensor

Einstein manifold

Tangent bundle

Endomorphism

Torsion

Local coordinates

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	Sectional curvature - Wikipedia, the free encyclopedia
		It is the Gaussian curvature of that section — the surface which has the plane σ
		Formally, the sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.
		The sectional curvature determines the curvature tensor completely, and is very useful geometric notion.
	en.wikipedia.org /wiki/Sectional_curvature (210 words)

	Sectional curvature (Site not responding. Last check: 2007-11-05)
		It is the Gauss curvature of that section - the surface which has the plane $\sigma_p as tangent plane at p obtained from geodesics which start at p in the directions of \sigma_p (in words the image of \sigma_p under the exponential map at p).$
		Sectional curvatures in all directions at p determine the curvature tensor completely and it is very useful notion.
		They are the only complete simply connected Riemannian manifolds of given sectional curvature all other complete constant curvature manifolds are of those by some group of isometries.
	www.freeglossary.com /Sectional_curvature (519 words)

	SPECIAL - Online Information article about SPECIAL (Site not responding. Last check: 2007-11-05)
		section is w impinge with a velocity v on a plane surface at rest, in a direction normal to the plane.
		Suppose the section AoAo taken at a point where the parallel motion has not begun to be disturbed, and A2A2 where the parallel motion is re-established.
		Also, for the sections AIAI and A2A2, allowing that the head due to the relative velocity vi —v is lost in shock: pl vl2 p2 v2 (v, —v)2.
	encyclopedia.jrank.org /SOU_STE/SPECIAL.html (6066 words)

	PlanetMath: sectional curvature
		This is a natural generalization of the classical Gaussian curvature for surfaces.
		Cross-references: surfaces, Gaussian, tensor, curvature, metric tensor, span, subspace, Riemannian manifold
		This is version 1 of sectional curvature, born on 2006-05-08.
	planetmath.org /encyclopedia/SectionalCurvature.html (62 words)

	Riemannian geometry - Wikipedia, the free encyclopedia
		Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M) where χ(M) denotes the Euler characteristic of M.
		The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric.
		If M is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of the fundamental group of M is isomorphic to Z.
	en.wikipedia.org /wiki/Riemannian_geometry (840 words)

	Craig Sutton Colloquium Abstract (Site not responding. Last check: 2007-11-05)
		Sectional curvature is perhaps the most fundamental geometric invariant in differential geometry.
		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)

	[No title] (Site not responding. Last check: 2007-11-05)
		Gauss found it remarkable that the value for his notion of curvature remained the same, so it is reasonable for you to find it a surprise.
		If you take a surface which initially has both of the sectional curvatures 1/R and isometrically bend it so that one sectional curvature increases, then the other sectional curvature decreases in such a way as to keep the product of the two the same.
		The Gaussian curvature remains 0, because although there is now a nonzero sectional curvature (which you can make whatever you like, in principle) the other sectional curvature is still 0.
	www.math.niu.edu /~rusin/known-math/99/sectional_curv (216 words)

	INI : Abstracts : GMR : Sectional Curvature and General Relativity (Site not responding. Last check: 2007-11-05)
		It is then shown that, with the exception of plane waves and spaces of constant curvature (and always for non-flat vacuum metrics),the sectional curvature function uniquely determines the space-time metric.
		Thus the suggestion is made that the sectional curvature function is a possible alternative variable for general relativity.
		Some of the properties of the sectional curvature function are then explored.
	www.newton.cam.ac.uk /programmes/GMR/hall.html (83 words)

	Sectional Curvature
		Sectional Curvature Bounds in Gravity: Regularisation of the Schwarzschild Singu...
		IngentaConnect Sectional Curvature Rigidity of Asymptotically Locally Hyperbolic...
		Elliptic genus and its rigidity/Green kernel with constant...
	www.scienceoxygen.com /math/721.html (126 words)

	TFCBooks -- Tesla Site Glossary
		In the near and intermediate zones, the field patterns are quite complicated, and the shape is, in general, a function of the radius and angular position (azimuth and elevation) in front of the antenna.
		AWG) The AWG copper wire gauge is a logarithmic scale base on a wire's cross section, each 3-gauge step in size corresponding to either a doubling or halving of the area.
		Power from the two-coil master oscillator section is fed to the lower end of the extra coil resonator through a heavy electrical conductor.
	www.tfcbooks.com /mainpage/glossary.htm (12260 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The total curvature of \begin_inset Formula \(\gamma \) \end_inset is defined as \begin_inset Formula \(\int _{\gamma }\kappa ds \) \end_inset.
		Similarly, any function which changes sign is the curvature of some metric on the torus, and any function which is negative somewhere is the curvature of a metric on a surface of genus \begin_inset Formula \(\geq 2.
		From that, it is not at all straightforward to establish a similar obstruction theory to the existence of metrics having, say, positive sectional curvature.
	www.lehigh.edu /dlj0/Desktop/dlj0/courses/424sp97-4.lyx (3530 words)

	Cohomogeneity one manifolds with positive sectional curvature (Site not responding. Last check: 2007-11-05)
		Manifolds with positive sectional curvature have been of much interest since the beginning of global Riemannian geometry.
		There are very few examples with positive sectional curvature, in fact in dim>24 only the rank one projective spaces are known, i.e.
		The goal of this project is to classify all cohomogeneity one manifolds which admit an invariant metric of positive curvature, which hopefully will produce new examples.
	www.pims.math.ca /birs/workshops/2004/04rit525 (222 words)

	Matter and Atoms (Site not responding. Last check: 2007-11-05)
		A cross section is just one face of the surface of an object, and so like surface area increases with the square of a linear increase in size.
		From this observation, it is also apparent after a moment's thought that if multiple pipes of different diameter are stuck into the top of a sealed vessel and the vessel is overfilled with water, that water will be at the same level no matter the diameter of any individual pipe.
		There is a trim tank in the forward section of the submarine, a trim tank in the rear section of the submarine, and trim is maintained by adjusting the relative amounts of water in these tanks.
	www.kosmoi.com /Technology/Engineering/Materials (6097 words)

	Curvature Microfiber Sectional
		Our sectional sofas offer the sleek, contemporary style to fit your modern home.
		Each sectional sofa is hand made to your exact color and configuration.
		This contemporary sectional sofa offers clean modern style that takes the durability and stain resistance of Microfiber fabric and creates furniture that is as comfortable as it is stylish.
	www.futoncreations.com /curvature-sectional.html (295 words)

	On Einstein Manifolds of Positive Sectional Curvature - Gursky, LeBrun (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M; g) is CP 2, with its standard Fubini-Study metric.
		Gursky and C. LeBrun, On Einstein manifolds of positive sectional curvature, Preprint (1998).
		On the Scalar Curvature of Einstein Manifolds - Catanese, LeBrun (1997)
	citeseer.ist.psu.edu /216328.html (251 words)

	Jef Raskin - Coanda Effect: Understanding Why Wings Work
		In the next few sections we look at attempts to explain two related phenomena--what makes a spinning ball curve and how a wing's shape influences lift--and see how the common explanation of lift has led a surprising number of scientists (including some famous ones) astray.
		The acceleration of the air around the sharper curvature near the front of the top of the wing also imparts a downward and forward component to the motion of the molecules of air (actually a slowing of their upward and backward motion, which is equivalent) and thus contributes to lift.
		On the B side, which is rotating opposite the direction of flight, the air is already moving (relative to the ball) in the same direction, and is thus more easily entrained.
	jef.raskincenter.org /published/coanda_effect.html (4907 words)

	FLUCTUATION OF SECTIONAL CURVATURE FOR CLOSED HYPERSURFACES (Site not responding. Last check: 2007-11-05)
		Liebmann proved in 1899 that the only closed surfaces in Euclidean three-space that have constant Gauss curvature are round spheres.
		Thus, if a closed surface in three-space is not a topological sphere, its Gauss curvature must fluctuate.
		We consider quantitative formulations of this fact, also in higher dimensions.
	math.la.asu.edu /~rmmc/rmj/vol32-1/OVE/OVE.html (48 words)

	Citebase - Collapsed Riemannian manifolds with bounded sectional curvature
		In the last two decades, one of the most important developments in Riemannian geometry is the collapsing theory of Cheeger-Fukaya-Gromov.
		A Riemannian manifold is called (sufficiently) collapsed if its dimension looks smaller than its actual dimension while its sectional curvature remains bounded (say a very thin flat torus looks like a circle in a bared eyes).
		Specifically, substantial progress has been made on manifolds with nonpositive curvature, on positively pinched manifolds, collapsed manifolds with an a priori diameter bound, and subclasses of manifolds whose members satisfy additional topological conditions e.g.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0304267 (1116 words)

	Citebase - Negative sectional curvature and the product complex structure
		We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature Kcurvature Ric > d, where c and d are constants.
		On the other hand, we observe that there are complete Kähler metrics with negative sectional curvature on C.
		Hence the upper sectional curvature bound is necessary.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0602289 (341 words)

	PROBLEMS
		8) A Riemannian submersion with totally geodesic fibers is called fat (Weinstein) if all vertizontal curvatures are positive.
		This would produce new examples of positive curvature, since the Eschenburg metrics requires that kl > 0.
		carry a metric with positive sectional curvature, e.g.
	www.lehigh.edu /~dlj0/Ziller-problems.html (311 words)

	diffgeom4.htm
		Comment: this is a special case of Hilbert’s theorem, which says that a complete surface with constant negative curvature cannot be isometrically immersed in
		both have nonnegative sectional curvature, does their product have nonnegative sectional curvature?
		both have positive sectional curvature, does their product have positive sectional curvature?
	www.math.sunysb.edu /~ktapp/568/M568HW4 (175 words)

	Sectional curvature and the energy--momentum tensor
		Many years ago Ehlers and Kundt showed that a spacetime M is an Einstein space if and only if the sectional curvatures of any pair of orthogonal non-null 2-spaces at any point of M are equal.
		This paper generalizes this result by first showing a very straightforward relation between the sectional curvatures of such orthogonal pairs of 2-spaces and the trace-free part of the Ricci tensor and then by establishing for each algebraic (Segre) type of the energy–momentum tensor precisely which orthogonal pairs of non-null 2-spaces have the same sectional curvature.
		The results are described in a manifold theoretic sense and are tabulated for each Segre type.
	stacks.iop.org /0264-9381/22/1493 (262 words)

	Encyclopedia :: encyclopedia : List of letters used in mathematics and science (Site not responding. Last check: 2007-11-05)
		K - Kaon, Kelvin, Potassium, Sectional curvature,a Spectral type,
		R - Curvature tensor, Electrical resistance, Radius, Set of real numbers,
		σ - Cross section, Electrical conductivity, Spin, Standard deviation, Stefan-Boltzmann constant,
	www.hallencyclopedia.com /List_of_letters_used_in_mathematics_and_science (644 words)

	Encyclopedia: Index
		Section One of the Canadian Charter of Rights and Freedoms
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	encyclopedie-en.snyke.com /ndx/page_1000.html (38 words)

	Education - Information - Educational Resources - Encyclopedia - Music - Se
		Secretary of State for War and the Colonies
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	education.music.us /Se.htm (119 words)

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