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Topic: Scalar curvature

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	Curvature of Riemannian manifolds - Wikipedia, the free encyclopedia
		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.
		It is possible to do this precisely because of the symmetries of the curvature tensor (namely antisymmetry in the first and last pairs of indices, and block-symmetry of those pairs).
		For a manifold of constant curvature, the Weyl tensor is zero.
	en.wikipedia.org /wiki/Curvature_of_Riemannian_manifolds (941 words)

	Scalar curvature - Wikipedia, the free encyclopedia
		The scalar curvature usually denoted by S (other notation are Sc, R).
		It is defined as the trace of the Ricci curvature tensor with respect to the metric:
		The three are distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices.
	en.wikipedia.org /wiki/Scalar_curvature (257 words)

	Scalar Curvature Estimates For Compact Symmetric Spaces - Goette, Semmelmann (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M = G=K of compact type with rk G \Gamma rk K 1.
		Let g be another metric with scalar curvature, such that g g on 2-vectors.
		4 Scalar curvature and conformal deformation of Riemannian str..
	citeseer.ist.psu.edu /416519.html (417 words)

	ipedia.com: Scalar curvature Article (Site not responding. Last check: 2007-10-22)
		In mathematics, the scalar curvature of a surface is the familiar Gaussian curvature.
		For higher-dimensional manifolds, it is double of the sum of all the sectional curvatures along all the 2-planes spanned by some orthonormal frame.
		It is also the full trace of the Ricci curvature as well as of the curvature tensor.
	www.ipedia.com /scalar_curvature.html (121 words)

	PlanetMath: Ricci tensor (Site not responding. Last check: 2007-10-22)
		It is also convenient to regard the Ricci tensor as a symmetric bilinear form.
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		It is the conformally invariant, trace-free part of the curvature tensor.
	planetmath.org /encyclopedia/RicciTensor.html (335 words)

	Z. Shen's papers on Finsler Geometry and Riemannian Geometry
		It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian).
		In this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain non-Riemannian quantities are isotropic.
		The flag curvature of a Finsler metric is a natural extension of the sectional curvature in Riemannian geometry, while the S-curvature is a non-Riemannian quantity.
	www.math.iupui.edu /~zshen/Research/preprint1.html (1284 words)

	[No title]
		Schoen, Richard M. Conformal deformation of a Riemannian metric to constant scalar curvature.
		Schoen, R.; Yau, Shing Tung Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature.
		Escobar, José F. Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary.
	www.math.ethz.ch /analysis+geometry/arb/arb-ws0001/yamabe.html (403 words)

	Citebase - Constant Scalar Curvature Metrics on Connected Sums
		There are 9 cases, depending on the signs of the scalar curvature on M' and M'' (positive, negative, or zero).
		We show that the constant scalar curvature metrics either develop small "necks" separating M' and M'', or one of M', M'' is crushed small by the conformal factor.
		In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous...
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0108022 (1274 words)

	Research (Site not responding. Last check: 2007-10-22)
		Assuming uniform lower bound for the scalar curvature, we find a sharp lower bound for the dilation constants in terms of the dimension of sphere.
		the total curvature of the sphere is 4 times the volume of the sphere.
		However, in higher dimension any relation that the volume and scalar curvature may have, requires a finer analysis.
	euphrates.wpunj.edu /faculty/llarullm/research.htm (627 words)

	Hertog, Horowitz and Maeda \| Musings
		This space contains (and this is HHM’s big point) metrics of both positive an negative scalar curvature, separated by a codimension-1 hypersuface of metrics of vanishing scalar curvature.
		The Ricci-flat metrics are found on some finite-dimensional hypersurface in the interior of the region of negative scalar curvature.
		But several of their arguments rely heavily on the restriction to constant scalar curvature metrics.
	golem.ph.utexas.edu /~distler/blog/archives/000147.html (612 words)

	scalar curvature
		Zhang, Dong Prescribed scalar curvature on the $n$-sphere.
		Min-Oo, Maung Scalar curvature rigidity of asymptotically hyperbolic spin manifolds.
		Kirchberg, K.-D. Compact six-dimensional Kähler Spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator.
	darkwing.uoregon.edu /~botvinn/psc1.html (1242 words)

	Scalar Curvature and Projective Embeddings, I, S.K. Donaldson
		We prove that a metric of constant scalar curvature on a polarised Kähler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo.
		This gives, as a Corollary, the uniqueness of constant scalar curvature Kähler metrics in a given rational cohomology class.
		The proof uses results in the literature on the asymptotics of the Bergman kernel.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.jdg/1090349449 (160 words)

	Christina Sormani's Research Interests (Site not responding. Last check: 2007-10-22)
		There are no almost rigidity theorems for manifolds with lower bounds on scalar curvature.
		Unlike Ricci curvature, scalar curvature does not provide a global comparison theorem on volume [6].
		While the Lichnerowitz Formula does relate the scalar curvature of a manifold to a laplacian, it is not the laplacian of a function but that of a spinor [15].
	comet.lehman.cuny.edu /sormani/research/interests.html (597 words)

	Citebase - On the flag curvature of Finsler metrics of scalar curvature
		Citebase - On the flag curvature of Finsler metrics of scalar curvature
		On the flag curvature of Finsler metrics of scalar curvature
		The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0303144 (712 words)

	Citebase - Role of the brane curvature scalar in the brane world cosmology (Site not responding. Last check: 2007-10-22)
		Authors: Kim, N. Lee, H. Myung, Y. We include the brane curvature scalar to study its cosmological implication in the brane world cosmology.
		We explore a new class of braneworld models in which the scalar curvature of the (induced) brane metric contributes to the brane action.
		The scalar curvature term arises generically on account of one-loop effects induced by matter fields residing on the brane.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0101091 (1109 words)

	Maung Min-Oo - Mass, scalar curvature and K-area
		This talk will give a short selected survey of some old and new results about scalar curvature rigidity of Riemannian manifolds.
		Many of the ideas have their origins in Witten's spinorial proof of the positive mass conjecture and in the classical work by Gromov and Lawson on obstruction to positive scalar curvature metrics.
		We will give ``sharp'' inequalities for the K-area of Hermitian symmetric spaces in terms of the scalar curvature.
	camel.math.ca /Events/summer98/s98-abs/node76.f (124 words)

	4a
		-curvature; it is simply the natural scalar field that turns up on the right hand side.
		There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives.
		Using the calculus naturally associated to tractor bundles (or equally effectively, using the ambient metric) it is in fact a simple matter to write down examples, and the possibilities increase with dimension.
	www.aimath.org /WWN/confstruct/articles/html/4a (2504 words)

	Closed Minimal Willimore Hypersurfaces of (1) with Constant Scalar Curvature, Tsasa Lusala, Mike Scherfner, Luiz ...
		Closed Minimal Willimore Hypersurfaces of (1) with Constant Scalar Curvature, Tsasa Lusala, Mike Scherfner, Luiz Amancio M. Souse, Jr.
		This gives a positive answer to the question made by Chern about the pinching of the scalar curvature for closed minimal Willmore hypersurfaces in dimension 4.
		We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
	projecteuclid.org /getRecord?id=euclid.ajm/1118689226 (147 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		A Priori estimates for prescribing scalar curvature equations, Annals of Math.
		Prescribing Gaussian curvature on surfaces with conical singularities, J.
		Prescribing scalar curvature on $S^n$, accepted for publication in Pacific J.
	math.smsu.edu /~wchen/chenpub.html (332 words)

	Talk 3575 data/Fall_2000/1120 (Site not responding. Last check: 2007-10-22)
		It is then natural to ask whether or not this is true for scalar curvature as well.
		, which can be scaled to have arbitrarily large scalar curvature and yet infinite volume.
		, greater scalar curvature does imply less total volume.
	www.math.duke.edu /mcal?abstract-3575 (186 words)

	Citebase - Positive scalar curvature and minimal hypersurfaces
		We show that the minimal hypersurface method of Schoen and Yau can be used for the ``quantitative'' study of positive scalar curvature.
		is the scalar curvature of of g, T any 2-tensor on M and W the Weyl tensor of g, then any closed orientable stable minimal (totally geodesic in the second case) hypersurface also admits a metric with the corresponding positivity of scalar curvature.
		[5] R. Schoen and S.T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math., 28 (1979), 159-183.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0308203 (280 words)

	BORIS BOTVINNIK - Spin manifolds with singularities and positive scalar curvature (Site not responding. Last check: 2007-10-22)
		Spin manifolds with singularities and positive scalar curvature
		To compute the coefficient groups and 2-local homotopy type of the corresponding classifying spectra in classical terms of cobordism theory, in particular, in terms of the Conner-Floyd characteristic numbers in the corresponding K-theory.
		To give an application of existence and classification of metrics with positive scalar curvature on Spin manifolds.
	camel.math.ca /CMS/Events/winter97/w97-abs/node14.html?nomenu=1 (102 words)

	Atlas: Prescribing scalar curvature on $S^3$ by Matthias Schneider (Site not responding. Last check: 2007-10-22)
		occur as scalar curvature of metrics g conformally equivalent to the standard round metric g
		The usual non-degeneracy assumption, \triangle K(y)\not = 0 at any critical point y of K, is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function K is a positive Morse function.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqm-68.
	atlas-conferences.com /cgi-bin/abstract/caqm-68 (144 words)

	AMCA: Compact Kahler surfaces with harmonic W^-. by Wlodzimierz Jelonek
		J. The conformal scalar curvature \kappa of a Hermitian manifold (M, g, J) is conformally covariant of weight -2 and is related to the Riemannian scalar curvature \tau of (M, g) by
		The author has proved that a Riemannian manifold (M, g) is an \AC manifold if and only if the Ricci endomorphism Ric of (M, g) is of the form Ric=S+\frac2n+2\tauId where S is a Killing tensor, \tau is the scalar curvature and n=dimM.
		Then (M, g) has constant scalar curvature or a scalar curvature of (M, g) is nonconstant and (M, J) is a ruled surface, (M, g, J) is an extremal Kähler surface and there exists an opposite Hermitian structure \bJ on M such that (M, g, \bJ) is a G
	at.yorku.ca /c/a/d/q/51.htm (767 words)

	Find in a Library: O(p + 1) x O(p + 1)-Invariant hypersurfaces with zero scalar curvature in euclidean space
		O(p + 1) x O(p + 1)-Invariant hypersurfaces with zero scalar curvature in euclidean space
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	worldcatlibraries.org /wcpa/ow/96922901260fbbdba19afeb4da09e526.html (84 words)

	Cartan curvature tensor (Site not responding. Last check: 2007-10-22)
		IngentaConnect Astrophysical and terrestrial probes to test Einstein-Cartan grav...
		CSDC : Frame vs. Metric connections, and their curvatures...
		The cosmological model with scalar, spin and torsion field...
	www.scienceoxygen.com /phys/192.html (117 words)

	Symplectic Fillings and Positive Scalar Curvature (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Abstract: Let X be a 4-manifold with contact boundary.
		We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_2 (X) > 0 or the boundary of X is disconnected.
		As an application we show that the Poincarandeacute; homology 3-sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semi-fillable...
	citeseer.ist.psu.edu /76128.html (450 words)

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