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Topic: Glossary of Riemannian and metric geometry

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Riemannian geometry

Bernhard Riemann

	Glossary of Riemannian and metric geometry - Wikipedia, the free encyclopedia
		This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
		For closed Riemannian manifold the injectivity radius is either half of minimal length of closed geodesic or minimal distance between conjugate points on a geodesic.
		Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.
	en.wikipedia.org /wiki/Glossary_of_Riemannian_and_metric_geometry (1246 words)

	METRIC SPACE FACTS AND INFORMATION (Site not responding. Last check: 2007-10-07)
		The geometry of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean_geometries such as those used in the theory of general_relativity.
		Similarly, in 3D, the metrics on the surface of a polyhedron include the ordinary metric, and the distance over the surface; a third metric on the edges of a polyhedron is one where the "paths" are the edges.
		An important consequence is that every metric space admits partitions_of_unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze_extension_theorem).
	www.gottagetflowers.com /metric_space (1521 words)

	Riemannian geometry - Wikipedia, the free encyclopedia
		In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i.e.
		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.
	en.wikipedia.org /wiki/Riemannian_geometry (842 words)

	Glossary - Wikipedia, the free encyclopedia
		Traditionally a glossary appears at the end a book to cover terms which are either newly introduced or relatively rare so the reader may easily get the linking knowledge to follow the argumentation in the book.
		A core glossary is a simple glossary or defining dictionary which enables definition of other concepts, especially for newcomers to a language or field of study.
		In computer science, a core glossary is a pre-requisite to a core ontology.
	www.wikipedia.org /wiki/Glossary (219 words)

	UC Davis Math: Glossary (Site not responding. Last check: 2007-10-07)
		In Riemannian geometry, usually means the intrinsic curvature of a manifold with a Riemannian metric.
		Riemannian geometry is a disproportionate part of differential geometry.
		In modern terms, hyperbolic geometry is the study of manifolds with Riemannian metrics with constant negative curvature.
	www.math.ucdavis.edu /profiles/glossary.html (9932 words)

	Glossary of Riemannian and metric geometry (Site not responding. Last check: 2007-10-07)
		Geodesic flow is a flow on a tangent bundle TM of a manifold M, generated by a vector field whose trajectories are of the form where is a geodesic.
		Injectivity radius at a point p of a Riemannian manifold is the largest radius for which the exponential map at p is a diffeomorphism.
		Radius of metric space is the infimum of radiuses of metric balls which contain the space completely.
	www.factsite.co.uk /en/wikipedia/g/gl/glossary_of_riemannian_and_metric_geometry.html (1150 words)

	Riemannian geometry
		There is an such that if n-dimensional Riemannian manifold has a metric with sectional curvature and diameter then its finite cover diffeomorphic to a nil manifold.
		The set of all Riemannian manifolds which with positive Ricci curvature and diameter at most D is pre-compact in Gromov-Hausdorff metric.
		If M is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of its fundamental group of M is isomorphic to Z.
	www.starrepublic.org /encyclopedia/wikipedia/r/ri/riemannian_geometry.html (728 words)

	Line (mathematics) - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points.
		If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's), then lines are not defined at all, but characterized axiomatically by their properties.
		Glossary of Riemannian and metric geometry#R for its meaning in Riemannian geometry.
	northmiami.us /project/wikipedia/index.php/Straight_line (571 words)

	Net (mathematics)
		See the glossary of Riemannian and metric geometry for its meaning for metric spaces.
		This article is about its meaning in topology, where the concept of a net is a generalization of that of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces.
		This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence, which is widely used in the theory of metric spaces.
	www.starrepublic.org /encyclopedia/wikipedia/n/ne/net__mathematics_.html (843 words)

	Riemannian geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		In (Click link for more info and facts about differential geometry) differential geometry, Riemannian geometry is the study of (Click link for more info and facts about smooth manifolds) smooth manifolds with (Click link for more info and facts about Riemannian metric) Riemannian metrics; i.e.
		The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is (Click link for more info and facts about pre-compact) pre-compact in the (Click link for more info and facts about Gromov-Hausdorff metric) Gromov-Hausdorff metric.
		If M is a complete Riemannian manifold with negative sectional curvature then any (Click link for more info and facts about abelian) abelian subgroup of the (Click link for more info and facts about fundamental group) fundamental group of M is (Click link for more info and facts about isomorphic) isomorphic to Z.
	www.absoluteastronomy.com /encyclopedia/R/Ri/Riemannian_geometry.htm (1315 words)

	Riemannian geometry - Encyclopedia, History, Geography and Biography
		Gauss-Bonnet Theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to $2\pi\chi(M)$ where $\chi(M)$ denotes the Euler characteristic of M.
		There is an $\epsilon_n>0$ such that if an n-dimensional Riemannian manifold has a metric with sectional curvature $K\le \epsilon_n$ and diameter $\le 1$ then its finite cover is diffeomorphic to a nil manifold.
		If the injectivity radius of a compact n-dimensional Riemannian manifold is $\ge \pi$ then the average scalar curvature is at most n(n-1).
	www.arikah.net /encyclopedia/Riemannian_geometry (897 words)

	RIEMANNIAN GEOMETRY FACTS AND INFORMATION (Site not responding. Last check: 2007-10-07)
		In differential_geometry, Riemannian geometry is the study of smooth manifolds with Riemannian_metrics; i.e.
		#'''Bishop's inequality.''' the volume of a metric ball of radius ''r'' in a complete ''n''-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius ''r'' in Euclidean space.
		#'''Gromov's_compactness_theorem.''' The set of all Riemannian manifolds with positive Ricci curvature and diameter at most ''D'' is pre-compact in the Gromov-Hausdorff metric.
	www.amysflowershop.com /Riemannian_geometry (802 words)

	Differential geometry and topology - Internet-Encyclopedia.com (Site not responding. Last check: 2007-10-07)
		Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions).
		Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e.
		Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space.
	www.internet-encyclopedia.com /ie/d/di/differential_geometry_and_topology.html (1157 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		The Euclidean metric of this space defines the distance between two points as the length of the straight line connecting them.
		The geometry of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.
		If M is a connected Riemannian manifold, then we can turn M into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
	www.homestayfinder.ca /Dictionary.aspx?q=Metric_space (1547 words)

	Read about Line (mathematics) at WorldVillage Encyclopedia. Research Line (mathematics) and learn about Line ... (Site not responding. Last check: 2007-10-07)
		Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points.
		Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties.
		Glossary of Riemannian and metric geometry#R for its meaning in
	encyclopedia.worldvillage.com /s/b/Line_segment (453 words)

	Glossary of Riemannian and metric geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		Dilation of a map between metric spaces is the infimum of numbers L such that the given map is L- (Click link for more info and facts about Lipschitz) Lipschitz.
		Riemannian submersion is a map between Riemannian manifolds which is (The act of wetting something by submerging it) submersion and submetry at the same time.
		Word metric on a group is a metric of the (Click link for more info and facts about Cayley graph) Cayley graph constructed using a set of generators.
	www.absoluteastronomy.com /encyclopedia/G/Gl/Glossary_of_Riemannian_and_metric_geometry.htm (1665 words)

	Riemannian geometry
		In mathematics, Riemannian geometry has at least two meanings, one which described in this article and an other also called elliptic geometry.
		If a compact Riemannian manifold has positive Ricci curvature then it has finite fundamental group.
		For any two points of complete simply connected Riemannian manifold with nonpositive sectional curvature are joint by unique geodesic.
	starrepublic.org /encyclopedia/wikipedia/r/ri/riemannian_geometry.html (728 words)

	Riemannian geometry
		---- In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i.e.
		#Metric tensor #Riemannian manifold #Levi-Civita connection #Curvature #Curvature tensor.
		In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.
	www.mcfly.org /wik/Riemannian_geometry (740 words)

	Glossary of Riemannian and metric geometry - Encyclopedia Glossary Meaning Explanation Glossary of Riemannian and ... (Site not responding. Last check: 2007-10-07)
		Glossary of Riemannian and metric geometry - Encyclopedia Glossary Meaning Explanation Glossary of Riemannian and metric geometry.
		Injectivity radius of a Riemannian manifold is the infimum of Injectivity
		For closed Riemannian manifold the injectivity radius is either half of minimal
	www.encyclopedia-glossary.com /en/Glossary-of-Riemannian-and-metric-geometry.html (1231 words)

	Insurance Glossary -- Recommendations and Resources (Site not responding. Last check: 2007-10-07)
		This is a glossary of some terms used in the branch of mathematics known as topology.
		An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.
		It seems that the current glossary is evolving into a "general topology" glossary (i.e.
	www.becomingapediatrician.com /health/79/insurance-glossary.html (909 words)

	iqexpand.com (Site not responding. Last check: 2007-10-07)
		If (M, d) is a metric space, a closed ball is a set of the form D(x; r) := {y in M : d(x, y) ≤ r}, where x is in M and r is a positive real number, the radius of the ball.
		If (M, d) is a metric space and S is a subset of M, the diameter of S is the supremum of the distances d(x, y), where x and y range over S.
		Topology glossary This is a glossary of some terms used in the branch of mathematics known as topology.
	topology_glossary.iqexpand.com (4558 words)

	eLibrary Project : Isometry (Site not responding. Last check: 2007-10-07)
		---- In the mathematical discipline of geometry and mathematical analysis, an "isometry", "isometric isomorphism" or "congruence mapping" is a distance-preserving isomorphism between metric spaces.
		The original space ''M'' is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace.
		Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
	elibraryproject.com /info/isometry.html (444 words)

	Riemannian geometry
		In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called
		In differential geometry, Riemannian geometry is the study of smooth manifolds with
		Riemannian manifold can be isometrically embedded in a Euclidean space R
	en.efactory.pl /Riemannian_geometry (762 words)

	Glossary of differential geometry and topology - Encyclopedia Glossary Meaning Explanation Glossary of differential ... (Site not responding. Last check: 2007-10-07)
		Glossary of differential geometry and topology - Encyclopedia Glossary Meaning Explanation Glossary of differential geometry and topology.
		This is a glossary of terms specific to differential geometry and differential topology.
		The orginal Glossary of differential geometry and topology article can be editet
	www.encyclopedia-glossary.com /en/Glossary-of-differential-geometry-and-topology.html (541 words)

	Glossary of Riemannian and metric geometry - Wikpedia (Site not responding. Last check: 2007-10-07)
		Glossary of Riemannian and metric geometry - Wikpedia
		A function f on a Riemannian manifold is a convex if for any geodesic $\gamma the function f\circ\gamma is convex .$
		A sub set S of a metric space X is called $\epsilon -net if for any point in X there is a point in S on the distance \le\epsilon .$
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Glossary_of_Riemannian_and_metric_geometry (1226 words)

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