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Topic: Riemannian metric


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In the News (Tue 16 Jul 19)

  
  riemannian metric   (Site not responding. Last check: 2007-10-06)
Riemannian geometry is a description of an important family of geometries, first put forward in generality by Bernhard Riemann in the nineteenth century.
The metric tensor g is a twice differentiable (at the very least) symmetric pointwise linear map of two sections of the tangent bundle, X and Y, g(X,Y) which is positive definite (i.e.
The metric tensor, conventionally notated as G, as a 2-dimensional tensor (making it a matrix), that is used to measure distance in a coordinate space or manifold.
www.yourencyclopedia.net /Riemannian_metric.html   (551 words)

  
 Glossary of Riemannian and metric geometry - Wikipedia, the free encyclopedia
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.
Word metric on a group is a metric of the Cayley graph constructed using a set of generators.
en.wikipedia.org /wiki/Glossary_of_Riemannian_and_metric_geometry   (1246 words)

  
 Riemannian manifold - Wikipedia, the free encyclopedia
In Riemannian geometry, a Riemannian manifold (M, g) is a real differentiable manifold M in which each tangent space is equipped with an inner product <, > in a manner which varies smoothly from point to point.
Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of positive-definite quadratic forms on the tangent bundle.
In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of the Hopf-Rinow theorem.
en.wikipedia.org /wiki/Riemannian_manifold   (522 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   (865 words)

  
 Geodesic - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-06)
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer.
The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.
The behaviour of the metric tensor under coordinate transformations implies that H is invariant under a change of variable.
www.bucyrus.us /project/wikipedia/index.php/Geodesic   (1184 words)

  
 Pseudo-Riemannian manifold - Open Encyclopedia   (Site not responding. Last check: 2007-10-06)
The key difference between a Riemannian metric and a pseudo-Riemannian metric is that a pseudo-Riemannian metric need not be positive-definite, merely nondegenerate.
Since every positive-definite form is also nondegenerate a Riemannian metric is a special case of a pseudo-Riemannian one.
The signature of a pseudo-Riemannian manifold is just the signature of the metric (one should insist that the signature is the same on every connected component).
www.open-encyclopedia.com /Lorentz_metric   (378 words)

  
 Riemannian Geometry
Therefore, the metrical relations on the manifold over any sufficiently small region approach arbitrarily close to flatness to the first order in the coordinate differentials.
In general, however, the metric components need not be constant to the second order of changes in position.
If there exists a coordinate system at a point on the manifold such that the metric components are constant in the first and second order, then the manifold is said to be totally flat at that point (not just asymptotically flat).
www.mathpages.com /rr/s5-07/5-07.htm   (2194 words)

  
 Differential geometry and topology - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-06)
Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e.
A Finsler metric is much more general structure than a Riemannian metric.
Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space.
encyclopedia.learnthis.info /d/di/differential_geometry_and_topology.html   (998 words)

  
 Glossary of Riemannian and metric geometry   (Site not responding. Last check: 2007-10-06)
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.
For complete manifolds, if the injectivity radius at p is a finite, say r, then either there is a geodesic of length 2r which starts and ends at p or there is a poit q conjugate to p and on the distance r from p.
A sub set S of a metric space X is called -net if for any point in X there is a point in S on the distance.
www.olive-oil-facts.info /en/wikipedia/g/gl/glossary_of_riemannian_and_metric_geometry.html   (1193 words)

  
 Glossary of Riemannian and metric geometry -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-06)
Injectivity radius of a Riemannian manifold is the infimum of Injectivity
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 with Maintaining the Notion \\ of Distant Parallelism
Riemannian Geometry has led to a physical description of the gravitational field in the theory of general relativity, but it did not provide concepts that can be assigned to the electromagnetic field.
Riemannian geometry is characterized by an Euclidean metric in an infinitesimal neighbourhood of any point P. Furthermore, the absolute values of the line elements which belong to the neighbourhood of two points P and Q of finite distance can be compared.
Yet metric and teleparallelism remain intact, if we substitute the n-beins of all points of the continuum with substitutability rotation invariance and note: Only those mathematical relations can claim or a real meaning that are rotational invariant.
www.lrz-muenchen.de /~u7f01bf/WWW/rep1.html   (1096 words)

  
 [No title]
Riemannian geometries in various fields have found various applications as different as in population dynamics and fractional distillation, just to mention the first and the most recent ones.
This leads to a natural Riemannian metric on the space of distributions in the thermodynamic limit of Gibbs statistical ensembles.
In this Chapter, we recapitulate basic ideas and results concerning the Riemannian metric structure of thermodynamics while we attempt to shed light on the underlying concept of statistical distance used in a much broader context.
www.rmki.kfki.hu /~diosi/springerabs.html   (1004 words)

  
 PlanetMath: Riemannian manifold
A Riemannian metric tensor is a covariant, type
should be called local coordinate components of a metric tensor, where as ``Riemannian metric'' should refer to the distance function defined above.
This is version 11 of Riemannian manifold, born on 2002-09-12, modified 2005-08-16.
www.planetmath.org /encyclopedia/MetricTensor2.html   (235 words)

  
 metric
In other realms, "metric" means other things, and then to be specific we say that the metric in GR is a "Lorentzian metric" if it has that minus sign in front of the time part, or a "Riemannian metric" if it's just for space.
If you have a tiny rigid rod moving along a path in space-time, the points in space-time occupied by its ends, which are simultaneous in the locally inertial frame for the rod, are a distance apart equal to the length of the rod.
The metric is assumed to be one which at any one point can be transformed by a change of coordinates into a Minkowski metric.
math.ucr.edu /home/baez/gr/metric.html   (994 words)

  
 RIEMANNIAN GEOMETRY FACTS AND INFORMATION   (Site not responding. Last check: 2007-10-06)
In differential_geometry, Riemannian geometry is the study of smooth manifolds with Riemannian_metrics; i.e.
This beautiful theorem has an intrinsic generalization (the ''Chern-Gauss-Bonnet Theorem'') to any compact even-dimensional Riemannian manifold, and an extrinsic generalization (part of the ''Allendoerfer-Fenchel Theorem'') to any compact Riemannian n-manifold that is isometric embedded in (n+1)-dimensional Euclidean_space.
#'''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.
www.amysflowershop.com /Riemannian_geometry   (826 words)

  
 Early Philosophical Interpretations of General Relativity
The empirical determination of the spacetime metric by measurement requires choice of some "metrical indicators": this can only be done by laying down a "coordinative definition" stipulating, e.g., that the metrical notion of a "length" is coordinated to some physical object or process.
Rather, the metric in the region around any observer O can be empirically determined from freely falling ideally small neutral test masses together with the paths of light rays.
More precisely stated, the spacetime metric results from the affine-projective structure of the behavior of neutral test particles of negligible mass and from the conformal structure of light rays received and issued by the observer.
plato.stanford.edu /entries/genrel-early   (11439 words)

  
 Definition of Riemannian geometry
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.wordiq.com /definition/Riemannian_geometry   (816 words)

  
 Riemannian geometry - Definition up Erdmond.Com
---- In differential_geometry, Riemannian geometry is the study of smooth manifolds with Riemannian_metrics; i.e.
the volume of a metric ball of radius ''r'' in a complete ''n''-dimensional Riemannian manifold manifold with positive Ricci curvature is at most as large as the volume of ball of the same radius ''r'' in Euclidean space.
The set of all Riemannian manifolds which with positive Ricci curvature and diameter at most ''D'' is pre-compact in Gromov-Hausdorff metric.
www.erdmond.com /Riemannian_geometry.html   (743 words)

  
 Amazon.com: Books: Metric Structures for Riemannian and Non-Riemannian Spaces : Based on Structures Metriques des ...   (Site not responding. Last check: 2007-10-06)
Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory.
This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman.
Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds.
images.spinics.net /am/0817638989   (1035 words)

  
 Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
Historical developments have conferred the name Riemannian geometry to this case while the general case, Riemannian geometry without the quadratic restriction (2), has been known as Finsler geometry.
In the Riemannian case this was the form problem, solved in 1870 by E. Christoffel and R. Lipschitz.
As is well known, Riemannian geometry can be handled, elegantly and efficiently, by tensor analysis on M. Its handicap with Finsler geometry arises from the fact that the latter needs more than one space, for instance PTM in addition to M, on which tensor analysis does not fit well.
www.math.iupui.edu /~zshen/Finsler/history/chern.html   (2855 words)

  
 GTR is founded on a Conceptual Mistake
By linking the metric of space with the gravitational field, the physical space is implied to be deformable and subjected to an incompatible set of strain components.
Therefore, the notion of Riemannian space may be directly associated with the non-zero value of Riemann tensor formed from the components of the metric tensor.
Therefore, the main postulate of GR and the Einstein’s Field Equations are found to be physically invalid, firstly on account of the ‘deformation of space’ induced by the Riemannian metric and secondly on account of the violation of essential compatibility conditions for the induced strain components in the space thus ‘deformed’.
www.geocities.com /ResearchTriangle/Forum/9850/gtr_mistake.htm   (4606 words)

  
 Advancing Front Mesh Generation in Three Dimensions Using a Riemannian Surface Definition   (Site not responding. Last check: 2007-10-06)
The blend of advancing front and a metric map is new since previous work with the advancing front method has been in parametric space (either directly meshing the parametric spaceor modifying the space) or directly on the 3D surface.
A second benefit is that the metric of a point in parametric space can be determined by a simple interpolation rather than a costly evaluation of the surface followed by the computation of the metric.
The details of the creation of the metric map used to determine the amount of distortion of the elements in parametric space were given along with the details of an advancing front algorithm that utilizes the metric map.
www.andrew.cmu.edu /user/sowen/riemann/riemann.html   (4063 words)

  
 Surface Evolver Documentation - Mathematical model
Other quantities that depend on the metric, such as volume, are up to the user to put in by hand with named quantities.
The ambient space can be endowed with a general Riemannian metric by putting the keyword METRIC in the datafile followed by the elements of the metric tensor.
The ambient space can be endowed with a conformal Riemannian metric by putting the keyword CONFORMAL_METRIC in the datafile followed by a formula for the conformal factor, i.e.
www.geom.uiuc.edu /software/evolver/html/model.htm   (3987 words)

  
 [No title]   (Site not responding. Last check: 2007-10-06)
With this norm, the metric is not proportional to the Hausdorff metric.
The distance from A to B to be the samllest number r such that each point in A is within r of some point in B and vice versa.
Then your metric is the distance between V & W, which is sqrt{2} sqrt{ 1 - cos(t) } But Tom's metric is the norm of the 2x2 matrix pi_W - pi_V the difference between the orthogonal projections onto these lines.
www.lehigh.edu /dmd1/public/www-data/pg928   (570 words)

  
 A Characterization Of Riemannian Flows (ResearchIndex)   (Site not responding. Last check: 2007-10-06)
We prove that a flow on a closed manifold is Riemannian if and only if it is locally generated by Killing vector fields for a Riemannian metric.
Then the flow F is Riemannian if and only if the tangent bundle of F is locally generated by Killing vector fields for a Riemannian metric g on M.
1 A tenseness theorem for Riemannian foliations (context) - Dom'inguez - 1995
citeseer.ist.psu.edu /122986.html   (261 words)

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