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Topic: Riemannian geometry


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  Riemannian geometry - Wikipedia, the free encyclopedia
Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century.
It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of Non-Euclidean geometry, spherical geometry and hyperbolic geometry, as well as Euclidean geometry itself.
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   (840 words)

  
 What Is Geometry?
Geometry is the study of shapes and configurations.
Differential geometry which is a natural extension of calculus and linear algebra, and is known in its simplest form as vector calculus.
Our geometry courses reflect this immense range from the study of configuration of points and lines in flat and curved spaces to the geometry of objects defined by polynomial equations.
www.math.uwaterloo.ca /PM_Dept/What_Is/geometry.shtml   (679 words)

  
 Glossary of Riemannian and metric geometry - Enpsychlopedia   (Site not responding. Last check: 2007-10-20)
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.
enpsych.liviant.com /psypsych/Shape_operator   (1308 words)

  
 2.1 Geometry   (Site not responding. Last check: 2007-10-20)
This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric.
Riemann-Cartan geometry is the subcase of a metric-affine geometry in which the metric-compatible connection contains torsion, i.e.
Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection.
relativity.livingreviews.org /Articles/lrr-2004-2/articlesu3.html   (2680 words)

  
 Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
It has been shown that modern differential geometry provides the concepts and tools to effect a treatment of Riemannian geometry, without the quadratic restriction, in a direct and elegant way so that all results, local and global, are included.
The key idea in Finsler geometry is to consider the projectivized tangent bundle PTM (i.e., the bundle of line elements) of the manifold M. The main reason is that all geometric quantities constructed from F are homogeneous of degree zero in y and thus naturally live on PTM, even though F itself does not.
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)

  
 What is non-Euclidean geometry?
A Riemannian geometry is also known as a spherical geometry.
The plane of a Riemannian geometry occurs on the surface of a spherical object.
The applications of this geometry, as well as its necessity, are best seen when thinking of a globe or the surface of some planet.
njnj.essortment.com /noneuclideange_risc.htm   (963 words)

  
 Riemannian Geometry, Kaluza-Klein Theories...
What the book is really about is Riemannian geometry of those spaces on which a group action is given (with a view on applications to physical theories -"unified theories"-).
A general study of the Riemannian geometry of "matter fields", i.e., vector valued functions (or forms) defined on a manifold (in particular the covariant derivative acting on tensors, spinors, p-forms valued in some vector space...) is made in Ch.6 (this chapter could be read independently of the rest of the book).
analyze the geometry of a manifold on which both metric and connection are given, along with the action of a symmetry group.
quantumfuture.net /quantum_future/kkintro.htm   (897 words)

  
 Riemannian Geometry and General Relativity   (Site not responding. Last check: 2007-10-20)
Riemannian geometry is designed to describe the universe of creatures who live on a curved surface or in a curved space and do not know about the world of higher dimensions or do not have any access to it.
One of the main notions of the Riemannian geometry is the notion of connection, which is, in fact, the key notion of the entire geometry, though it is not always explicitly formulated.
Earlier, in the 1910's, A.Einstein discovered that the Riemannian geometry can be successfully used to describe General Relativity which is in fact a classical theory of gravitation.
mystic.math.neu.edu /courses/diffgeom/intro.htm   (356 words)

  
 Riemannian Geometry
The main text for the course is "Riemannian Geometry" by Gallot, Hulin and Lafontaine (Second Edition) published by Springer.
This course is intended as an introduction at the graduate level to the venerable subject of Riemannian geometry.
Also central to geometry this century has been the relation between analysis on manifolds (for example properties of the Laplace operators) and their topology and geometry.
www.math.jhu.edu /~mhaskin/teaching/grad_geom/grad_geom.html   (615 words)

  
 Graduate Study in Geometry and Topology
Modern differential geometry is concerned with the spaces on which calculus of several variables applies (differentiable manifolds) and the various geometrical structures which can be defined on them.
In a Riemannian manifold a neighborhood of each point is given a Euclidean structure to a first order approximation.
Classical differential geometry considers the second order effects of such a structure locally, that is, on an arbitrarily small piece.
www.math.uiuc.edu /GraduateProgram/researchmath/gradgeomtop.html   (1213 words)

  
 Preface to Semi-Riemannian Geometry   (Site not responding. Last check: 2007-10-20)
The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry.
After establishing the requisite language of manifolds and tensors (Chs.1 and 2), the plan of the book is to develop the foundations of semi-Riemannian geometry in the simplest way and without regard to signature, allowing the Riemannina and Lorentz cases to appear as needed (Chs.3-5 and 7).
The fact that relativity theory is expressed in terms of Lorentz geometry is lucky for geometers, who can thus penetrate surprisingly quickly into cosmology (redshift, expanding universe, and big bang) and, a topic no less interesting geometrically, the gravitation of a single star (perihelion advance, bending of light, and fl holes).
www.math.ucla.edu /~bon/srgpreface.html   (402 words)

  
 PlanetMath: Riemannian manifold
A Riemannian metric tensor is a covariant, type
Indeed, it is possible to define a Riemannian structure on a manifold
This is version 12 of Riemannian manifold, born on 2002-09-12, modified 2006-04-19.
planetmath.org /encyclopedia/RiemannianMetric.html   (234 words)

  
 Riemannian Geometry   (Site not responding. Last check: 2007-10-20)
An N-dimensional Riemannian manifold is characterized by a second-order metric tensor g
However, it’s interesting to observe that the ratio of the coefficients of the Riemannian part to the square of the coefficient of the Euclidean part is precisely the Gaussian curvature on the surface
The numerator and denominator are both determinants of 2x2 matrices, representing different "ground forms" of the surface.
www.mathpages.com /rr/s5-07/5-07.htm   (2194 words)

  
 About "Papers on Riemannian Geometry"   (Site not responding. Last check: 2007-10-20)
Papers written by graduate students in the course Math 240, Riemannian Geometry, in the Spring Semester of 1995 at the Univ. of California at Berkeley.
Each paper is either a survey of an area or a tutorial essay in a topic related to riemannian geometry.
Nevertheless, there are also papers in traditional areas of riemannian geometry (such as minimal surface theory) and topics related to operator algebras, symplectic geometry, and mathematical physics.
mathforum.org /library/view/2579.html   (118 words)

  
 Riemannian Geometry   (Site not responding. Last check: 2007-10-20)
His talk with the title "Über die Hypothesen, welche der Geometrie zu Grunde liegen" is often said to be the most important in the history of differential geometry.
Riemann's revolutionary ideas generalized the geometry of surfaces which had been studied earlier by Gauss, Bolyai and Lobachevsky.
This course is an introduction to the beautiful theory of Riemannian Geometry, a subject with no lack of interesting examples.
www.matematik.lu.se /matematiklu/personal/sigma/MAT3/Riemannian-Geometry.html   (313 words)

  
 Z. Shen's papers on Finsler Geometry and Riemannian Geometry
We show that if two Riemannian metrics g* and g are pointwise projectively equivalent, i.e., they have common geodesics as point sets, and the Ricci curvature of g* is less than or equal to that of g, then the projective equivalence is trivial provided that g is complete.
In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics.
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)

  
 Non-Riemannian Geometry
Non-Riemannian Geometry deals basically with manifolds dominated by the geometry of paths co-developed by the distinguished mathematician Luther Pfahler Eisenhart, who is also the author of this text.
Discussions of the projective geometry of paths follow; and in the final chapter, Eisenhart explores the geometry of sub-spaces.
Stimulating and thought-provoking treatment of geometry’s crucial role in a wide range of mathematical applications, for students and mathematicians.
store.doverpublications.com /0486442438.html   (184 words)

  
 Math 256: Riemannian Geometry   (Site not responding. Last check: 2007-10-20)
The principal goal of the course is to study the "intrinsic" geometry of surfaces as opposed to the "extrinsic" geometry studied in chapters 1 through 3 in MTH255.
The intrinsic properties are those which can be determined soley by measuring distances along the surface, while the extrinsic properties are those determined from the way the surface sits in space.
Gauss curvature, described as an extrinsic quantity in the first chapters, can also be determined entirely from the distance function and its derivatives, hence it is also an intrinsic function.
www.math.rochester.edu /courses/256/home/mth256-sp06-description.html   (201 words)

  
 3 Riemannian Geometry   (Site not responding. Last check: 2007-10-20)
However, this motion requires a notion of parallel transport or rigid motion--which may be different for our geometry from the Euclidean one provided by the coordinates; the Riemannian metric provides the correct notion of angles and distances and hence of rigid motion.
Riemann introduced the Riemannian curvature as a measure of this.
This is by far the most important invariant associated with a Riemannian manifold.
www.imsc.res.in /~kapil/papers/krp/node3.html   (1041 words)

  
 RIEMANNIAN GEOMETRY, FIBER BUNDLES, KALUZA-KLEIN THEORIES AND ALL THAT.....
RIEMANNIAN GEOMETRY, FIBER BUNDLES, KALUZA-KLEIN THEORIES AND ALL THAT.....
The ten chapters cover topics from the differential and Riemannian manifolds to the reduction of Einstein-Yang-Mills action.
Riemannian Geometry of a Bundle with Fibers G/H and a Given Action of a Lie Group G
www.worldscibooks.com /physics/0488.html   (161 words)

  
 53: Differential geometry
Differential geometry is the language of modern physics as well as an area of mathematical delight.
A metric in the sense of differential geometry is only loosely related to the idea of a metric on a metric space.
Ricci, A Mathematica package for doing tensor calculations in differential geometry GRG 3.2 is the computer algebra system designed for the calculations in differential geometry and field theory.
www.math.niu.edu /~rusin/known-math/index/53-XX.html   (461 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.
I believe many researchers with interests in Riemannian geometry as well as those who appreciate or want to learn what's current in Riemannian geometry may find this book beneficial.
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)

  
 What is Riemannian Geometry?
Euclidean Geometry is the study of flat space.
One of the basic topics in Riemannian Geometry is the study of curved surfaces.
One kind of theorem Riemannian Geometers are looking for today is a relationship between the curvature of a space and its shape.
comet.lehman.cuny.edu /sormani/research/riemgeom.html   (1402 words)

  
 Open Directory - Science: Math: Geometry: Differential Geometry   (Site not responding. Last check: 2007-10-20)
Differential Geometry - Lecture notes for a course at the Weizmann Institute of Science by Sergei Yakovenko.
Riemannian Geometry - A set of postscript lecture notes for a graduate level course on Riemannian geometry.
Riemannian Geometry - Mostly a definition with a few equations.
dmoz.org /Science/Math/Geometry/Differential_Geometry   (256 words)

  
 Amazon.com: Riemannian Geometry: Books: Luther Pfahler Eisenhart   (Site not responding. Last check: 2007-10-20)
Riemannian Geometry: A Modern Introduction (Cambridge Tracts in Mathematics) by Isaac Chavel
Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) by M.
I bought the Russian translation of this book in 1954 and found that this is the best source of the Riemannian geometry, not only for a beginner (as I was at that time), but also for every specialist.
www.amazon.com /exec/obidos/tg/detail/-/0691023530?v=glance   (691 words)

  
 Riemannian Geometry   (Site not responding. Last check: 2007-10-20)
In the early 1800s Gauss asked how much of the geometry of a surface is independent of how it bends in space.
Riemannian geometry is designed to describe the universe of creatures who live on a curved surface and who are unaware of space outside and can only measure distances and areas on the surface.
I plan to cover basic Riemannian geometry from ch 1-6 in Do Carmo (or ch 3-5 in O'Neill or ch 2-3(5) in Gallot et al) and some Semi-Riemannian geometry and general relativity from O'Neill:
math.ucsd.edu /~lindblad/250b/250b.html   (290 words)

  
 Eisenhart, L.P.: Riemannian Geometry.
In his classic work of geometry, Euclid focused on the properties of flat surfaces.
In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces.
The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results.
pup.princeton.edu /titles/486.html   (155 words)

  
 Citebase - Discrete Riemannian Geometry
Authors: Dimakis, A. Muller-Hoissen, F. Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets.
If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (non-local) tensor product over the algebra of functions.
In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/9808023   (1907 words)

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