Curvature of Riemannian manifolds (8410 bytes)
1: ...ntial geometry]], the infinitesimal geometry of [[Riemannian manifold]]s with dimension at least 3 is too comp...
3: ...e on [[curvature]] which discusses the curvature of curves and surfaces in 2 and 3 dimensions.
5: Curvature of [[Pseudo-Riemannian manifold]] can be expressed on the same way with ...
7: == Ways to express the curvature of a Riemannian manifold ==
9: ===The [[curvature tensor]]===

List of manifolds (669 bytes)
1: ...[[manifold]]s, by Wikipedia page. See also [[list of geometric topology topics]].
28: [[Category:Topological spaces|Manifolds]]

Sub-Riemannian manifold (2102 bytes)
1: .... Roughly speaking, to measure distances in a sub-Riemannian manifold,
4: ...[topological dimension]] (unless it is actually a Riemannian manifold).
8: ...we mean a [[subbundle]] of the [[tangent bundle]] of <math>M</math>.
15: ... presented as a [[linear combination]] of vectors of the following types
18: A '''sub-Riemannian manifold''' is a triple <math>(M, H, g)</math>,

Riemannian submanifold (275 bytes)
1: A '''Riemannian submanifold''' ''N'' of a [[Riemannian manifold]] ''M'' is a [[smooth manifold]]
2: equipped with the induced [[Riemannian metric]] from ''M''.
4: The image of an [[isometric immersion]] is a Riemannian submanifold.
6: [[Category:Riemannian geometry]]

Riemannian geometry (6826 bytes)
1: ...annian geometry''' has at least two meanings, one of which is described in this article and another al...
3: ...rticular local ideas of [[angle]], [[curve|length of curves]], and [[volume]]. From those some other g...
5: ...l treated on the same basis, as are a broad range of geometries whose metric properties vary from poin...
7: ...s, which (in dimension four) are the main objects of [[general relativity]] theory.
9: ...ere; it is usually done by doing enormous amounts of calculations. The following articles might serve ...

Riemannian manifold (3944 bytes)
1: ...]], [[gradient]]s of functions and [[divergence]] of [[vector field]]s.
3: ... The [[integral]] of this length gives the length of a curve on the manifold. To pass from a linear a...
5: ...om the [[Nash embedding theorem]], all Riemannian manifolds can be realized this way.
6: ...eful to build the first geometric intuitions in [[Riemannian geometry]].
8: ...ed as a smooth manifold with a [[smooth section]] of positive-definite quadratic forms on the [[tangen...

Pseudo-Riemannian manifold (2956 bytes)
1: ...n the manifold. This tensor is called a '''pseudo-Riemannian metric''' or, simply, a '''(pseudo-)[[metric tens...
3: ... manifolds can be considered generalizations of [[Riemannian manifold]]s.
5: ...ath> is the dimension of the manifold. Riemannian manifolds are simply those with signature <math>(n,0)</math...
7: ...acetime]] can be modeled as a Lorentzian manifold of signature <math>(3,1)</math>.
9: ... the model space for a pseudo-Riemannian manifold of signature <math>(p,q)</math> is <math>\mathbf R^{...

Glossary of Riemannian and metric geometry (10196 bytes)
1: ...d vocabulary or provide more detailed expositions of the definitions given below.
4: * [[Curvature]]
6: * [[Riemannian manifold]]
8: * [[Glossary of general topology]]
9: * [[Glossary of differential geometry and topology]]

Fundamental theorem of Riemannian geometry (4346 bytes)
1: ...that given a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) there is a unique [[torsion]]-free [[...
7: [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]])
9: ...where <math>Xg(Y,Z)</math> denotes the derivative of function <math>g(Y,Z)</math> along vector field <...
12: ...e quicker and simpler methods to obtain the Christoffel symbols for a given metric, e.g. using the [[a...
14: ==Proof==

Mean curvature (683 bytes)
1: In [[mathematics]], '''mean curvature''' of a [[surface]] is a notion from [[differential geo...
3: ...intersection]] is a [[curve]]. This curve has a [[curvature]] ''C'' at ''P''.
5: ... [[average]] is the '''mean curvature''' at ''P'' of the surface.

Curvature (6436 bytes)
1: '''Curvature''' is the amount by which a geometric object devi...
5: ...e consider the most basic examples: the curvature of a plane
6: curve and the curvature of a surface in Euclidean space.
9: == Curvature of plane curves ==
13: ...ection of that circle's center. The magnitude of curvature at points on physical curves can be measured in [...

Weyl curvature (2166 bytes)
1: ...ensor]] with the extra condition that its [[Ricci curvature]] must vanish.
3: ...ishes identically. In dimensions &ge; 4, the Weyl curvature is generally nonzero.
5: The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is...
7: ...all>''k'' denotes the [[Kulkarni-Nomizu product]] of two symmetric (0,2) tensors:
16: ...n given by contracting the above with the inverse of the metric.

Degree of curvature (1436 bytes)
1: ...'' or '''degree of curvature''' is a measure of [[curvature]] used in [[civil engineering]] for its easy use ...
3: ...eet]] of [[Chord (geometry)|chord]], but 100 feet of [[arc]] is also common, and other lengths&mdash;s...
11: ...org/freemoslo/Modules/Tips-and-Techniques/degrees_of_curve.htm
12: ...tries.com/standardmodule/circurve.htm *http://ceprofs.tamu.edu/rbruner/curves/circularcrvs.htm
14: *http://www.cityoffrederick.com/departments/Planning/APPENDIX_C_SUBD...

Sectional curvature (1892 bytes)
1: [[Category:Riemannian geometry]]
3: ... <math>\sigma_p</math> (in other words, the image of <math>\sigma_p</math> under the [[exponential map...
5: ...atures in all directions at ''p'' determine the [[curvature tensor]] completely, and it is very useful geomet...
7: ...re manifolds are quotients of those by some group of [[isometry|isometries]] .
11: ...math>f_p(x)=dist^2(p,x)</math> is 1-[[Glossary of Riemannian and metric geometry|concave]] for all points ''p'...

Geodesic curvature (1142 bytes)
1: [[Category:Riemannian geometry]]
3: ...h]] distance along each [[infinitesimal]] segment of its length.
5: ...he [[curvature vector]] '''k''' of the projection of the curve ''C'' onto the [[tangent plane]] at ''P...
7: ... <math>k_g</math>. A curve for which the geodesic curvature is everywhere vanishing is called a [[geodesic]].
9: ...urve along a surface is an [[intrinsic property]] of the surface. -->

Ricci curvature (4952 bytes)
1: ...the [[Riemannian metric]] tensor in the case of [[Riemannian manifold]]s.
3: ... by ''Ric''(''u'',''v''), is defined as the trace of the [[endomorphism]]
5: where ''R'' is the [[Riemann curvature tensor]]. In [[local coordinates]], one can write...
10: ...Ricci tensor is the only independent trace of the curvature tensor; all others vanish or are equal to the Ric...
11: on a Riemannian manifold is symmetric in its arguments

Curvature form (2782 bytes)
3: ... the '''curvature form''' describes [[curvature]] of [[principal bundle]] with [[Cartan connection|con...
4: ...sidered as an alternative or generalization of [[curvature tensor]].
8: .... Let us denote by <math>g</math> the Lie algebra of ''G''.
11: Then the '''curvature form''' is the 2-form with values in ''g'' define...
23: .../math> is a vector bundle then one can also think of <math>\omega</math> as

Curvature tensor (2395 bytes)
1: ...torsion]]. The curvature tensor is given in terms of a [[Levi-Civita connection]] (more generally, an ...
5: ...> is a linear transformation of the tangent space of the manifold; it is linear in each argument.
7: '''NB.''' Some authors define the curvature tensor with the opposite sign.
13: ...the curvature tensor measures ''anticommutativity of the covariant derivative''.
15: ...ath>w\mapsto R(u,v)w</math> is also called the '''curvature transformation'''.

Scalar curvature (1117 bytes)
1: ...l number]] characterizing the intrinsic curvature of the manifold at that point.
3: ...ormation is needed. See [[curvature of Riemannian manifolds]] for a complete discussion.
5: ...vature is defined as the [[trace]] of the [[Ricci curvature]] with respect to the [[metric tensor|metric]]:
7: ...he trace (see [[musical isomorphisms]]). In terms of [[local coordinates]] one can write
12: [[Category:Riemannian geometry]]

Gromov's compactness theorem (284 bytes)
1: In [[Riemannian geometry]] is '''Gromov's compactness theorem''' ...
2: the set of Riemannian manifolds with [[Ricci curvature]] &ge; ''c'' and [[diameter]] &le; ''D'' is [[pre...
4: [[Category:Riemannian geometry]]

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