Riemannian geometry (6826 bytes)
1: ...n this article and another also called [[elliptic geometry]].''
3: ...the study of [[manifold|smooth manifolds]] with [[Riemannian metric]]s; i.e. a choice of [[positive-definite]]...
5: ...[Non-Euclidean geometry]], as well as [[Euclidean geometry]] itself. These are all treated on the same basis...
7: ...el for the more complicated structure of [[pseudo-Riemannian manifold]]s, which (in dimension four) are the ma...
9: There is no easy introduction to Riemannian geometry. One should work quite a while to build some geom...

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]])
82: [[Category:Riemannian geometry]]

Glossary of Riemannian and metric geometry (10196 bytes)
1: ...erms used in [[Riemannian geometry]] and [[metric geometry]] — it doesn't cover the terminology of [[d...
6: * [[Riemannian manifold]]
9: * [[Glossary of differential geometry and topology]]
10: * [[List of differential geometry topics]]
12: ...' below denote metric spaces, ''M'', ''N'' denote Riemannian manifolds, |''xy''| or <math>|xy|_X</math> denote...

Algebraic geometry and analytic geometry (3533 bytes)
1: ...lso in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this ar...
3: ...hisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings. Since it is usu...
9: ...y the use of topological techniques for algebraic geometry over any [[algebraically closed field]] ''K'' of ...
11: ...c methods within the classical parts of algebraic geometry.
17: [[Category:Algebraic geometry]]

Riemannian manifold (3944 bytes)
1: In [[Riemannian geometry]], a '''Riemannian manifold''' is a real differentiable [[manifold]]...
5: ... follows from the [[Nash embedding theorem]], all Riemannian manifolds can be realized this way.
6: ...ld the first geometric intuitions in [[Riemannian geometry]].
8: Usually a Riemannian manifold is defined as a smooth manifold with a [...
10: ...is a continuously differentiable [[curve]] in the Riemannian manifold ''M'', then we define its length ''L''(&...

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).
18: A '''sub-Riemannian manifold''' is a triple <math>(M, H, g)</math>,
23: Any '''sub-Riemannian manifold''' carries the natural [[intrinsic metri...
29: [[Category:Metric geometry]]

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]]

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: ... + q = n</math> is the dimension of the manifold. Riemannian manifolds are simply those with signature <math>(...
7: ...an manifolds, form the most important subclass of Riemannian manifolds. They are important because of their ph...
9: ... manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature <math>(p,q)</math> is <math...

Curvature of Riemannian manifolds (8410 bytes)
1: ...ave found applications everywhere in differential geometry.
5: Curvature of [[Pseudo-Riemannian manifold]] can be expressed on the same way with ...
7: == Ways to express the curvature of a Riemannian manifold ==
11: The curvature of Riemannian manifold can be described by various ways; the mo...
37: ...h satisfis the identities above, one could find a Riemannian manifold with such a curvature tensor at some poi...

Geometry (3543 bytes)
1: ...ht lines]], [[curve]]s, [[surface]]s, and [[solid geometry|solids]] to draw logical conclusions.
3: ...until a millennium later, and that was [[analytic geometry]], in which [[coordinate system]]s are introduced...
5: ...[reflection]]s, [[rotation]]s, and [[translation (geometry)|translation]]s.
7: ...invariants there are. As an example, in [[affine geometry]] any linear transformation is allowed, and so th...
9: A [[discrete]] form of geometry is treated under [[Pick's theorem]].

Square (geometry) (2865 bytes)
3: In [[plane geometry]], a '''square''' is a [[polygon]] with four equa...
12: ...gle with two squares ] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
13: ...le with two squares I] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
14: ...e with two squares II] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
15: ...ith three squares III] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"

Cupola (geometry) (650 bytes)
4: In [[geometry]], a '''cupola''' is a solid formed by joining tw...

Distance geometry (1996 bytes)
1: '''Distance Geometry''' is the characterization and study of [[set]]s ...
3: Therefore distance geometry has immediate relevance where distance values are...

Parallel (geometry) (1544 bytes)
1: '''Parallel''' is a term in [[geometry]] and in everyday life that refers to a property ...
11: [[Category:Geometry]]

Defect (geometry) (2110 bytes)
1: In [[geometry]], the '''defect''' of a vertex of a [[polyhedron...
5: The defect of any of the vertices of a [[cube (geometry)|cube]] is a [[right angle]].
17: ...is tempting to think (and has even been stated in geometry textbooks) that every non-convex polyhedron has s...

Warped geometry (683 bytes)
1: In [[physics]], '''warped geometry''' is a special type of [[metric]] in [[general r...
5: ...rtesian product]] of the "y" geometry and the "x" geometry - except that the "x"-part is warped, i.e. it is ...

Quantum geometry (1816 bytes)
1: ...hematical concepts generalizing the concepts of [[geometry]] whose understanding is necessary to describe th...
3: ... geometrical intuition. More technically, quantum geometry refers to the shape of the spacetime manifold as ...
5: ...m]]. It has also been shown that the loop quantum geometry is non-commutative.
7: ...ry will be consistent with the quantum picture of geometry arising from string theory.

Scaling (geometry) (2624 bytes)
1: In [[Euclidean geometry]], '''scaling''' is an [[affine transformation|af...
29: ...[[homogeneous coordinates]], since [[translation (geometry)|translation]] cannot be accomplished with a 3-by...

Taxicab geometry (1307 bytes)
1: ... the usual [[metric space|metric]] of [[Euclidean geometry]] is replaced by a new metric in which the distan...
9: [[Category:Metric geometry]]

Descriptive geometry (9247 bytes)
1: [[Category:Geometry]]
3: [[Gaspard Monge]] is the father of descriptive geometry. He developed a graphical protocol which creates ...
5: Descriptive geometry uses the image-creating technique of imaginary, p...
17: ... Side; Left Side; Top; Bottom; Rear), Descriptive Geometry strives to yield three basic solution views: the ...
19: ...lp solve engineering type problems posed by solid-geometry principles.

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