- Riemannian geometry
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Elliptic geometry is also sometimes called**Riemannian geometry**."**Riemannian geometry**is the branch ofdifferential geometry that studiesRiemannian manifold s, smooth manifolds with a "Riemannian metric", i.e. with aninner product on thetangent space at each point which varies smoothly from point to point. This gives in particular local notions ofangle , length of curves,surface area , andvolume . From those some other global quantities can be derived by integrating local contributions.Riemannian geometry originated with the vision of

Bernhard Riemann expressed in his inaugurational lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (German: On the hypotheses on which geometry is based). It is a very broad and abstract generalization of thedifferential geometry of surfaces in**R**^{3}. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior ofgeodesic s on them, with techniques that can be applied to the study ofdifferentiable manifold s of higher dimensions. It inspired Einstein'sgeneral relativity theory , made profound impact ongroup theory andrepresentation theory , as well as analysis, and spurred the development of algebraic anddifferential topology .**Introduction**Riemannian geometry was first put forward in generality by

Bernhard Riemann in thenineteenth century . It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types ofNon-Euclidean geometry ,spherical geometry andhyperbolic geometry , as well asEuclidean geometry itself.Any smooth manifold admits a

Riemannian metric , which often helps to solve problems ofdifferential topology . It also serves as an entry level for the more complicated structure ofpseudo-Riemannian manifold s, which (in four dimensions) are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry and spray spaces.There is no easy introduction to Riemannian geometryFact|date=July 2008. It is generally recommendedWho|date=July 2008 that one should work in the subject for quite a while to build some geometric intuition, usually by doing enormous amounts of calculations. The following articles might serve as a rough introduction:

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Metric tensor

#Riemannian manifold

#Levi-Civita connection

#Curvature

#Curvature tensor .The following articles might also be useful:

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List of differential geometry topics

#Glossary of Riemannian and metric geometry **Classical theorems in Riemannian geometry**What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance, beauty, and simplicity of formulation. Most of the results can be found in the classic monograph by

Jeff Cheeger and D. Ebin (see below).The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.

**General theorems**#

The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to $2pichi(M)$ where $chi(M)$ denotes theGauss–Bonnet theorem Euler characteristic of "M". This theorem has a generalization to any compact even-dimensional Riemannian manifold, seegeneralized Gauss-Bonnet theorem .

#also called fundamental theorems of Riemannian geometry. They state that everyNash embedding theorem sRiemannian manifold can be isometrically embedded in aEuclidean space **R**^{"n"}.**Local to global theorems**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.

**Pinched sectional curvature**#

**Brendle and Schoen's differential sphere theorem.**If "M" is a compact "n"-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then "M" is diffeomorphic to a spherical space form. This is sharp: complex projective space has curvature non-strictly pinched between 1/4 and 1. If strict pinching is replaced by weak pinching (i.e. if the sectional curvature of "M" lies in the closed interval $[1/4,1]$), then "M" is diffeomorphic to a spherical space form or isometric to a locally symmetric space. For more information see the article on theSphere theorem .

#**Cheeger's finiteness theorem.**Given constants "C" and "D" there are only finitely many (up to diffeomorphism) compact "n"-dimensional Riemannian manifolds with sectional curvature $|K|le\; C$ and diameter $le\; D$.

#**Gromov's almost flat manifolds.**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 anil manifold .**Positive sectional curvature**#

If "M" is a non-compact complete positively curved "n"-dimensional Riemannian manifold then it is diffeomorphic toSoul theorem .**R**^{n}.

#**Gromov's Betti number theorem.**There is a constant "C=C(n)" such that if "M" is a compact connected "n"-dimensional Riemannian manifold with positive sectional curvature then the sum of itsBetti number s is at most "C".**Positive**Ricci curvature #

If a compact Riemannian manifold has positive Ricci curvature then itsMyers theorem .fundamental group is finite.

#If a complete "n"-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic which minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete ("n"-1)-dimensional Riemannian manifold which has nonnegative Ricci curvature.Splitting theorem .

#**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.**Positive scalar curvature**#The "n"-dimensional torus does not admit a metric with positive scalar curvature.

#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).**Non-positive sectional curvature**#The

states that a completeCartan–Hadamard theorem simply connected Riemannian manifold M with nonpositive sectional curvature isdiffeomorphic to theEuclidean space R^n with n = dim M via theexponential map at any point. It implies that any two points of a simply connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.**Negative sectional curvature**#The

geodesic flow of any compact Riemannian manifold with negative sectional curvature isergodic .

#If "M" is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant "k" then it is a CAT("k") space. Consequently, itsfundamental group "Γ" = π_{1}("M") is Gromov hyperbolic. This has many implications for the structure of the fundamental group:::* it is finitely presented;::* the word problem for "Γ" has a positive solution;::* the group "Γ" has finite virtualcohomological dimension ;::* it contains only finitely manyconjugacy class es of elements of finite order;::* the abelian subgroups of "Γ" are virtually cyclic, so that it does not contain a subgroup isomorphic to**Z**×**Z**.**Negative Ricci curvature**#The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.

#Any smooth manifold of dimension $n\; geq\; 3$ admits a Riemannian metric with negative Ricci curvature [*Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature.*] . ("This is not true for surfaces".)**ee also***

Shape of the universe

*Basic introduction to the mathematics of curved spacetime

*Normal coordinates

*Systolic geometry **Notes****References**;Books

*. "(Provides a historical review and survey, including hundreds of references.)"*; Revised reprint of the 1975 original.

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;Papers

*citation|first1=Simon|last1=Brendle|first2=Richard M.|last2=Schoen|authorlink2=Richard Schoen|title=Classification of manifolds with weakly 1/4-pinched curvatures|journal=Preprint|year=2007 arxiv | id= 0705.3963

**External links*** [

*http://mathworld.wolfram.com/RiemannianGeometry.html MathWorld: Riemannian Geometry*]

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