RIEMANNI

AN

GEOMETRYBy: Winme Catchonite

Belonio

Riemannian geometry

is sometimes called

Elliptic geometry.

It 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.

It is the branch of

differential geometry that

studies Riemannian

manifolds, smooth

bmanifolds with

a Riemannian metric.

It is the study

of manifolds having a

complete Riemannian

metric.

Riemannian geometry is a general space based on the line element ds=F(x1,…,xn; d x1,…, d xn), with F(x,y)>0 for y≠0 a function on the tangent bundle TM. In addition, F is homogeneous of degree 1 in y and of the formF2= gij(x)d xi d xj

CLASSICAL THEOREMS

IN

RIEMANNIAN GEOMETRY

General theorems

Gauss–Bonnet theorem It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

General theorems

Nash embedding theorems

also called fundamental

theorems of Riemannian

geometry named after

John Forbes Nash.

Pinched sectional curvature

Sphere theorem

Cheeger's finiteness theorem

Gromov's almost flat

manifolds

Sectional curvature bounded below

Cheeger-Gromoll's Soul theorem

Gromov's Betti number theorem

Grove–Petersen's finiteness theorem

Sectional curvature bounded above

The Cartan–Hadamard

theorem

The geodesic flow

Ricci curvature bounded below

Myers theorem

Splitting theorem

Bishop–Gromov inequality

Gromov's compactness

theorem

The development of the 20th century has turned Riemannian geometry into one of the most important parts of modern mathematics.

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