riemannian geometry
TRANSCRIPT
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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