Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry1/4-pinched sphere theoremabstract generalization of geometryclassical theoremsclassical theorems of Riemannian geometrycontributions to differential geometrygeometryRiemann spacetime
Elliptic geometry is also sometimes called "Riemannian geometry".wikipedia
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Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture " Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based").
Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.

Volume

volumetriccapacityOrders of magnitude (volume)
This gives, in particular, local notions of angle, length of curves, surface area and volume.
In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.

Differential geometry of surfaces

surfaceshape operatorsmooth surface
It is a very broad and abstract generalization of the differential geometry of surfaces in R 3.
This point of view was extended to higher-dimensional spaces by Riemann and led to what is known today as Riemannian geometry.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.
The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

General relativity

general theory of relativitygeneral relativity theoryrelativity
It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.
The 19th century mathematician Bernhard Riemann's non-Euclidean geometry, called Riemannian Geometry, provided the key mathematical framework which Einstein fit his physical ideas of gravity on, and enabled him to develop general relativity.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.
In Riemannian geometry, the Levi-Civita connection serves a similar purpose.

Glossary of Riemannian and metric geometry

injectivity radiusgeodesic metric spaceproper
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

Non-Euclidean geometry

non-Euclideannon-Euclidean geometriesalternative geometries
It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry.
Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.

Fundamental theorem of Riemannian geometry

fundamental theorems of Riemannian geometry
In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

Angle

acute angleobtuse angleoblique
This gives, in particular, local notions of angle, length of curves, surface area and volume.
In Riemannian geometry, the metric tensor is used to define the angle between two tangents.

Sphere theorem

1/4-pinched sphere theoremdifferentiable sphere theoremquarter pinched manifold
In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound.

Grigori Perelman

PerelmanG. PerelmanPerelman, Grigori
He has made contributions to Riemannian geometry and geometric topology.

Soul theorem

soul conjecturesouls
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case.

Cartan–Hadamard theorem

Cartan-Hadamard theorem
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature.

Exponential map (Riemannian geometry)

exponential mapexponential map of this Riemannian metricExponential map, Riemannian geometry
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.

Embedding

embeddedisometric embeddingtopological embedding
In Riemannian geometry:

Myers's theorem

Myers theoremMyers' theoremBonnet–Myers theorem
The Myers theorem, also known as the Bonnet–Myers theorem, is a classical theorem in Riemannian geometry.

Geodesic

geodesicsgeodesic flowgeodesic equation
Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.
Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry.

Splitting theorem

The splitting theorem is a classical theorem in Riemannian geometry.

Gromov's compactness theorem (geometry)

Gromov's compactness theoremgeometry
In Riemannian geometry, Gromov's (pre)compactness theorem states that the set of compact Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric.

Einstein–Cartan theory

Einstein-CartanEinstein-Cartan gravityEinstein–Cartan
The theory of general relativity was originally formulated in the setting of Riemannian geometry by the Einstein-Hilbert action, out of which arise the Einstein field equations.

Jeff Cheeger

CheegerCheeger, Jeff
Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below).

Isometry

isometriesisometricisometrically
Thus, isometries are studied in Riemannian geometry.