Riemannian manifolds

E3649

differential geometric object geometric structure mathematical object

Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.

Aliases (5)

Statements (47)

Predicate	Object
instanceOf	differential geometric object → geometric structure → mathematical object →
allows	definition of curvature → definition of geodesics → measurement of angles between tangent vectors → measurement of areas and volumes → measurement of lengths of curves →
contrastsWith	Finsler manifold → pseudo-Riemannian manifold →
definedAs	smooth manifold equipped with an inner product on each tangent space →
enables	definition of distance function → definition of divergence and Laplace–Beltrami operator → definition of gradient of functions → integration of scalar fields and differential forms →
field	Riemannian geometry → differential geometry →
generalizes	Euclidean space → curved surfaces →
hasComponent	Levi-Civita connection → Riemann curvature tensor → metric tensor → tangent bundle →
hasDimension	any positive integer →
hasHistoricalOrigin	19th century →
hasPart	Riemannian metric → smooth manifold →
hasProperty	locally Euclidean as a topological space → positive-definite metric tensor → smooth structure →
hasVariant	Einstein manifold → Kähler manifold → Riemannian surface → compact Riemannian manifold → complete Riemannian manifold →
introducedBy	Bernhard Riemann →
namedAfter	Bernhard Riemann →
requires	smoothness of metric tensor → smoothness of transition maps →
specialCaseOf	smooth manifold with additional structure →
studiedIn	comparison geometry → global Riemannian geometry → spectral geometry →
usedIn	general relativity → geometric analysis → global analysis → topology via metric methods →

Referenced by (21)

Subject (surface form when different)	Predicate
Bianchi identities ("Riemannian geometry") → Gauss map ("Riemannian geometry") → Gaussian curvature ("Riemannian geometry") → Gauss–Bonnet theorem (early form) ("Riemannian geometry") → Riemannian manifold ("Riemannian geometry") →	field
Über die Hypothesen, welche der Geometrie zu Grunde liegen ("Riemannian geometry") → Über die Hypothesen, welche der Geometrie zu Grunde liegen ("Riemannian manifold") → Über die Hypothesen, welche der Geometrie zu Grunde liegen ("Riemannian metric") →	introduces
Friedrich Bernhard Riemann ("Riemannian geometry") → Georg Friedrich Bernhard Riemann ("Riemannian geometry") →	notableWork
Kullback–Leibler divergence ("information geometry") → Rényi divergence ("information geometry") →	usedIn
Bianchi identities →	appliesTo
Nash embedding theorem →	concerns
Ricci scalar ("Riemannian manifold") →	definedOn
Ricci scalar ("Riemannian geometry") →	fieldOfStudy
Riemannian manifold ("Riemannian surface") →	hasVariant
Can one hear the shape of a drum? ("Riemannian geometry") →	influencedField
Bernhard Riemann ("Riemannian metric") →	knownFor
general relativity ("Riemannian geometry") →	mathematicalFramework
Friedrich Bernhard Riemann ("Riemannian manifold") →	notableConcept