Levi-Civita connection

E22817

affine connection geometric structure mathematical concept

The Levi-Civita connection is the unique torsion-free affine connection on a Riemannian manifold that is compatible with its metric, enabling the definition of parallel transport and covariant differentiation.

Aliases (2)

Christoffel symbols ×1
Koszul formula ×1

Statements (48)

Predicate	Object
instanceOf	affine connection → geometric structure → mathematical concept →
actsOn	tangent bundle → tensor fields →
appearsIn	general relativity →
characterizedBy	covariant derivative of metric equals zero → zero torsion tensor →
codomain	smooth vector fields →
compatibleWith	Riemannian metric → pseudo-Riemannian metric →
definedByFormula	Koszul formula →
definedOn	Riemannian manifold → pseudo-Riemannian manifold →
determinedBy	Christoffel symbols →
domain	smooth vector fields →
enables	covariant differentiation → geodesic equation → parallel transport →
field	Riemannian geometry → differential geometry →
introducedBy	Tullio Levi-Civita →
isUnique	true →
localExpression	Christoffel symbols of the second kind →
namedAfter	Tullio Levi-Civita →
property	metric-compatible → torsion-free →
relatedTo	exponential map on a manifold → geodesic spray →
roleInGeneralRelativity	connection compatible with spacetime metric → defines geodesics of free-falling particles →
satisfies	Leibniz rule for covariant derivative → linearity in vector field arguments → metric-compatibility condition ∇g = 0 → tensoriality in lower argument → torsion tensor T = 0 →
specialCaseOf	metric connection → torsion-free connection →
typeOf	linear connection on tangent bundle →
usedFor	defining Ricci curvature → defining Riemann curvature tensor → defining curvature tensors → defining scalar curvature →
usedIn	Riemannian submanifold theory → comparison theorems in Riemannian geometry → holonomy theory → study of symmetric spaces →
yearIntroducedApprox	early 20th century →

Referenced by (13)

Subject (surface form when different)	Predicate
Einstein tensor → Ricci curvature tensor → Ricci curvature tensor ("Christoffel symbols") → Ricci scalar → Riemann curvature tensor →	dependsOn
Christoffel symbols →	associatedWith
Levi-Civita connection ("Koszul formula") →	definedByFormula
Riemannian manifold →	hasComponent
Tullio Levi-Civita →	knownFor
Tullio Levi-Civita →	notableConcept
Christoffel symbols →	relatedConcept
Bianchi identities →	relatesTo
Einstein–Maxwell equations →	uses