Proofs involving covariant derivatives

proofProofs involving Christoffel symbols

This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.wikipedia

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection

This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.

Metric connection Affine connection Covariant derivative Levi-Civita connection Geodesic

List of formulas in Riemannian geometry

formulas in Riemannian geometryreturn to article

This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols. : Q.E.D. (return to article) :The expression in parentheses is the Einstein tensor, so : Q.E.D. (return to article)

The "gradient" of the scalar curvature follows from the Bianchi identity (proof):

Proofs involving covariant derivatives Christoffel symbols Riemann curvature tensor Covariant derivative Formula

Curvature form

Bianchi identityBianchi identitiescurvature

Start with the Bianchi identity

Connection form Curvature Ehresmann connection Flat vector bundle Riemann curvature tensor

Tensor contraction

contractioncontractingContract

Contract both sides of the above equation with a pair of metric tensors:

Tensor Multilinear algebra Trace (linear algebra)Covariance and contravariance of vectors Interior product

Metric tensor

metricmetricsround metric

Contract both sides of the above equation with a pair of metric tensors:

Metric (mathematics)Differential geometry Tensor field Line element Tensor

Q.E.D.

quod erat demonstrandum∎QED

: Q.E.D. (return to article) :The expression in parentheses is the Einstein tensor, so : Q.E.D. (return to article)

Tombstone (typography)Paul Halmos Theorem List of Latin abbreviations Ipso facto

Kronecker delta

Kronecker delta functiongeneralized Kronecker deltadelta function

:where δ is the Kronecker delta.

Iverson bracket Impulse response Tensor Leopold Kronecker Dirac delta function

Covariant derivative

covariant differentiationtensor derivativecovariant differential

:and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then

Connection (mathematics)Affine connection Connection (principal bundle)Christoffel symbols Directional derivative

Einstein tensor

Einstein curvature tensor

:The expression in parentheses is the Einstein tensor, so : Q.E.D. (return to article)

Einstein field equations General relativity Contracted Bianchi identities Mathematics of general relativity Differential geometry

Coordinate system

coordinatescoordinateaxis

Starting with the local coordinate formula for a covariant symmetric tensor field, the Lie derivative along a vector field is

Analytic geometry Geometry Curvilinear coordinates Orthogonal coordinates Skew coordinates

Lie derivative

Lie bracketLie commutatorcommuting vector fields

Starting with the local coordinate formula for a covariant symmetric tensor field, the Lie derivative along a vector field is

Lie bracket of vector fields Differential geometry Tensor field Sophus Lie Differentiable manifold

Vector field

vector fieldsvectorgradient flow

Starting with the local coordinate formula for a covariant symmetric tensor field, the Lie derivative along a vector field is

Vector calculus Magnetic field Line integral Divergence Curl (mathematics)

Partial derivative

partial derivativespartial differentiationpartial differential

:here, the notation means taking the partial derivative with respect to the coordinate x^a.

Derivative Vector calculus Total derivative ∂Gradient

Proofs involving covariant derivatives

Christoffel symbols

List of formulas in Riemannian geometry

Curvature form

Tensor contraction

Metric tensor

Q.E.D.

Kronecker delta

Covariant derivative

Einstein tensor

Coordinate system

Lie derivative

Vector field

Partial derivative

Four-gradient

Palatini identity

Ricci calculus

Contracted Bianchi identities

Proofs involving covariant derivatives

Christoffel symbols100% (1/1)

List of formulas in Riemannian geometry100% (1/1)

Curvature form50% (1/1)

Tensor contraction50% (1/1)

Metric tensor50% (1/1)

Q.E.D.50% (1/1)

Kronecker delta50% (1/1)

Covariant derivative50% (1/1)

Einstein tensor50% (1/1)

Coordinate system50% (1/1)

Lie derivative50% (1/1)

Vector field50% (1/1)

Partial derivative50% (1/1)

Four-gradient50% (1/1)

Palatini identity50% (1/1)

Ricci calculus50% (1/1)

Contracted Bianchi identities50% (1/1)

Christoffel symbols

List of formulas in Riemannian geometry

Curvature form

Tensor contraction

Metric tensor

Q.E.D.

Kronecker delta

Covariant derivative

Einstein tensor

Coordinate system

Lie derivative

Vector field

Partial derivative

Four-gradient

Palatini identity

Ricci calculus

Contracted Bianchi identities