List of formulas in Riemannian geometry

formulas in Riemannian geometryreturn to article
This is a list of formulas encountered in Riemannian geometry.wikipedia
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Proofs involving covariant derivatives

proofProofs involving Christoffel symbols
The "gradient" of the scalar curvature follows from the Bianchi identity (proof):
This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
If one defines the curvature operator as and the coordinate components of the (1,3)-Riemann curvature tensor by, then these components are given by:
(see also the list of formulas in Riemannian geometry).

Covariant derivative

covariant differentiationtensor derivativecovariant differential
The covariant derivative of a vector field with components v^i is given by:

Formula

formulaeformulasmathematical formulae
This is a list of formulas encountered in Riemannian geometry.

Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
This is a list of formulas encountered in Riemannian geometry.

Topological manifold

coordinate charttopological2-manifold
In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

Invertible matrix

invertibleinversenonsingular
Here g^{ij} is the inverse matrix to the metric tensor g_{ij}.

Manifold

manifoldsboundarymanifold with boundary
is the dimension of the manifold.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
the second of which is equivalent to the torsion-freeness of the Levi-Civita connection.

Determinant

determinantsdetmatrix determinant
where |g| is the absolute value of the determinant of the metric tensor g_{ik}.

Vector field

vector fieldsvectorgradient flow
The covariant derivative of a vector field with components v^i is given by:

Tensor field

tensor analysistensortensor bundle
and similarly the covariant derivative of a (0,1)-tensor field with components v_i is given by:

Geodesic

geodesicsgeodesic flowgeodesic equation
:The geodesic X(t) starting at the origin with initial speed v^i has Taylor expansion in the chart:

Curvature form

Bianchi identityBianchi identitiescurvature
The (second) Bianchi identity is

Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
The Ricci curvature tensor is essentially the unique (up to sign) nontrivial way of contracting the Riemann tensor:

Scalar curvature

Ricci scalarcurvaturecurvature scalar
The scalar curvature is the trace of the Ricci curvature,

Einstein tensor

Einstein curvature tensor
The Einstein tensor G ab is defined in terms of the Ricci tensor R ab and the Ricci scalar R,

Weyl tensor

Weyl curvatureWeyl curvature tensorWeyl
The Weyl tensor is given by

Gradient

gradientsgradient vectorvector gradient
The gradient of a function \phi is obtained by raising the index of the differential, whose components are given by:

Divergence

divergence operatorconverge or divergeConvergence
The divergence of a vector field with components V^m is

Laplace–Beltrami operator

Laplace-Beltrami operatorLaplacianLaplace–de Rham operator
The Laplace–Beltrami operator acting on a function f is given by the divergence of the gradient:

Antisymmetric tensor

antisymmetricantisymmetrizationcompletely antisymmetric tensor
The divergence of an antisymmetric tensor field of type (2,0) simplifies to

Kulkarni–Nomizu product

Kulkarni-Nomizu product
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold.

Inertial frame of reference

inertial frameinertialinertial reference frame
An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations and (but these may not hold at other points in the frame).