# List of formulas in Riemannian geometry

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:

### 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

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

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).