# List of formulas in Riemannian geometry

**formulas in Riemannian geometryreturn to article**

This is a list of formulas encountered in Riemannian geometry.wikipedia

36 Related Articles

### 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).