List of formulas in Riemannian geometry

Christoffel symbols, covariant derivative

In a smooth coordinate chart, the Christoffel symbols

Christoffel symbols

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...

are given by:

Here

is the inverse

Inverse

Inverse may refer to:* Inverse , a type of immediate inference from a conditional sentence* Inverse , a program for solving inverse and optimization problems...

matrix to the metric tensor

. In other words,

and thus

is the dimension of the manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

.

Christoffel symbols satisfy the symmetry relation

which is equivalent to the torsion-freeness of the Levi-Civita connection

Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...

.

The contracting relations on the Christoffel symbols are given by

and

where |g| is the absolute value of the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of the metric tensor

. These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

of a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

with components

is given by:

and similarly the covariant derivative of a

-tensor field

Tensor field

In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

with components

is given by:

For a

-tensor field

Tensor field

with components

this becomes

and likewise for tensors with more indices.

The covariant derivative of a function (scalar)

is just its usual differential:

Because the Levi-Civita connection

Levi-Civita connection

is metric-compatible, the covariant derivatives of metrics vanish,

The geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

starting at the origin with initial speed

has Taylor expansion in the chart:

Riemann curvature tensor

If one defines the curvature operator

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

and the coordinate components of the

-Riemann curvature tensor

Riemann curvature tensor

, then these components are given by:

where n denotes the dimension of the manifold. Lowering indices with

one gets

The symmetries of the tensor are

and

That is, it is symmetric in the exchange of the first and last pair of indices, and antisymmetric in the flipping of a pair.

The cyclic permutation sum (sometimes called first Bianchi identity) is

The (second) Bianchi identity is

that is,

which amounts to a cyclic permutation sum of the last three indices, leaving the first two fixed.

Ricci and scalar curvatures

Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information.

The Ricci curvature

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

tensor is essentially the unique nontrivial way of contracting the Riemann tensor:

The Ricci tensor

is symmetric.

By the contracting relations on the Christoffel symbols, we have

The scalar curvature

Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

is the trace of the Ricci curvature,

.

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

that is,

Einstein tensor

The Einstein tensor

Einstein tensor

In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...

G^ab is defined in terms of the Ricci tensor R^ab and the Ricci scalar R,

where g is the metric tensor.

The Einstein tensor is symmetric, with a vanishing divergence (proof) which is due to the Bianchi identity:

Weyl tensor

The Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...

is given by

where

denotes the dimension of the Riemannian manifold.

Gradient, divergence, Laplace–Beltrami operator

The gradient of a function

is obtained by raising the index of the differential

, that is:

The divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of a vector field with components

The Laplace–Beltrami operator acting on a function

is given by the divergence of the gradient:

The divergence of an antisymmetric tensor field of type

simplifies to

Kulkarni–Nomizu product

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

and

be symmetric covariant 2-tensors. In coordinates,

Then we can multiply these in a sense to get a new covariant 4-tensor, which we denote

. The defining formula is

Often the Kulkarni–Nomizu product is denoted by a circle with a wedge that points up inside it. However, we will use

instead throughout this article. Clearly, the product satisfies

Let us use the Kulkarni–Nomizu product to define some curvature quantities.

Weyl tensor

The Weyl tensor

is defined by the formula

Each of the summands on the righthand side have remarkable properties. Recall the first (algebraic) Bianchi identity that a tensor

can satisfy:

Not only the Riemann curvature tensor on the left, but also the three summands on the right satisfy this Bianchi identity. Furthermore, the first factor in the second summand has trace zero. The Weyl tensor is a symmetric product of alternating 2-forms,

just like the Riemann tensor. Moreover, taking the trace over any two indices gives zero,

In fact, any tensor that satisfies the first Bianchi identity can be written as a sum of three terms. The first, a scalar multiple of

. The second, as

where

is a symmetric trace-free 2-tensor. The third, a symmetric product of alternating two-forms which is totally traceless, like the Weyl tensor described above.

The most remarkable property of the Weyl tensor, though, is that it vanishes (

)if and only if a manifold

of dimension

is locally conformally flat. In other words,

can be covered by coordinate systems in which the metric

satisfies

This is essentially because

is invariant under conformal changes.

In an inertial 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).
In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only.

Under a conformal change

Let

be a Riemannian metric on a smooth manifold

, and

a smooth real-valued function on

. Then

is also a Riemannian metric on

. We say that

is conformal to

. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with