List of formulas in Riemannian geometry


This is a list of formulas encountered in Riemannian geometry.

Christoffel symbols, covariant derivative

In a smooth coordinate chart, the Christoffel symbols of the first kind are given by
and the Christoffel symbols of the second kind by
Here is the inverse matrix to the metric tensor. In other words,
and thus
is the dimension of the manifold.
Christoffel symbols satisfy the symmetry relations
the second of which is equivalent to the torsion-freeness of the Levi-Civita connection.
The contracting relations on the Christoffel symbols are given by
and
where |g| is the absolute value of the determinant of the metric tensor. These are useful when dealing with divergences and Laplacians.
The covariant derivative of a vector field with components is given by:
and similarly the covariant derivative of a -tensor field with components is given by:
For a -tensor field with components this becomes
and likewise for tensors with more indices.
The covariant derivative of a function is just its usual differential:
Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,
as well as the covariant derivatives of the metric's determinant
The geodesic starting at the origin with initial speed has Taylor expansion in the chart:

Curvature tensors

Definitions

(3,1) [Riemann curvature tensor]

See Proofs involving Christoffel symbols for some proofs

Basic symmetries

The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
Equivalently:
If is a vector field then
which is just the definition of the Riemann tensor. If is a one-form then
More generally, if is a -tensor field then

Remarks

A classical result says that if and only if is locally conformally flat, i.e. if and only if can be covered by smooth coordinate charts relative to which the metric tensor is of the form for some function on the chart.

Gradient, divergence, Laplace–Beltrami operator

The gradient of a function is obtained by raising the index of the differential, whose components are given by:
The divergence of a vector field with components is
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
The Hessian of a map is given by

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 is often denoted. The defining formula is
Clearly, the product satisfies

In an inertial frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations and . These coordinates are also called normal coordinates.
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.

Conformal change \widetilde{g}=e^{2\varphi}g

Let be a Riemannian or pseudo-Riemanniann 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.
Levi-Civita connection
Using the Kulkarni–Nomizu product:
Scalar curvature
Suppose is Riemannian and is a twice-differentiable immersion. Recall that the second fundamental form is, for each a symmetric bilinear map which is valued in the -orthogonal linear subspace to Then
Here denotes the -orthogonal projection of onto the -orthogonal linear subspace to
Mean curvature of an immersion
In the same setting as above, recall that the mean curvature is for each an element defined as the -trace of the second fundamental form. Then
Let be a smooth manifold and let be a one-parameter family of Riemanannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives exist and are themselves as differentiable as necessary for the following expressions to make sense. Denote as a one-parameter family of symmetric 2-tensor fields.
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.