This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.

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 g^{ij} is the inverse matrix to the metric tensor g_{ij}. 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 | is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor g_{ik}. These are useful when dealing with divergences and Laplacians (see below). The covariant derivative of a vector field with components v^i is given by: and similarly the covariant derivative of a (0,1)-tensor field with components v_i is given by: For a (2,0)-tensor field with components v^{ij} this becomes and likewise for tensors with more indices. The covariant derivative of a function (scalar) \phi is just its usual differential: Because the Levi-Civita connection is metric-compatible, the covariant derivative of the metric vanishes, as well as the covariant derivatives of the metric's determinant (and volume element) The geodesic X(t) starting at the origin with initial speed v^i has Taylor expansion in the chart:

Curvature tensors

Definitions

(3,1) Riemann curvature tensor

(3,1) Riemann curvature tensor

Ricci curvature

Scalar curvature

Traceless Ricci tensor

(4,0) Riemann curvature tensor

(4,0) Weyl tensor

Einstein tensor

Identities

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:

First Bianchi identity

Second Bianchi identity

Contracted second Bianchi identity

Twice-contracted second Bianchi identity

Equivalently:

Ricci identity

If X is a vector field then which is just the definition of the Riemann tensor. If \omega is a one-form then More generally, if T is a (0,k)-tensor field then

Remarks

A classical result says that W=0 if and only if (M,g) is locally conformally flat, i.e. if and only if M can be covered by smooth coordinate charts relative to which the metric tensor is of the form for some function \varphi on the chart.

Gradient, divergence, Laplace–Beltrami operator

The gradient of a function \phi is obtained by raising the index of the differential, whose components are given by: The divergence of a vector field with components V^m is The Laplace–Beltrami operator acting on a function f is given by the divergence of the gradient: The divergence of an antisymmetric tensor field of type (2,0) 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 A and B 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 (but these may not hold at other points in the frame). 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

Let g be a Riemannian or pseudo-Riemanniann metric on a smooth manifold M, and \varphi a smooth real-valued function on M. Then is also a Riemannian metric on M. We say that \tilde g is (pointwise) conformal to g. 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 \tilde g, while those unmarked with such will be associated with g.)

Levi-Civita connection

(4,0) Riemann curvature tensor

Using the Kulkarni–Nomizu product:

Ricci tensor

Scalar curvature

Traceless Ricci tensor

(3,1) Weyl curvature

Volume form

Hodge operator on p-forms

Codifferential on p-forms

Laplacian on functions

Hodge Laplacian on p-forms

The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian.

Second fundamental form of an immersion

Suppose (M,g) is Riemannian and is a twice-differentiable immersion. Recall that the second fundamental form is, for each p\in M, a symmetric bilinear map which is valued in the g_{F(p)}-orthogonal linear subspace to Then Here denotes the g_{F(p)}-orthogonal projection of onto the g_{F(p)}-orthogonal linear subspace to

Mean curvature of an immersion

In the same setting as above (and suppose \Sigma has dimension n), recall that the mean curvature vector is for each p\in\Sigma an element defined as the g-trace of the second fundamental form. Then Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature H in the hypersurface case is where \eta is a (local) normal vector field.

Variation formulas

Let M be a smooth manifold and let g_t be a one-parameter family of Riemannian 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. is a one-parameter family of symmetric 2-tensor fields.

Principal symbol

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.