This is a list of formulas encountered in Riemannian geometry.
This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
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
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 matrix of scalar coefficients of the metric tensor
The covariant derivative of a vector field with components
The covariant derivative of a function (scalar)
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
starting at the origin with initial speed
is a vector field then which is just the definition of the Riemann tensor.
can be covered by smooth coordinate charts relative to which the metric tensor is of the form
is obtained by raising the index of the differential
is given by The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold.
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 An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations
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.
be a Riemannian or pseudo-Riemanniann metric on a smooth manifold
Evidently, conformality of metrics is an equivalence relation.
Here are some formulas for conformal changes in tensors associated with the metric.
Using the Kulkarni–Nomizu product: 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.
Then Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature
is a (local) normal vector field.
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.
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.