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 satisfy the symmetry relations
or, respectively,
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 . These are useful when dealing with divergences and Laplacians (see below).
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 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 (0,k)-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 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 (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 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 (pointwise) 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 , while those unmarked with such will be associated with .)
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 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
for all
Here denotes the -orthogonal projection of onto the -orthogonal linear subspace to
Mean curvature of an immersion
In the same setting as above (and suppose has dimension ), recall that the mean curvature vector is for each an element defined as the -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 in the hypersurface case is
where is a (local) normal vector field.
Variation formulas
Let be a smooth manifold and let 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.
The principal symbol of the map assigns to each a map from the space of symmetric (0,2)-tensors on to the space of (0,4)-tensors on given by
The principal symbol of the map assigns to each an endomorphism of the space of symmetric 2-tensors on given by
The principal symbol of the map assigns to each an element of the dual space to the vector space of symmetric 2-tensors on by
Arthur L. Besse. "Einstein manifolds." Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. Springer-Verlag, Berlin, 1987. xii+510 pp. ISBN3-540-15279-2
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