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List of formulas in Riemannian geometry

This is a list of formulas encountered in Riemannian geometry.

Contents

Christoffel symbols, covariant derivativeEdit

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

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

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 (scalar)   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 (and volume element)

 

The geodesic   starting at the origin with initial speed   has Taylor expansion in the chart:

 

Curvature tensorsEdit

Riemann curvature tensorEdit

If one defines the curvature operator as   and the coordinate components of the  -Riemann curvature tensor by  , then these components are given by:

 

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 curvaturesEdit

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

The Ricci curvature tensor is essentially the unique (up to sign) nontrivial way of contracting the Riemann tensor:

 

Due to the symmetries of the Riemann tensor, contracting on the 4th instead of the 3rd index yields the same tensor, but with the sign reversed - see sign convention (contracting on the 1st lower index results in an array of zeros). The Ricci tensor   is symmetric.

By the contracting relations on the Christoffel symbols, we have

 

The scalar curvature is the trace of the Ricci curvature,

 .

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

 

that is,

 

Einstein tensorEdit

The Einstein tensor Gab is defined in terms of the Ricci tensor Rab 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 tensorEdit

The Weyl tensor is given by

 

where   denotes the dimension of the Riemannian manifold.

The Weyl tensor satisfies the first (algebraic) Bianchi identity:

 

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,

 

The Weyl tensor 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.

Gradient, divergence, Laplace–Beltrami operatorEdit

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 productEdit

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 frameEdit

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.

 
 

Under a conformal changeEdit

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  , while those unmarked with such will be associated with  .)

 
 

Note that the difference between the Christoffel symbols of two different metrics always form the components of a tensor.

We can also write this in a coordinate-free manner as

 

whenever   is locally a conformal diffeomorphism, i.e.   (  in our case), and   are vector fields.

 

Here   is the Riemannian volume element.

 

Here   is the Kulkarni–Nomizu product defined earlier in this article. The symbol   denotes partial derivative, while   denotes covariant derivative. From this formula and the orthogonal decomposition of the Riemann tensor, it follows that

 

and so we see that the (1,3) Weyl tensor is invariant under conformal changes:

 
 

Beware that here the Laplacian   is minus the trace of the Hessian on functions,

 

Thus the operator   is elliptic because the metric   is Riemannian.

 
 

If the dimension  , then the formula for   simplifies to

 

Given an immersed hypersurface   with a local unit normal vector field  , its second fundamental form   (defined by   for local vector fields   on  ) becomes

 

while its mean curvature   (i.e. the trace of   with respect to the induced metric on  , divided by  ) changes as

 

Let   be a differential  -form. Let   be the Hodge star, and   the codifferential. Under a conformal change, these satisfy

 
 

See alsoEdit