I guess I will use the relation between curvature tensors of metrics lying in a conformal class frequently so I decide to post something related to this stuff which may be helpful and we can use later. Actually, I have used it when we proved conformal Laplacian operator is invariant. Let us briefly recall some terminologies

Definition(conformal). Two pseudo-Riemannian metrics and on a manifold are said to be

- (pointwise) conformal if there exists a function on such that
;

- conformally equivalent if there exists a diffeomorphism of such that and are pointwise conformal.

Note that, if and are conformally equivalent, then is an isometry from onto . So we will only study below the case . Our aim is to compare Riemann curvature, Scalar curvature, Ricci curvature,… of and .

Definition(the Kulkarni–Nomizu product). This product is defined for two -tensors and gives as a result a -tensor. Precisely,or

.

**Levi-Civita connection**. On , the Levi-Civita connection is an affine connection which is torsion free

and satisfies the rule

for any vector fields . We now have

.

**Weyl tensor**. This tensor is defined to be

.

Thus we have the rule

.

**Ricci tensor**. This is a -tensor defined by

.

In local coordinates, it has the form

.

So we have the following rule

.

**Traceless Ricci tensor**. This tensor is defined by

.

A simple calculation shows that its trace, , equals zero. So

.

**Scalar curvature**. This tensor is defined to be the trace of Ricci tensor, that is

.

So

.

In practice, this conformal change is not useful, we usually use the following conformal change

.

With this, we simply have

or

.

**Riemann curvature tensor**. This tensor is defined to be

.

In local coordinates, we get

.

So

.

**Volume element**. This, , is the unique density such that, for any orthonormal basis of ,

.

In local coordinates,

.

So

.

**Hodge operator on -forms** (if is oriented). The Hodge star operator on an oriented inner product space is a linear operator on the exterior algebra of , interchanging the subspaces of -vectors and -vectors where , for . It has the following property, which defines it completely: given an oriented orthonormal basis we have

.

One can repeat the construction above for each cotangent space of an -dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the Hodge dual -form, of a -form. The Hodge star then induces an -norm inner product on the differential forms on the manifold. One writes

for the inner product of sections and of . (The set of sections is frequently denoted as ). Elements of are called exterior -forms). For example, for a positively oriented orthogonal cofram , one has

.

So

.

**Codifferential on -forms**. This notion is usually defined through the exterior derivative

by the following rule (also called the formal adjoint of exterior derivative)

,

i.e.

.

In other words, for a -form ,

.

So

where denotes the interior product (the contraction of a differential form with a vector field).

**(pseudo-) Laplacian on -forms**. This is known as the Hodge Laplacian and also known as the Laplace–de Rham operator. It is defined by

.

An important property of the Hodge Laplacian is that it commutes with the operator, i.e.

.

So

.

See also: Arthur L. Besse, *Einstein manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1987.

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