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
![\displaystyle \nabla_XY+\nabla_YX=[X,Y]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cnabla_XY%2B%5Cnabla_YX%3D%5BX%2CY%5D&bg=ffffff&fg=333333&s=0&c=20201002)
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
![\displaystyle\widetilde\delta \beta = {e^{ - 2f}}\left[ {\delta \beta - (n - 2p){\iota_{{\rm grad}f}}\beta } \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cwidetilde%5Cdelta+%5Cbeta+%3D+%7Be%5E%7B+-+2f%7D%7D%5Cleft%5B+%7B%5Cdelta+%5Cbeta+-+%28n+-+2p%29%7B%5Ciota_%7B%7B%5Crm+grad%7Df%7D%7D%5Cbeta+%7D+%5Cright%5D&bg=ffffff&fg=333333&s=0&c=20201002)
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.