# Warped Products and Conformal Rescalings

## Conventions

In case there's any confusion, the following computations belong to a setting where

and of course

## Warped Procucts

Suppose you have a space with coordinates of the form

so that schematically

This is a so-called *warped product*. A common question is "What is the form of the Ricci tensor?"

Let the dimension of the *x* subspace - that is to say, how many values *i* can range over - be denoted by *D*. Then one obtains the result

where is the Ricci tensor of the metric and is the Ricci tensor of the metric . Note that is the *covariant* derivative with respect to the metric.

Observe that the form of the modification to is a multiple of the metric tensor for the *x* subspace.

## Conformal Rescalings

Another very common computation involves the rescaling

A direct computation in local inertial coordinates for the new metric gives the result

where all covariant derivatives are measured in the unaltered metric , and all indices are raised and lowered using this metric too.