Today, let us discuss a very simple question. Assume that is a Riemannian manifold and is constant. We study the following so-called pullback metric of under the scaling . Precisely, we aim to compare and .
For simplicity, we follow definition of the pullback metric. Besides, by we mean the map .
The pushforward by . This is the first step. Suppose for some . By definition, we have which is just
By the definition of derivation, in local coordinates, we clearly have
The pullback metric . By definition, we have
Thus, we can conclude that