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

Obviously,

Thus, we can conclude that

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What do you mean by “the” map x \rightarrow \lambda x ? I don’t think this map is well defined (nor unique) on an arbitrary manifold. The best that I could think of is: fix a local coordinate system with image being the whole of \mathbb{R}, then define a multiplication map on the domain of this local coordinate system via the local coordinate system. But I don’t see the interest.

Alternatively, you have to assume that their exist an action of R on your manifold. Your map is then dependent on fixing this action.

Comment by me — March 5, 2012 @ 17:39

Thanks for your interest in my post. I agree with you that the whole topic is somewhat ridiculous and probably doesn’t make any sense. Besides, thanks for your suggestion, I think that should be the best and that matches my thinking.

At this moment, I am interested in some connection between and where is a smooth map. In other words, how much we understand the pullback metric . But, I guess this is far way from what I really need and what I really can. I just need, as you pointed out, to deal with the case is the local coordinate map, i.e., . To be precise, .

If I have time, I will write a note where I make use of this trivial stuff.

Comment by Ngô Quốc Anh — March 5, 2012 @ 18:00