Push-forward of vector fields by local isometries

Question

I am studying Riemannian Manifolds by John Lee, and there is this lemma:

Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if $\varphi:(M,g)\to(\widetilde{M},\tilde{g})$ is a local isometry, then $$\varphi^\ast\widetilde{Rm}=Rm;$$ $$\widetilde{R}(\varphi_\ast X,\varphi_\ast Y)\varphi_\ast Z=\varphi_\ast(R(X,Y)Z)\tag{1}.$$

But a general local isometry need not be a diffeomorphism, so how to make sense of the push-forward of vector fields (e.g. $\varphi_\ast X$) in this case? How to interpret equation $(1)$?

As far as I know, the push-forward is defined for a vector field $X$ on $M$ by $$\varphi_\ast X:\widetilde{M}\to T\widetilde{M},\quad (\varphi_\ast X)_q=d\varphi_{\varphi^{-1}(q)}(X_{\varphi^{-1}(q)}),$$ so $\varphi^{-1}$ must exist and be smooth.

The proof of Lemma 7.2 is left as an exercise to the reader, so I don't have much clue to understand what the author meant.

Answer 1

@JamesS.Cook is right -- that equation is meant to be interpreted pointwise. It's confusing, because in that book I was using the same notation ($\varphi_*X$) for the global pushforward of a vector field and the pointwise pushforward (or differential) of $\varphi$ acting on a vector at a point. I've added a correction to my online correction list, clarifying that this is just a pointwise equation.

In the second edition of my Smooth Manifolds book, I've switched to the clearer notation $d\varphi_p(X)$ for the pointwise differential, and I reserve $\varphi_*X$ for situations in which there is a global pushforward map on vector fields (namely, when $\varphi$ is either a diffeomorphism or a Lie group homomorphism). I'll be switching to this convention in the second edition of Riemannian Manifolds, whenever it comes out (hopefully within the next two years).

Answer 2

I would assume that the key word here is "local". We know for a local isometry the map $\phi$ is regular (meaning the local coordinate representative has full rank) hence $\phi$ suitably restricted gives a diffeomorphism. In particular, if $\phi$ has an injective differential at a point $p \in M$ then by the inverse function theorem for manifolds there exists an open set $\mathcal{U}\subseteq M$ containing $p$ and an open set $\mathcal{V} \subseteq \widetilde{M}$ for which $\phi|_{\mathcal{U}}: \mathcal{U} \rightarrow \mathcal{V}$ is a diffeomorphism. See page 79, Theorem 4.5 of John M. Lee's Introduction to Smooth Manifolds for a clear statement and proof of the inverse function theorem for manifolds. I couldn't find it in the text Riemannian Manifolds by Lee (although, perhaps it's in there somewhere)

So, how should we understand the push-forward? Well, in short, point-wise. Observe, locally, the push-forward does give vector fields on an open set and that is enough to calculate the local Riemann data. Indeed, globally we can have $M$ and $\widetilde{M}$ which are not diffeomorphic, yet, locally they are diffeomorphic. For example, see this related MSE question

This much we can say, if we have a bijection of $\psi:M \rightarrow \widetilde{M}$ and $\psi_*$ is injective at each $p \in M$ then $\psi$ is a diffeomorphism. We just weave together the local inverses to show smoothness of $\psi^{-1}$.