I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
- Gromov, Mikhael; Lawson, H.Blaine jun., The classification of simply connected manifolds of positive scalar curvature, Ann. Math. (2) 111, 423-434 (1980). ZBL0463.53025.
More specifically, I want to ask about the proof of Lemma 2 on page 430. I don't understand the first sentence in the proof. (I think I can follow the rest of the proof.) That means that I do not understand what the precise meaning of the first sentence is and that I do not understand how to prove it.
I will try to restate this first sentence as I understand it in a slightly more general setting.
Setup: Let $(M,g)$ be a compact Riemannian manifold and $S\subset M$ be an embedded compact submanifold. We denote by $TS^\bot$ the normal bundle of $S$ in $M$. For small $\varepsilon >0$ we consider the sphere bundle of radius $\varepsilon$ of $TS^\bot$, denoted by $S_\varepsilon (TS^\bot)$. On this sphere bundle, we have two Riemannian metrics.
- We pull back the metric $g$ on $M$ via the normal exponential map $exp^\bot$, $\tilde{g}:=(exp^\bot)^*g$.
- On the tangent bundle $TM$ we have the Saskaki metric $g_S$. The Sasaki metric turns the projection $\pi\colon (TM,g_S)\to (M,g)$ into a Riemannian submersion with totally geodesic flat fibers. It is given as follows: Let $(p,v)\in TM$, $X,Y\in T_{(p,v)}TN$. We can write $X$ and $Y$ as equivalence classes of curves, $X=[(c(t),v(t))]$, $Y=[(k(t),w(t))]$. Then $g_S(X,Y)=g(d\pi X, d\pi Y) + g(\frac{\nabla}{dt}|_{t=0}v(t),\frac{\nabla}{dt}|_{t=0}w(t))$. The second metric on the sphere bundle $S_\varepsilon (TS^\bot)$ is given by restricting the Sasaki metric, $\hat{g}:=g_s|_{S_\varepsilon (TS^\bot)}$.
Claim: $\tilde{g}$ converges in $C^2$ to $\hat{g}$ as $\varepsilon\to 0$.
This claim is not very precise, since the manifold on which $\tilde{g}$ and $\hat{g}$ are defined depends on $\varepsilon$. So I think that one first has to pull back the two metrics to the unit spehre bundle via the map $m_\varepsilon\colon S_1(TS^\bot)\to S_\varepsilon (TS^\bot)$, $m_\varepsilon(p,v):=(p,\varepsilon v)$. Then the claim would be that $(m_\varepsilon)^*\tilde{g}-(m_\varepsilon)^*\hat{g}$ converges to zero in $C^2$. Maybe one also needs to rescale $(m_\varepsilon)^*\tilde{g}$ or $(m_\varepsilon)^*\hat{g}$ (or both) by $\frac{1}{\varepsilon^2}$ that the claim is true. (Here you can see my troubles pinning down the precise statement that Gromov and Lawson made in the first sentence of their proof of Lemma 2.) As for the proof, I'm lost a little at the moment. I can somewhat see that this convergence should be true if one "leaves the basepoint $p\in S$ fixed, by looking at the usual expansion for normal coordinates and using that the fibers of the Sasaki metric are flat.
Questions: I would be very grateful if someone could help me pin down the precise statement that I am looking for. I also would be very happy about any hints on how to prove that statement or any reference that would be helpful.
Thanks in advance!
