Equivariant tubular and slice neighborhoods for compact torus actions

For a point $x\in M$, let $H=T_x$. By [F2], $O=T\cdot x$ has the intrinsic manifold $T\mathrm{/}H$ and its orbit map into $M$ is a continuous injective immersion. The domain $T\mathrm{/}H$ is compact as a continuous image of compact $T$, and $M$ is Hausdorff, so [F5] makes the orbit map a homeomorphism onto $O$; together with immersion this makes $O$ an embedded submanifold.

Now let $S\subseteq M$ be any compact invariant embedded submanifold and write $E(p,v)={\exp }_p(v)$ on the open set where the geodesic exists up to time $1$. The geodesic theorem in [F3] makes this a smooth map on an open neighborhood of the zero section. At $(p,0)$, its differential on $T_{p}S\oplus {\nu }_p(S)$ is $(u,w)\mapsto u+w$: on the zero section $E(p,0)=p$, while $\frac{d}{ds}{\mathrm{\mid }}_{s=0}E(p,sw)=w$ by the initial-velocity definition and time rescaling of the geodesic. This is an isomorphism because $T_{p}M=T_{p}S\oplus {\nu }_p(S)$, so [F4] makes $E$ a local diffeomorphism at every point of the zero section.

The zero section of $\nu (S)$ is compact. A finite cover by the local diffeomorphism neighborhoods in step 1.2, followed by shrinking in local bundle charts, gives ${ϵ}_0>0$ such that $E$ is defined and a local diffeomorphism on $D_{{ϵ}_0}(\nu (S))$. Suppose no smaller uniform disk is injective. For each sufficiently large $n$, choose distinct $(p_n,v_n),(q_n,w_n)$ with $\mathrm{\parallel }{v}_n\mathrm{\parallel },\mathrm{\parallel }{w}_n\mathrm{\parallel }<1\mathrm{/}n$ and $E(p_n,v_n)=E(q_n,w_n)$. Compactness and first countability of $S$ give a subsequence with $p_n\to p$ and then one with $q_n\to q$: compactness gives a cluster point and a countable neighborhood basis gives the subsequence. Continuity and the vanishing fiber norms imply $E(p_n,v_n)\to p$, $E(q_n,w_n)\to q$, hence $p=q$ by Hausdorffness of $M$. Both bundle points then converge to $(p,0)$, contradicting local injectivity near $(p,0)$. Thus $E$ is injective on some $D_{ϵ}(\nu (S))$, and since it is a local diffeomorphism there, it is a diffeomorphism onto its open image.

Each $t\in T$ is an isometry of $g$, so [F3] and uniqueness of geodesics give $E(t\cdot p,(dt{)}_{p}v)=t\cdot E(p,v)$. The norm of the normal vector is preserved, so the disk $D_{ϵ}(\nu (S))$ and its image are $T$-invariant, proving the equivariant tubular assertion.

Apply the tube construction to $S=O$ from step 1.1. The normal bundle of $O$ is $T$-equivariantly isomorphic to $T{\times }_{H}V$ by $[t,v]\mapsto (t\cdot x,(dt{)}_{x}v)$; this is well defined because replacing $(t,v)$ by $(th,h^{-1}v)$ leaves the vector unchanged. The isomorphism is smooth in the homogeneous charts of [F2], and it identifies the normal disk bundle with $T{\times }_H{D}_{ϵ}(V)$. If $g_0$ fixes a point $[t,v]$, then $g_{0}tH=tH$, hence $t^{-1}{g}_{0}t\in H$; because $T$ is abelian this means $g_0\in H$. Thus every stabilizer in the slice neighborhood is contained in $H$.

If $S=\mathrm{\varnothing }$ the zero bundle and neighborhood are empty. If $S=M$, the normal bundle has rank zero and $E$ is the identity. For the orbit of a fixed point, $H=T$ and the displayed slice is the ordinary invariant normal disk.