Fixed loci and normal weights of compact torus actions

Choose a smooth Riemannian metric $g_0$ by [F3], let $\mu$ be the normalized Haar probability measure from [F2], and define

for $p\in M$ and $v,w\in T_{p}M$.

In every pair of local tangent-bundle frames the integrand in step 1.1 is smooth in $p$ and the torus coordinates; repeated application of [F4] on $[0,1{]}^r$ shows that the coefficients of $g$ are smooth. Each integrand is a symmetric bilinear form, and for $v\mathrm{\ne }0$ its value on $(v,v)$ is strictly positive because $d{a}_{t^{-1}}$ is invertible. Its continuous positive value has a positive minimum on compact $T$ by [F5], so the integral is positive and $g$ is a Riemannian metric. For $s\in T$, translation invariance of $\mu$, with the change of variable $t=su$, gives $g_{sp}(d{a}_{s}v,d{a}_{s}w)=g_p(v,w)$. Thus $g$ is $T$-invariant.

For each $t\in T$, the equalizer $\{p:a(t,p)=p\}$ is closed: if $a(t,p)\mathrm{\ne }p$, Hausdorffness gives disjoint neighborhoods of $p$ and $a(t,p)$, and continuity gives a neighborhood of $p$ disjoint from the equalizer. Hence $F$, their intersection, is closed in compact $M$, and is compact.

Fix $p\in F$. Each $a_t$ is a $g$-isometry fixing $p$; its differential ${\rho }_p(t)=(d{a}_t{)}_p$ is an orthogonal linear action on $T_{p}M$. The geodesic exponential ${\exp }_p$ is defined and smooth near zero by [F6], has differential the identity at zero by the initial-velocity definition of a geodesic, and is a local diffeomorphism there by [F8]. Isometries preserve geodesics by [F7], so uniqueness in [F6] gives $a_t({\exp }_{p}v)={\exp }_p({\rho }_p(t)v)$ wherever both sides are defined. Choose a sufficiently small ${\rho }_p(T)$-invariant ball about zero on which ${\exp }_p$ is injective; in its image the fixed set is exactly the image of the linear fixed subspace $(T_{p}M{)}^T$. These charts show that $F$ is an embedded submanifold near every $p\in F$ and $T_{p}F=(T_{p}M{)}^T$. Since $F$ is compact, its connected components are finite in number: its components are open in the manifold $F$, so compactness gives a finite subcover by component neighborhoods.

The orthogonal complement ${\nu }_{F,p}=(T_{p}F{)}^{\perp }$ has no fixed vector, since the fixed subspace is precisely $T_{p}F$ by step 4.1. Complexify ${\nu }_{F,p}$; the commuting orthogonal transformations ${\rho }_p(t)$ become commuting unitary transformations, so simultaneous diagonalization decomposes the complexified fiber into character eigenspaces. Every character is nontrivial because the fixed eigenspace has been removed. A self-inverse character of the connected group $T$ has image in $\{1,-1\}$ and is therefore trivial; hence complex conjugation pairs the remaining $\chi$ and ${\chi }^{-1}$ eigenspaces. Using the fixed lattice identification, retain the character whose first nonzero integer coordinate is positive from each pair; this equips the real normal fiber with a $T$-invariant complex structure and orientation. For a character $\chi$, the operator $P_{\chi }={\int }_T\overline{\chi (t)}{\rho }_p(t)d\mu (t)$ is the orthogonal projection onto its $\chi$-eigenspace: on a $\psi$-eigenvector it acts by the scalar ${\int }_T\overline{\chi (t)}\psi (t)d\mu (t)$. If $\psi =\chi$, this integral is $1$. If $\psi \mathrm{\ne }\chi$, then $\theta =\bar{\chi }\psi$ is a nontrivial character; choose $t_0\in T$ with $\theta (t_0)\mathrm{\ne }1$. Translation invariance gives $I={\int }_T\theta (t)d\mu (t)={\int }_T\theta (t_{0}t)d\mu (t)=\theta (t_0)I$, so $I=0$. The character averages are therefore exactly the eigenspace projections. Their coefficients vary smoothly with $p$ by [F4]. Their ranks are integer-valued and continuous, hence locally constant, so they form smooth subbundles. At each $p$, finitely many weight projections have ranks summing to the full fiber dimension; their ranks remain constant on a neighborhood, leaving no dimension for any other weight there. Thus the finite weight list is locally constant, and compactness of $F$ gives a finite cover by such neighborhoods, so only finitely many weights occur globally.

If $r=0$, the action is trivial, $F=M$, and ${\nu }_F=0$, so the weight assertion is vacuous. If $M$ is empty, all asserted submanifolds and bundles are empty. Otherwise the construction applies componentwise and uses no assumption on connectedness of $M$.