| \chapter{An explicit family with one fixed soul} |
| \section{The quotient construction} |
| Let $G=\SO(d)$ act on $G\times G\times\R^d$ on the right by |
| \begin{equation}\label{eq:action} |
| (g,h,v)\cdot k=(gk,hk,k^{-1}v). |
| \end{equation} |
| The action is free because its first component is free, and it is proper because |
| $G$ is compact. Equip the product with the metric |
| \begin{equation}\label{eq:productmetric} |
| aQ\oplus bQ\oplus\ip{\cdot}{\cdot},\qquad |
| a=\frac1{1-\lambda},\quad b=\frac1\lambda. |
| \end{equation} |
| The action is isometric. Let $M_\lambda$ be the resulting smooth Riemannian |
| quotient. Its dimension is $d(d-1)/2+d$. |
|
|
| There is a global diffeomorphism |
| \begin{equation}\label{eq:trivialization} |
| [g,h,v]\longmapsto(gh^{-1},hv)\in G\times\R^d. |
| \end{equation} |
| Indeed, the displayed pair is unchanged by \eqref{eq:action}, and the inverse |
| maps $(x,w)$ to $[x,\Id,w]$. Thus no bundle-topology change is hidden in the |
| parameter. The geometry, not the differentiable bundle, is changing. |
|
|
| \begin{proposition}[Completeness and nonnegative curvature]\label{prop:quotient} |
| The quotient $M_\lambda$ is complete and noncompact, with nonnegative sectional |
| curvature. |
| \end{proposition} |
| \begin{proof} |
| Both group factors with their bi-invariant metrics are complete. Their |
| sectional curvatures are nonnegative, since their curvature numerators are |
| positive multiples of $\norm{[X,Y]}_Q^2$. The Euclidean factor is complete and |
| flat, and the product has nonnegative sectional curvature. |
|
|
| For a Riemannian submersion, the sectional curvature of a horizontal plane in |
| the quotient equals the sectional curvature of its horizontal lift plus a |
| nonnegative term involving the vertical part of the bracket. This is |
| O'Neill's formula \cite{oneill}. It proves the curvature assertion. A quotient |
| of a complete Riemannian manifold by a proper isometric action is complete: |
| a finite-length curve in the quotient can be horizontally lifted from a chosen |
| starting point, and the lift has the same length; the completeness of the |
| upstairs metric gives the required extension. Finally $|v|$ is an unbounded |
| continuous invariant of the action, so the quotient is noncompact. |
| \end{proof} |
|
|
| \section{The exact metric in global coordinates} |
| Use coordinates $(x,v)$ from \eqref{eq:trivialization}. Write a tangent vector |
| as $(X,W)$, where $X=x^{-1}\dot x\in\so(d)$ and $W=\dot v\in\R^d$. |
| Set |
| \begin{equation}\label{eq:Hv} |
| B_v=\Id+\coup\left(|v|^2\Id-vv^{\mathsf T}\right), |
| \qquad \coup=\lambda(1-\lambda). |
| \end{equation} |
| This is positive definite. Its radial eigenvalue is one and each of its |
| $d-1$ tangential eigenvalues is $1+\coup|v|^2$. |
|
|
| \begin{theorem}[Global metric formula]\label{thm:metric} |
| In the coordinates above, |
| \begin{equation}\label{eq:metric} |
| g_\lambda\bigl((X,W),(X,W)\bigr) |
| =\norm X_Q^2+ |
| \ip{W+\lambda Xv}{B_v^{-1}(W+\lambda Xv)}. |
| \end{equation} |
| \end{theorem} |
| \begin{proof} |
| At the representative $(x,\Id,v)$, adding a vertical tangent vector indexed by |
| $Z\in\so(d)$ replaces $(X,0,W)$ by $(X+Z,Z,W-Zv)$. The squared norm of the |
| quotient tangent is the minimum |
| \begin{equation}\label{eq:minmetric} |
| \inf_Z\left\{a\norm{X+Z}_Q^2+b\norm Z_Q^2+|W-Zv|^2\right\}. |
| \end{equation} |
| Since $a+b=1/\coup$ and $a/(a+b)=\lambda$, put $Z=-\lambda X+Y$. |
| The first two terms become |
| \[ |
| \norm X_Q^2+\coup^{-1}\norm Y_Q^2. |
| \] |
| Define $L_v:\so(d)\to\R^d$ by $L_v(Y)=Yv$, and put |
| $\zeta=W+\lambda Xv$. The remaining minimization is |
| \[ |
| \inf_Y\{\coup^{-1}\norm Y_Q^2+|\zeta-L_vY|^2\} |
| =\ip{\zeta}{(\Id+\coup L_vL_v^*)^{-1}\zeta}. |
| \] |
| The identity follows either from the normal equation or by completing the |
| square. For the normalization \eqref{eq:Q}, |
| \[ |
| L_v^*w=wv^{\mathsf T}-vw^{\mathsf T},\qquad |
| L_vL_v^*w=|v|^2w-v\ip vw. |
| \] |
| Substitution gives \eqref{eq:metric}. |
| \end{proof} |
|
|
| The metric formula is more informative than a bare quotient existence proof. |
| It fixes every normalization in the subsequent curvature and loop-length |
| calculations. In particular, the horizontal lift of $X$ has fiber velocity |
| $W=-\lambda Xv$, while the restriction to a fiber is the rotationally |
| symmetric metric |
| \begin{equation}\label{eq:fibermetric} |
| \mathrm dr^2+f_\lambda(r)^2g_{\mathbb S^{d-1}},\qquad |
| f_\lambda(r)=\frac r{\sqrt{1+\coup r^2}}. |
| \end{equation} |
| The Euclidean metric is recovered at the fiber origin. Radial lines have unit |
| speed, and the metric is smooth there because $f_\lambda(r)$ has an odd smooth |
| extension with derivative one at zero. |
|
|
| \section{The zero section is a soul} |
| At $v=0$, formula \eqref{eq:metric} restricts on the zero section to |
| $Q(X,X)$. This restriction is independent of $\lambda$. The zero section is |
| compact and diffeomorphic to $G$. |
|
|
| \begin{proposition}[Soul and normal coordinates]\label{prop:soul} |
| The zero section $S=G\times\{0\}$ is totally geodesic and totally convex. It is |
| a soul, and its normal exponential map is $(x,v)\mapsto(x,v)$ in the global |
| coordinates of \eqref{eq:trivialization}. |
| \end{proposition} |
| \begin{proof} |
| The map $[g,h,v]\mapsto[g,h,-v]$ is an isometry. Its fixed-point set is exactly |
| the zero section, because a group element fixing $(g,h)$ is the identity. |
| A fixed-point set of an isometry is totally geodesic. |
|
|
| For total convexity, the invariant upstairs function $F(g,h,v)=|v|^2$ has |
| nonnegative Hessian. It descends to $M_\lambda$. The Hessian of a basic |
| function along a horizontal vector equals its upstairs Hessian along the |
| horizontal lift. Hence the descended function has nonnegative Hessian. Along |
| any geodesic segment whose endpoints are in its zero set, the function is |
| convex, nonnegative, and zero at both ends. It is therefore zero throughout. |
| This proves total convexity in the geodesic sense, not merely convexity for |
| some chosen minimizing segments. |
|
|
| The curve $(x,tv)$ is the projection of the horizontal product geodesic |
| $(x,\Id,tv)$. Horizontality follows because |
| $\ip{v}{Z(tv)}=0$ for every skew $Z$. Thus it is a geodesic with the claimed |
| initial normal velocity. It is defined for all real $t$, and the global |
| trivialization makes the normal exponential a diffeomorphism. The compact, |
| boundaryless, totally convex zero section is therefore a soul. |
| \end{proof} |
|
|
| \section{The normal connection and its curvature} |
| The principal bundle $G\times G\to G$, $(g,h)\mapsto gh^{-1}$, has the |
| mechanical connection determined by $aQ\oplus bQ$. At the section |
| $x\mapsto(x,\Id)$ its connection form is |
| \begin{equation}\label{eq:connection} |
| A_\lambda=\frac a{a+b}\theta=\lambda\theta, |
| \qquad \theta=x^{-1}\mathrm dx. |
| \end{equation} |
| This is also the connection of the horizontal distribution read from |
| \eqref{eq:metric}. The fibers have an $\SO(d)$-invariant metric independent |
| of the base, so the horizontal parallel maps are fiber isometries. The |
| linearization at the zero section is the normal metric connection. |
|
|
| The Maurer--Cartan identity is |
| $\mathrm d\theta+\theta\wedge\theta=0$, where |
| $(\theta\wedge\theta)(X,Y)=[X,Y]$. Consequently |
| \begin{equation}\label{eq:curvature} |
| F_\lambda(X,Y)=(\lambda^2-\lambda)[X,Y] |
| =-\coup[X,Y]. |
| \end{equation} |
| All tangent variables in this formula are measured with the fixed metric $Q$. |
| No unmentioned rescaling of the soul occurs. |
|
|
| \begin{proposition}[Full normal holonomy]\label{prop:fullhol} |
| For $d\ge3$ and $0<\lambda<1$, the normal holonomy group is $\SO(d)$. |
| \end{proposition} |
| \begin{proof} |
| The curvature values in \eqref{eq:curvature} span $\so(d)$. For example, for |
| distinct indices $i,j,k$, the commutator of the elementary rotations in the |
| $i,j$ and $j,k$ planes is, up to sign, the elementary rotation in the $i,k$ |
| plane. Thus all elementary skew matrices are spanned. The Ambrose--Singer |
| theorem places these curvature values in the restricted holonomy algebra |
| \cite{ambrose-singer}; hence that algebra is all of $\so(d)$. The restricted |
| holonomy is then the connected group $\SO(d)$. All transport matrices already |
| lie in $\SO(d)$, so the full group cannot be larger. |
| \end{proof} |
|
|
| The general use of associated bundles to obtain nonnegative curvature and |
| large normal holonomy has established precedents \cite{tapp}. The special |
| feature needed here is the simultaneous explicit connection parameter and |
| fixed base normalization, which permit quantitative observation comparisons. |
|
|
| \Needspace{17\baselineskip} |
| \section{A noncollapsing approach to trivial transport} |
| Formula \eqref{eq:metric} extends smoothly at $\lambda=0$ by |
| \[ |
| g_0((X,W),(X,W))=\norm X_Q^2+|W|^2. |
| \] |
| On every compact subset of $G\times\R^d$, all derivatives of the coefficients |
| of $g_\lambda$ converge to those of $g_0$. Indeed, $B_v^{-1}$ is a rational |
| smooth function of $\coup$ and $v$, with denominator bounded away from zero. |
| The coefficients $\lambda Xv$ also converge smoothly to zero. |
|
|
| This proves more than a formal vanishing of curvature components. The entire |
| family converges in $C^\infty$ on each fixed compact tube. Metric ellipticity, |
| local volume bounds, and curvature bounds on that tube can be chosen uniformly |
| for sufficiently small $\lambda$. Nevertheless the normal holonomy group is |
| full for every positive parameter and trivial at the limiting product metric. |
| The discontinuity is qualitative holonomy versus quantitative accessibility, |
| not a collapse of the fixed soul. |
|
|
| Swapping the two group factors is an isometry between $M_\lambda$ and |
| $M_{1-\lambda}$ covering inversion on the soul. We retain both parameters |
| because the formulas are symmetric in $\coup$, but one may restrict to |
| $0<\lambda\le1/2$ when avoiding this duplication. |
|
|
