research/sections/02_geometry.tex

\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.