| \chapter{Finite-radius distances without an artificial noise floor} |
| \section{From a normal vector to an actual point} |
| A scalar normal echo is a tangent-space quantity. To connect it with a |
| Riemannian distance, fix $x$ in the soul and use the normal exponential map. |
| In the global coordinates of Chapter~2, |
| \[ |
| \exp_x(tu)=(x,tu),\qquad |
| \exp_x(tHu)=(x,tHu). |
| \] |
| Define the normalized squared-distance signal |
| \begin{equation}\label{eq:radiusdef} |
| \mathcal D_t(u,H)= |
| \frac{d_{M_\lambda}((x,tu),(x,tHu))^2}{2t^2},\qquad t>0. |
| \end{equation} |
| For a general smooth metric one expects its limit to be $1-u^{\mathsf T}Hu$. |
| The explicit quotient family permits a multiplicative bound, uniform even |
| when this echo deficit is very small. |
|
|
| \section{The normal fibers are totally geodesic} |
| Recall the metric |
| \[ |
| g_\lambda=Q+ h_v(\mathrm dv+\lambda\theta v, |
| \mathrm dv+\lambda\theta v), |
| \qquad |
| h_v(W,W)=\ip W{B_v^{-1}W}. |
| \] |
| The fiber metric $h$ is invariant under the standard $\SO(d)$ action. The |
| horizontal flow of the connection acts on the fiber by such rotations and |
| therefore preserves its metric. |
|
|
| \begin{lemma}[Totally geodesic fibers]\label{lem:geodesicfibers} |
| Every fiber $\{x\}\times\R^d$ is totally geodesic in $M_\lambda$. |
| \end{lemma} |
| \begin{proof} |
| Take vertical vector fields $V,W$ and a horizontal field $X$. The Koszul |
| formula expresses $2g(\nabla_VW,X)$ as the negative horizontal derivative |
| of $g(V,W)$ plus the two correction terms involving the vertical parts of |
| $[X,V]$ and $[X,W]$. Their sum is the negative Lie derivative of the fiber |
| metric along the horizontal flow. That derivative is zero because the flow |
| is a fiber isometry. The other Koszul terms vanish by horizontal--vertical |
| orthogonality and because $[V,W]$ is vertical. Hence the horizontal part of |
| $\nabla_VW$ is zero. This is exactly the vanishing of the second fundamental |
| form of the fiber. |
| \end{proof} |
|
|
| Total geodesy alone does not say that arbitrary distant fiber points are |
| joined by ambient minimizing geodesics staying in the fiber. We use it only |
| inside a sufficiently small common convex neighborhood of the zero section. |
| For such points the fiber geodesic is an ambient geodesic and, by local |
| uniqueness, is the ambient minimizer. |
|
|
| \section{Uniform local distance comparison} |
| In polar fiber coordinates, |
| \[ |
| h=\mathrm dr^2+\frac{r^2}{1+\coup r^2}\,g_{\mathbb S^{d-1}}. |
| \] |
| Consequently, on the Euclidean fiber ball $|v|\le t$, |
| \begin{equation}\label{eq:fibercomparison} |
| \frac1{1+\coup t^2}g_{\rm Eucl}\le h\le g_{\rm Eucl}. |
| \end{equation} |
| The function $r^2$ is geodesically convex for this fiber metric. For example, |
| the radial Hessian is positive and the angular Hessian has the sign of |
| $f'(r)$ for $f(r)=r/\sqrt{1+\coup r^2}$; here $f'(r)>0$. Thus a |
| minimizing fiber segment with endpoints of radius $t$ remains in that ball. |
|
|
| \begin{theorem}[Multiplicative finite-radius calibration]\label{thm:radius} |
| There is a radius $t_0(d)>0$ such that, for $0<t<t_0(d)$, every unit $u$, |
| every $H\in\SO(d)$, and every $0\le\lambda\le1$ satisfy |
| \begin{equation}\label{eq:radiusbound} |
| \frac{1-u^{\mathsf T}Hu}{1+\coup t^2} |
| \le \mathcal D_t(u,H)\le 1-u^{\mathsf T}Hu. |
| \end{equation} |
| At the pure-gauge metric endpoints, interpret the metric by its smooth |
| extension. In particular, at $\lambda=0$ it is the product metric and both |
| inequalities are equalities. |
| \end{theorem} |
| \begin{proof} |
| On a compact tube the metric coefficients depend smoothly on |
| $\lambda\in[0,1]$, and remain uniformly positive definite. The zero section |
| is compact and homogeneous. Standard local existence and uniqueness for |
| the geodesic equation, together with a finite cover of the zero section and |
| the compact parameter interval, therefore provide a common sufficiently |
| small convex neighborhood radius. Choose $t_0$ inside it. By |
| Lemma~\ref{lem:geodesicfibers}, the short fiber geodesic between the two |
| points is also the ambient minimizing geodesic. |
|
|
| For a lower bound, this segment stays within radius $t$ by convexity. Its |
| length is at least the Euclidean chord length divided by |
| $\sqrt{1+\coup t^2}$, by \eqref{eq:fibercomparison}. For an upper bound, |
| the Euclidean straight chord remains in the same fiber ball and its |
| $h$-length is at most its Euclidean length. Squaring and using |
| $|tu-tHu|^2=2t^2(1-u^{\mathsf T}Hu)$ proves the claim. |
| \end{proof} |
|
|
| The bound is relative to the true deficit. It does not introduce an absolute |
| bias of order $t^2$ when the signal itself is much smaller. In this family, |
| a very weak echo remains proportionally weak in the finite-radius geometry: |
| \[ |
| 0\le(1-\phi)-\mathcal D_t |
| \le\frac{\coup t^2}{1+\coup t^2}(1-\phi). |
| \] |
| This matters in the joint weak-connection and short-loop limit. |
|
|
| \section{A direct curvature check inside the fibers} |
| For completeness, the radial and tangential sectional curvatures of the |
| fiber warped metric can be read from $f$: |
| \begin{align} |
| K_{\rm radial}(r)&=-\frac{f''(r)}{f(r)} |
| =\frac{3\coup}{(1+\coup r^2)^2},\label{eq:radialcurv}\\ |
| K_{\rm tangential}(r)&=\frac{1-f'(r)^2}{f(r)^2} |
| =\frac{\coup(3+3\coup r^2+\coup^2r^4)}{(1+\coup r^2)^2}. |
| \label{eq:tangentcurv} |
| \end{align} |
| The limits at $r=0$ are both $3\coup$. These equations verify the |
| nonnegative fiber curvature directly. They are not a computation of every |
| ambient mixed sectional curvature; the ambient nonnegative-curvature proof |
| is the Riemannian quotient argument. |
|
|
| \section{Detector noise still has to be charged} |
| Suppose an actual instrument measures the raw squared distance with additive |
| standard deviation $\sigma_D$. Dividing by $2t^2$ gives normalized signal |
| noise $\sigma_D/(2t^2)$. Shrinking $t$ improves the multiplicative geometric |
| bias but amplifies that raw detector noise. The Gaussian theorem of |
| Chapter~10 assumes a stated normalized noise level $\sigma$; it does not |
| claim that $\sigma$ is independent of the physical radius in every detector. |
|
|
| A practical radius choice therefore needs both a detector model and a bias |
| budget. The present result supplies the exact geometric side of that |
| tradeoff. No universally optimal radius is asserted without the missing |
| measurement law. In particular, a mathematical tangent limit is not a method |
| for obtaining infinitely precise curvature from a finite apparatus. |
|
|
