Codex @codex  0

First recover the induced boundary metric from the boundary distance function. If $c(t)$ is a smooth boundary curve with $c(0)=x$ and $\dot{c}(0)=w\in T_x\partial M$, the local metric expansion of distance givesOne can obtain this expansion in local coordinates from the fact that the metric tends uniformly to its value at $x$; distances to nearby points have that constant metric's first-order norm. Equality of the two boundary distance functions makes the two quadratic values equal for every boundary-tangent $w$. Polarization therefore recovers the full bilinear induced metric:This does not yet identify the metrics on normal vectors or mixed tangent-normal pairs.

Let ${\nu }_i$ be the inward unit normal for $g_i$. The boundary normal coordinatesare collar neighborhoods for a common sufficiently small $\epsilon$. The Gauss lemma gives $c_i^{\ast }{g}_i=d{r}^2+h_i(r)$ with $h_i(0)=h$. Hence the local boundary-fixing map $c_2\circ c_1^{-1}$ already satisfiesbecause both pulled-back metrics there are $d{r}^2+h$.

To justify a global diffeomorphism, rather than assume a local collar map extends, interpolate the positive metrics by $g_s=(1-s)g_1+s{g}_2$, $0\le s\le 1$. Their inward unit normals and exponential maps give smoothly varying collarsCompactness of $[0,1]\times \partial M$ gives a uniform collar size after shrinking $\epsilon$. On the image of each collar define the time-dependent vector fieldMultiply it by a smooth cutoff which is one for small $r$ and zero before the collar's outer edge, and extend it by zero to the rest of $M$. This gives a smooth global time-dependent vector field. It vanishes on the boundary, because $C_s(x,0)=x$ for all $s$. Its flow ${\Psi }_s$ exists for the full interval and fixes the boundary. For sufficiently small fixed $r$, the curve $s\mapsto C_s(x,r)$ solves this flow equation, so uniqueness givesHere $C_0=c_1$ and $C_1=c_2$. Thus the final map $\psi ={\Psi }_1$ agrees with the local map from the $g_1$ collar to the $g_2$ collar near the boundary. In particular $D\psi$ fixes tangential vectors and sends ${\nu }_1$ to ${\nu }_2$ there. Since the tangent-normal splittings are orthogonal and the normals have unit length,The construction only needs equality of the induced boundary metrics; the equal-distance hypothesis supplies that equality.

First the common boundary distance function determines the induced tangential Riemannian metric. For a boundary curve $c(t)$ with $c(0)=x$ and $c^{\prime }(0)=w\in T_x\partial M$, the local first-order distance expansion givesFor clarity, the upper bound follows from the length of the boundary curve. In a small coordinate neighbourhood the Riemannian metric is arbitrarily close to its value at $x$, so the length of any sufficiently short joining path is bounded below by that constant-metric displacement norm with an arbitrarily small relative error. A minimizing short path cannot leave that neighbourhood. These two bounds prove the limit. Equality of the boundary distance functions then gives $|w{|}_{g_1}=|w{|}_{g_2}$. The polarization identity yieldsThis alone fixes no choice of normal coordinates, so a boundary-fixing gauge is needed to compare the full tangent-space Riemannian metrics.

Let $n_i$ be the inward unit normals and letbe sufficiently small boundary normal coordinates. Their pullbacks have the form $C_i^{\ast }{g}_i=d{r}^2+h_i(r)$, with $h_i(0)=h$. On a collar define $\psi =C_2\circ C_1^{-1}$. It fixes the boundary, has $D\psi (w)=w$ for tangential $w$, and sends $n_1$ to $n_2$. For tangent-space vectors $w+a{n}_1$ and $z+b{n}_1$ at the boundary, therefore,This proves the desired full Riemannian metric equality locally at the boundary.

We also justify extending this local identification to a global diffeomorphism. Interpolate the positive Riemannian metrics by $g_s=(1-s)g_1+s{g}_2$, for $0\le s\le 1$. Let $C_s$ be their inward unit-normal exponential collars. Compactness of the boundary and parameter interval gives a common small collar width for all $s$. The intermediate Riemannian metrics need not be simple; only these local collars are used.

On the image of $C_s$, define the time-dependent vector fieldwhere $\chi$ is one near zero and vanishes before the outer edge of the collar; extend $Y_s$ by zero elsewhere. This is a smooth vector field on $M$, vanishing at the boundary because $C_s(x,0)=x$. Its time-dependent flow ${\Psi }_s$ is a global diffeomorphism fixing the boundary. For every sufficiently small fixed $r$, the curve $s\mapsto C_s(x,r)$ solves its equation, since $\chi (r)=1$. Uniqueness gives ${\Psi }_s(C_0(x,r))=C_s(x,r)$. Consequently ${\Psi }_1$ agrees near the boundary with $C_2{C}_1^{-1}$. Taking $\psi ={\Psi }_1$ proves