= Solution
First the common <boundary distance function> determines the induced tangential <Riemannian metric>. For a boundary curve $c(t)$ with $c(0)=x$ and $c'(0)=w\in T_x\partial M$, the local first-order distance expansion gives
$$
\lim_{t\to0}\frac{d_{g_i}(x,c(t))}{|t|}=|w|_{g_i}.
$$
For 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> yields
$$
g_1|_{T\partial M}=g_2|_{T\partial M}=:h.
$$
This 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 let
$$
C_i(x,r)=\exp_x^{g_i}(r n_i(x))
$$
be sufficiently small <boundary normal coordinates>. Their pullbacks have the form $C_i^*g_i=dr^2+h_i(r)$, 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,
$$
(\psi^*g_2)(w+a n_1,z+b n_1)=h(w,z)+ab
=g_1(w+a n_1,z+b n_1).
$$
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 $g_s=(1-s)g_1+sg_2$, for $0\le s\le1$. 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 field>
$$
Y_s(C_s(x,r))=\chi(r)\,\partial_s C_s(x,r),
$$
where $\chi$ 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 $C_2C_1^{-1}$. Taking $\psi=\Psi_1$ proves
$$
\boxed{\psi|_{\partial M}=\mathrm{id},\qquad
\psi^*g_2=g_1\text{ on every }T_xM\ (x\in\partial M)}.
$$
Back to article page
