Distance hessian in euclidean space

The radial direction. Put $\rho :=\parallel q-p{\parallel }_2$ and $u:=(q-p)\mathrm{/}\rho$. By [F3] $\rho >0$, and $q=p+\rho u$ with $\parallel u{\parallel }_2=1$. Under the canonical identification of [F2], $u$ is a unit tangent vector at $p$, $\mathrm{\mid }u{\mathrm{\mid }}_{g_{\mathrm{E}}}=1$, and $\gamma (t):=p+tu$ is, by [F4], the geodesic with $\gamma (0)=p$ and $\dot{\gamma }(0)=u$; its velocity is the constant vector $u$, so $\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_{g_{\mathrm{E}}}=1$ and $T:=\dot{\gamma }(\rho )=u$ has $g_{\mathrm{E},q}(T,T)=1$.

The Euclidean Riemannian distance. Let $x\in {\mathbb{R}}^n$. If $x=p$ both sides of $d_g(p,x)=\parallel x-p{\parallel }_2$ are $0$, because the constant curve has length $0$ [F7]. Let $x\mathrm{\ne }p$, put $s:=\parallel x-p{\parallel }_2>0$ and $w:=(x-p)\mathrm{/}s$, so $\parallel w{\parallel }_2=1$ and $x=p+sw$. The segment $c(t)=p+tw$, $t\in [0,s]$, is a piecewise $C^1$ curve from $p$ to $x$ of constant speed $\mathrm{\mid }\dot{c}{\mathrm{\mid }}_{g_{\mathrm{E}}}=\parallel w{\parallel }_2=1$ by [F2], so [F7] and [F11] give $L_g(c)={\int }_0^{s}1dt=s$; hence $d_g(p,x)\le s$. Conversely, for an arbitrary piecewise $C^1$ curve $c$ from $p$ to $x$, [F2] and [F7] identify its Riemannian length with ${\sum }_j\int \parallel c^{\mathrm{\prime }}(t){\parallel }_{2}dt$, which is the polygonal arc length $L(c)$ of [F8], and the chord bound of [F8] gives $L_g(c)=L(c)\ge \parallel x-p{\parallel }_2=s$. Thus $s$ is a real lower bound of the nonempty set of curve lengths whose infimum is $d_g(p,x)$ [F7], so [F9] gives $s\le d_g(p,x)$. Therefore $d_g(p,x)=\parallel x-p{\parallel }_2(x\in {\mathbb{R}}^n)\mathrm{.}$

Geodesic completeness and the exponential map. By [F1] and [F10] the manifold $({\mathbb{R}}^n,g_{\mathrm{E}})$ is a boundaryless connected Riemannian manifold. Let ${\gamma }_{x,v}$ be the maximal geodesic with ${\gamma }_{x,v}(0)=x$ and ${\gamma }_{x,v}^{\mathrm{\prime }}(0)=v$ on its open domain $I$ [F5]. On $I$, [F4] writes ${\gamma }_{x,v}(t)=a+tb$ with constant $a,b$, so $a={\gamma }_{x,v}(0)=x$ and $b={\gamma }_{x,v}^{\mathrm{\prime }}(0)=v$, that is ${\gamma }_{x,v}(t)=x+tv$. The curve $\mathrm{\ell }(t)=x+tv$, $t\in \mathbb{R}$, is a geodesic by [F4] with the same initial data; since its domain cannot be enlarged to a longer interval, it is a maximal geodesic with these data, and the uniqueness in [F5] forces ${\gamma }_{x,v}=\mathrm{\ell }$ and $I=\mathbb{R}$. Hence every maximal geodesic has domain $\mathbb{R}$ and $({\mathbb{R}}^n,g_{\mathrm{E}})$ is geodesically complete [F6]; by the equivalence of [F12] it satisfies the completeness hypothesis used by the cut-time and distance-Hessian results, and in particular ${\exp }_x(w)={\gamma }_{x,w}(1)=x+w(w\in T_x{\mathbb{R}}^n\cong {\mathbb{R}}^n)\mathrm{.}$

The cut time in every direction is infinite. Fix the unit direction $u$ of 1.1. For every $t>0$, 1.3 gives ${\exp }_p(tu)=p+tu$, and 1.2 gives $d_g(p,{\exp }_p(tu))=\parallel tu{\parallel }_2=t$ (using $\parallel u{\parallel }_2=1$ from 1.1). Hence the set $\{t>0:d_g(p,{\exp }_p(tu))=t\}$ is all of $(0,\mathrm{\infty })$; since $({\mathbb{R}}^n,g_{\mathrm{E}})$ is complete, connected and boundaryless by 1.3, the cut-time definition [F13] assigns $c_p(u)=+\mathrm{\infty }$, the value allowed there when every positive radial segment minimizes. In particular $0<\rho <c_p(u)$ with $\rho >0$ from 1.1.

The radial Jacobi fields. Let $X\in T_q{\mathbb{R}}^n$ satisfy $g_{\mathrm{E},q}(X,T)=0$, with $T=u$ the unit vector of 1.1. By 1.1 and 2.1 the hypotheses of [F15] hold with $v=u$ and $t=\rho$, so there is exactly one Jacobi field $J_X$ along $\gamma$ with $J_X(0)=0$ and $J_X(\rho )=X$. The field $J(s):=(s\mathrm{/}\rho )X$ — the vector $X$ being extended as a constant field in the Cartesian trivialization — has the affine form $A+sB$ with $A=0$ and $B=X\mathrm{/}\rho$, so it is a Jacobi field by [F14] and satisfies $J(0)=0$, $J(\rho )=X$; by the uniqueness in F15, $J_X=J$ and $D_s{J}_X(\rho )=X\mathrm{/}\rho \mathrm{.}$ Consequently the endomorphism $S=\mathrm{\nabla }\mathrm{grad}{r}_p$ of F15 satisfies $S(T)=0$ and $S(X)=X\mathrm{/}\rho$ for every $X\perp T$.

The gradient of the distance. By [F16], applied with the unit direction $u$ and $0<\rho <c_p(u)$ from 2.1, $\mathrm{grad}{r}_p(q)=\dot{\gamma }(\rho )=T$ and $\mathrm{\mid }\mathrm{grad}{r}_p(q){\mathrm{\mid }}_{g_{\mathrm{E}}}=1$. The gradient characterization [F17] therefore gives $d{r}_p(Z)=g_{\mathrm{E},q}(\mathrm{grad}{r}_p(q),Z)=g_{\mathrm{E},q}(T,Z)(Z\in T_q{\mathbb{R}}^n),$ and in particular $d{r}_p(T)=g_{\mathrm{E},q}(T,T)=1$.

The Hessian identity. Let $Z\in T_q{\mathbb{R}}^n$ and write $Z=Z^{\perp }+g_{\mathrm{E},q}(Z,T)T$ with $Z^{\perp }:=Z-g_{\mathrm{E},q}(Z,T)T$ orthogonal to $T$; this is the orthogonal decomposition because $g_{\mathrm{E},q}(T,T)=1$. By linearity of $S$ and step 3.1, $S(Z)=S(Z^{\perp })=Z^{\perp }\mathrm{/}\rho$. Since $S$ represents the Hessian by F15, for every $W\in T_q{\mathbb{R}}^n$ one has $({\mathrm{\nabla }}^2{r}_p{)}_q(Z,W)=g_{\mathrm{E},q}(S(Z),W)=\frac{1}{\rho }{g}_{\mathrm{E},q}(Z^{\perp },W)=\frac{1}{\rho }(g_{\mathrm{E},q}(Z,W)-g_{\mathrm{E},q}(Z,T)g_{\mathrm{E},q}(T,W))=\frac{1}{\rho }(g_{\mathrm{E},q}(Z,W)-d{r}_p(Z)d{r}_p(W)),$ the third equality by bilinearity and the definition of $Z^{\perp }$, the fourth by step 3.2. Since $r_p(q)=d_g(p,q)=\parallel q-p{\parallel }_2=\rho$ by 1.2, this is the asserted identity at $q$; the point $q\mathrm{\ne }p$ was arbitrary, so the identity holds at every point of ${\mathbb{R}}^n\setminus \{p\}$.

Audit. The hypothesis $n\ge 2$ of the statement is respected; the argument uses only $n\ge 1$ (for the connectedness supplied by [F10]) and no case outside the stated hypothesis is claimed. Since $q\mathrm{\ne }p$, [F3] gives $\rho >0$, so the divisions by $\rho$ and by $r_p(q)=\rho$ are legitimate; the point $p$ itself is excluded and no differentiability or Hessian value at $p$ is asserted. The zero vector is harmless: $Z=0$ gives $S(0)=0$ and both sides of the identity vanish, while the radial vector $T$ is unit. There is no endpoint in the parameter range: $0<\rho <c_p(u)=+\mathrm{\infty }$ by 2.1, so no cut point occurs along this ray and the formula is interior. All constructions are explicit: $u=(q-p)\mathrm{/}\rho$ and $\gamma (t)=p+tu$ involve no selection, and [A1] is inherited exactly through the declared suppliers that carry it, namely the geodesic existence, uniqueness and smooth-dependence theorem [F5], the cut-time definition [F13], Hopf--Rinow [F12], and the Hessian and gradient results [F15, F16]. No biconditional is asserted: the conclusion is a tensor identity off $p$, the distance computation of 1.2 is proved by the two inequalities, and the Jacobi classification [F14] is used in the direction from the affine form to the Jacobi equation.