Hessian of distance in terms of radial jacobi fields

Setting and the distance sphere. By [F1] and [F2], ${\mathcal{E}}_p=T_{p}M$ and ${\exp }_p$ is smooth on all of $T_{p}M$; by [F3], $D_p$ is open, $U$ is an open submanifold of $M$, and ${\exp }_p{\mathrm{\mid }}_{D_p}$ is a diffeomorphism onto $U$. Since $\mathrm{\mid }v{\mathrm{\mid }}_g=1$ and $0<t<c_p(v)$ by [F5], the vector $tv$ lies in $D_p$, hence $q={\exp }_p(tv)\in U$ and $d({\exp }_p{)}_{tv}:T_{p}M\to T_{q}M$ is a linear isomorphism by [F3] and [F4]. By [F6], $r_p$ is smooth on $U$ and $r_p({\exp }_p(w))=\mathrm{\mid }w{\mathrm{\mid }}_g$ on $D_p$; by [F7], $\mathrm{grad}{r}_p(q)=T=\dot{\gamma }(t)$ and $\mathrm{\mid }\mathrm{grad}{r}_p(x){\mathrm{\mid }}_g=1$ for all $x\in U$, so $d{r}_p$ never vanishes on $U$, because $d{r}_p(Y)=g_q(\mathrm{grad}{r}_p,Y)$ [F8] and $g_q$ is positive definite [F35]. Put $S_X:={\mathrm{\nabla }}_X\mathrm{grad}{r}_p$; by [F27] and [F35], $g_q(S_X,Y)=({\mathrm{\nabla }}^2{r}_p{)}_q(X,Y)=X(Y{r}_p)-({\mathrm{\nabla }}_{X}Y)r_p$ for vector fields $X,Y$, and [F28] gives $S_{fX}=f{S}_X$ for smooth functions $f$, so $X\mapsto S_X$ is a smooth endomorphism field on $U$. Since $\mathrm{\mid }v{\mathrm{\mid }}_g=1$ and $\gamma$ is the unit-speed geodesic with $\dot{\gamma }(0)=v$, [F16] gives $\mathrm{\mid }T{\mathrm{\mid }}_g=1$. Finally, $t$ is a regular value of the smooth function $r_p{\mathrm{\mid }}_U:U\to \mathbb{R}$ by [F29], because every $x\in (r_p{\mathrm{\mid }}_U{)}^{-1}(t)$ has $d{r}_p(x)\mathrm{\ne }0$; the fibre is nonempty since it contains $q$, so ${\mathrm{\Sigma }}_t:=\{x\in U:r_p(x)=t\}$ is an embedded hypersurface of $U$ by [F29] and [F30], and $T_q{\mathrm{\Sigma }}_t=\ker d(r_p{)}_q=\{Z\in T_{q}M:g_q(T,Z)=0\}=T^{\perp }$ by [F30] and [F8]. Moreover $\nu :=\mathrm{grad}{r}_p{\mathrm{\mid }}_{{\mathrm{\Sigma }}_t}$ is a unit normal field along ${\mathrm{\Sigma }}_t$: it is unit by [F7], and at every point it is orthogonal to the tangent space of the level set by [F8]. The sign convention of [F31] is the one used below.

The radial pairing identity. We claim that $g_q(T,d({\exp }_p{)}_{tv}(Z))=g_p(v,Z)\text{for every }Z\in T_{p}M\mathrm{.}$ Indeed $tv\in {\mathcal{E}}_p=T_{p}M$, so Gauss's lemma [F9] applied at the base vector $tv$ and the vector $Z$ gives $g_q(d({\exp }_p{)}_{tv}(tv),d({\exp }_p{)}_{tv}(Z))=g_p(tv,Z)$. By [F10], $d({\exp }_p{)}_{tv}(tv)=td({\exp }_p{)}_{tv}(v)$, and $d({\exp }_p{)}_{tv}(v)=\mathrm{grad}{r}_p(q)=T$ by [F7]; bilinearity [F35] gives $g_q(d({\exp }_p{)}_{tv}(tv),d({\exp }_p{)}_{tv}(Z))=t{g}_q(T,d({\exp }_p{)}_{tv}(Z))$ and $g_p(tv,Z)=t{g}_p(v,Z)$. Since $t>0$, division by $t$ proves the claim.

The radial direction. For $0<\sigma \le t$ the point $\gamma (\sigma )={\exp }_p(\sigma v)$ has $\mathrm{grad}{r}_p(\gamma (\sigma ))=\dot{\gamma }(\sigma )$ by [F7], since $0<\sigma <c_p(v)$. Hence the section $(\mathrm{grad}{r}_p)\circ \gamma$ agrees on $(0,t]$ with $\dot{\gamma }$, and [F15] with [F16] gives $S(T)={\mathrm{\nabla }}_T\mathrm{grad}{r}_p=D_{\sigma }((\mathrm{grad}{r}_p)\circ \gamma ){\mid }_{\sigma =t}=D_t\dot{\gamma }(t)=0,$ the first equality by [F28] (function-linearity in the direction slot).

Radial Jacobi fields and the endpoint map. First record the reparametrization rules used below. If $c$ is a smooth curve, $\mathrm{\ell }(\tau )=a\tau +b$ with $a\mathrm{\ne }0$, and $V$ a smooth section along $c$, then the pullback functoriality [F20] applied to the two pullbacks of $\mathrm{\nabla }$ along $c$ and along $c\circ \mathrm{\ell }$, together with the direction-linearity [F28] and the definition of $D$ [F13], gives $D_{\tau }^{c\circ \mathrm{\ell }}(V\circ \mathrm{\ell })(\tau )=a(D_t^{c}V)(\mathrm{\ell }(\tau )),D_{\tau }^{c\circ \mathrm{\ell }}{}^2(V\circ \mathrm{\ell })(\tau )=a^2(D_t^c{}^{2}V)(\mathrm{\ell }(\tau ))\mathrm{.}$ If in addition $V$ is a Jacobi field along a geodesic $c$ and $c\circ \mathrm{\ell }$ is the geodesic of [F21], then [F17] and the $C^{\mathrm{\infty }}$-linearity of the curvature in its second and third slots [F22] give $D_{\tau }^2(V\circ \mathrm{\ell })+R(V\circ \mathrm{\ell },(c\circ \mathrm{\ell }{)}^{\mathrm{\prime }})(c\circ \mathrm{\ell }{)}^{\mathrm{\prime }}=a^2(D_t^{2}V+R(V,\dot{c})\dot{c})\circ \mathrm{\ell }=0,$ so $V\circ \mathrm{\ell }$ is again a Jacobi field. Now fix $W\in T_{p}M$ and consider the geodesic $\theta (\tau )={\exp }_p(\tau tv)=\gamma (t\tau )$ on $[0,1]$, a reparametrization of $\gamma$ by the affine map $\sigma \mapsto t\sigma$ which is a geodesic by [F21] and [F34]; here $tv\in {\mathcal{E}}_p=T_{p}M$ by [F1]. By [F18] and [F19], the Jacobi field $K_W$ along $\theta$ with $K_W(0)=0$ and $D_{\tau }{K}_W(0)=tW$ satisfies $K_W(\tau )=d({\exp }_p{)}_{\tau tv}(\tau tW)(\tau \in [0,1])\mathrm{.}$ Put $J_W(\sigma ):=K_W(\sigma \mathrm{/}t)$ for $\sigma \in [0,t]$. By the reparametrization rule just proved, $J_W$ is a Jacobi field along $\gamma$ with $J_W(0)=0,D_{\sigma }{J}_W(0)=\frac{1}{t}{D}_{\tau }{K}_W(0)=W,$ and evaluating at $\sigma =t$ gives $J_W(t)=K_W(1)=d({\exp }_p{)}_{tv}(tW),$ while for general $\sigma \in [0,t]$, $J_W(\sigma )=d({\exp }_p{)}_{\sigma v}(\sigma W)$. In particular the endpoint map $\mathrm{\Phi }:\{v{\}}^{\perp }⟶T^{\perp },W⟼J_W(t)=d({\exp }_p{)}_{tv}(tW)$ is $\mathbb{R}$-linear by [F10], injective by [F4] and the invertibility of $d({\exp }_p{)}_{tv}$, and its image lies in $T^{\perp }$ by step 1.2, since $g_q(T,d({\exp }_p{)}_{tv}(tW))=t{g}_p(v,W)=0$ for $W\perp v$; conversely, if $X\in T^{\perp }$ and $Z\in T_{p}M$ is the unique vector with $d({\exp }_p{)}_{tv}(Z)=X$, then step 1.2 gives $g_p(v,Z)=g_q(T,X)=0$, so $Z\perp v$ and $X=\mathrm{\Phi }(Z\mathrm{/}t)$ lies in the image. Hence $\mathrm{\Phi }$ is a linear isomorphism onto $T^{\perp }$. Finally let $X\in T^{\perp }$ and put $W:=\frac{1}{t}d({\exp }_p{)}_{tv}^{-1}(X)$, so that $J_W(t)=X$ by [F10]; then $W\perp v$ and $J_W$ is a Jacobi field along $\gamma$ with $J_W(0)=0$ and $J_W(t)=X$, and we write $J_X:=J_W$. It is unique with these properties: if $J$ is any Jacobi field along $\gamma$ with $J(0)=0$ and $J(t)=X$, then $\tilde{J}(\tau ):=J(t\tau )$ is a Jacobi field along $\theta$ with $\tilde{J}(0)=0$ by the reparametrization rule above, so [F19] gives $X=J(t)=\tilde{J}(1)=d({\exp }_p{)}_{tv}(D_{\tau }\tilde{J}(0))=d({\exp }_p{)}_{tv}(t{D}_{\sigma }J(0)),$ and injectivity of $d({\exp }_p{)}_{tv}$ by [F4] gives $D_{\sigma }J(0)=W$, so $J=J_W=J_X$ by the initial-data uniqueness [F18].

Normality of the radial Jacobi fields. Let $X\in T^{\perp }$ and $J=J_X$ as in step 2.1. Put $\phi (\sigma ):=g_{\gamma (\sigma )}(J(\sigma ),\dot{\gamma }(\sigma ))$. By [F12], ${\phi }^{\mathrm{\prime }\mathrm{\prime }}=g(D_{\sigma }^{2}J,\dot{\gamma })+2g(D_{\sigma }J,D_{\sigma }\dot{\gamma })+g(J,D_{\sigma }^2\dot{\gamma })\mathrm{.}$ The second summand vanishes and $D_{\sigma }^2\dot{\gamma }=0$ by [F16] and [F13], while $D_{\sigma }^{2}J=-R(J,\dot{\gamma })\dot{\gamma }$ by [F17]. Hence ${\phi }^{\mathrm{\prime }\mathrm{\prime }}=-g(R(J,\dot{\gamma })\dot{\gamma },\dot{\gamma })=-\mathrm{Rm}(J,\dot{\gamma },\dot{\gamma },\dot{\gamma })=0$ by [F23] and the skew symmetry in the last two slots [F24]. Moreover $\phi (0)=g(J(0),\dot{\gamma }(0))=0$ and, since $D_{\sigma }J(0)=W$ and $D_{\sigma }\dot{\gamma }(0)=0$ by [F16], ${\phi }^{\mathrm{\prime }}(0)=g(W,v)=0,$ because $W\perp v$. By [F26] applied to ${\phi }^{\mathrm{\prime }}$ and then to $\phi$, $\phi$ is affine; with vanishing value and slope at $0$ it vanishes identically. Thus $g(J_X,\dot{\gamma })\equiv 0$, and evaluating the derivative of $\phi \equiv 0$ at $\sigma =t$ gives $g(D_t{J}_X(t),T)=0$.

The variation of the radial family. Fix $X\in T^{\perp }$ and let $W:=\frac{1}{t}d({\exp }_p{)}_{tv}^{-1}(X)\in T_{p}M$, so that $W\perp v$ and $J_X=J_W$ with $J_W(t)=X$ by step 2.1. Define $v(s):=\frac{v+sW}{\mathrm{\mid }v+sW{\mathrm{\mid }}_g}(\mathrm{\mid }s\mathrm{\mid }<\epsilon )\mathrm{.}$ Since $g_p(v,W)=0$, the identity $\mathrm{\mid }v+sW{\mathrm{\mid }}_g^2=1+s^2\mathrm{\mid }W{\mathrm{\mid }}_g^2\ge 1$ holds by [F35], so $v(s)$ is defined, smooth in $s$, lies in $S_{p}M$, and $v(0)=v$; differentiating the quotient at $s=0$ gives $v^{\mathrm{\prime }}(0)=W-({\mathrm{\partial }}_s\mathrm{\mid }v+sW{\mathrm{\mid }}_g)(0)v=W-g_p(v,W)v=W\mathrm{.}$ Define $F(s,\tau ):={\exp }_p(\tau tv(s))$ for $\mathrm{\mid }s\mathrm{\mid }<\epsilon$ and $\tau \in [0,1]$. This is smooth by [F2] and [F1]; for each fixed $s$ the curve $\tau \mapsto F(s,\tau )={\gamma }_{p,v(s)}(\tau t)$ is a geodesic on $[0,t]$ by [F34], and since ${\exp }_p(tv(s))={\gamma }_{p,v(s)}(t)$, the map $s\mapsto F(s,1)$ is the endpoint curve. For fixed $\tau$ the curve $s\mapsto \tau tv(s)$ has velocity $\tau tW$ at $s=0$, so [F33] and [F10] give ${\mathrm{\partial }}_{s}F(0,\tau )=d({\exp }_p{)}_{\tau tv}(\tau tW)=J_W(\tau t),$ the last equality being the formula of step 2.1; at $\tau =1$ this is ${\mathrm{\partial }}_{s}F(0,1)=J_W(t)=X$. Since $D_p$ is open by [F3] and $tv\in D_p$, there is $\delta >0$ such that $tv(s)\in D_p$ for $\mathrm{\mid }s\mathrm{\mid }<\delta$; by the definition of $D_p$ this means $t<c_p(v(s))$, so [F7] gives $\mathrm{grad}{r}_p(F(s,1))=d({\exp }_p{)}_{tv(s)}(v(s))=\frac{1}{t}{\mathrm{\partial }}_{\tau }F(s,1),$ where the second equality uses [F10] and ${\mathrm{\partial }}_{\tau }F(s,1)=d({\exp }_p{)}_{tv(s)}(tv(s))$. Let $\alpha (s):=F(s,1)$, so that ${\alpha }^{\mathrm{\prime }}(0)=X$. By [F15], ${\mathrm{\nabla }}_X\mathrm{grad}{r}_p=D_s((\mathrm{grad}{r}_p)\circ \alpha ){\mid }_{s=0}=\frac{1}{t}{D}_s{\mathrm{\partial }}_{\tau }F(0,1),$ the scalar $\frac{1}{t}$ being constant in $s$ and [F13] justifying the differentiation of the product with the constant function. It remains to commute the two covariant derivatives. In a chart on the target about $q$ write ${\mathrm{\partial }}_{\tau }F={\sum }_k({\mathrm{\partial }}_{\tau }{F}^k)({\mathrm{\partial }}_k\circ F)$ and ${\mathrm{\partial }}_{s}F={\sum }_j({\mathrm{\partial }}_s{F}^j)({\mathrm{\partial }}_j\circ F)$, with $F^i:=x^i\circ F$. By the real-linearity and coefficient rule of $D$ [F13], the pullback evaluation [F20] on the local sections ${\mathrm{\partial }}_k$ with the direction-linearity [F28], and the Christoffel-symbol identity ${\mathrm{\nabla }}_{{\mathrm{\partial }}_j}{\mathrm{\partial }}_k={\sum }_i{\mathrm{\Gamma }}^i{}_{jk}{\mathrm{\partial }}_i$ [F36], one has $(D_s{\mathrm{\partial }}_{\tau }F{)}^i={\mathrm{\partial }}_s{\mathrm{\partial }}_{\tau }{F}^i+{\mathrm{\Gamma }}^i{}_{jk}(F){\mathrm{\partial }}_s{F}^j{\mathrm{\partial }}_{\tau }{F}^k,$ and the same argument with $s,\tau$ exchanged gives $(D_{\tau }{\mathrm{\partial }}_{s}F{)}^i={\mathrm{\partial }}_{\tau }{\mathrm{\partial }}_s{F}^i+{\mathrm{\Gamma }}^i{}_{jk}(F){\mathrm{\partial }}_{\tau }{F}^j{\mathrm{\partial }}_s{F}^k$. The mixed partials of $F$ commute and the Levi-Civita connection is torsion free [F11], so its Christoffel symbols are symmetric, ${\mathrm{\Gamma }}^i{}_{jk}={\mathrm{\Gamma }}^i{}_{kj}$, by [F14]; hence both component formulas agree and $D_s{\mathrm{\partial }}_{\tau }F=D_{\tau }{\mathrm{\partial }}_{s}F$ on the parameter rectangle, the displayed identity being intrinsic and the local computation independent of the chart. Therefore ${\mathrm{\nabla }}_X\mathrm{grad}{r}_p=\frac{1}{t}{D}_{\tau }{\mathrm{\partial }}_{s}F(0,1)=\frac{1}{t}{D}_{\tau }[J_W(\tau t)]{\mid }_{\tau =1}=\frac{1}{t}\cdot t\cdot D_{\sigma }{J}_W(t)=D_t{J}_X(t),$ where the third equality is the reparametrization rule of step 2.1 applied to the section $J_W$ and the affine map $\tau \mapsto t\tau$, and where $J_W(\tau t)={\mathrm{\partial }}_{s}F(0,\tau )$ for every $\tau \in [0,1]$.

The Hessian formula. For $X\in T^{\perp }$ and arbitrary $Y\in T_{q}M$, [F27] with the endomorphism $S_X={\mathrm{\nabla }}_X\mathrm{grad}{r}_p$ of step 1.1 gives $({\mathrm{\nabla }}^2{r}_p{)}_q(X,Y)=g_q(S_X,Y)$, and step 3.2 identifies $S_X=D_t{J}_X(t)$; hence $({\mathrm{\nabla }}^2{r}_p{)}_q(X,Y)=g_q(D_t{J}_X(t),Y),$ which is assertion (b). Taking $Y=T$ gives zero by step 3.1, and replacing $X$ by $T$ gives $({\mathrm{\nabla }}^2{r}_p{)}_q(T,Y)=g_q(S_T,Y)=0$ for every $Y$ by step 1.3. In particular ${\mathrm{\nabla }}^2{r}_p$ is determined by the normal radial Jacobi fields $J_X$, and its radial direction is annihilated.

The normal shape operator and the second fundamental form. By step 3.2, $S(X)=D_t{J}_X(t)$ for every $X\in T^{\perp }$, and by step 1.3, $S(T)=0$; these two prescriptions determine $S$ on all of $T_{q}M$. Given $X\in T^{\perp }$ write $X=J(t)W$ with $W\perp v$ as in step 2.1; then $J_X=J_W$ and $S(X)=D_t{J}_W(t)=(D_{t}J(t))(W)=(D_{t}J(t)\circ J(t{)}^{-1})(X),$ so on $T^{\perp }$ the endomorphism $S$ is $D_{t}J(t)\circ J(t{)}^{-1}$. It is $g$-self-adjoint on $T^{\perp }$: for $X=J(t)W$ and $X^{\mathrm{\prime }}=J(t)W^{\mathrm{\prime }}$ with $W,W^{\mathrm{\prime }}\perp v$, the Wronskian [F25] of the Jacobi fields $J_W,J_{W^{\mathrm{\prime }}}$ is constant, and at $\sigma =0$ it equals $g(D_{\sigma }{J}_W(0),J_{W^{\mathrm{\prime }}}(0))-g(J_W(0),D_{\sigma }{J}_{W^{\mathrm{\prime }}}(0))=g(W,0)-g(0,W^{\mathrm{\prime }})=0$; hence $g_q(SX,X^{\mathrm{\prime }})=g_q(D_t{J}_W(t),J_{W^{\mathrm{\prime }}}(t))=g_q(J_W(t),D_t{J}_{W^{\mathrm{\prime }}}(t))=g_q(X,S{X}^{\mathrm{\prime }}),$ since $S{X}^{\mathrm{\prime }}=D_t{J}_{W^{\mathrm{\prime }}}(t)$, the symmetry of $g$ being [F35]. Finally, ${\mathrm{\Sigma }}_t$ is an embedded hypersurface of the open Riemannian manifold $U$ with unit normal field $\nu =\mathrm{grad}{r}_p{\mathrm{\mid }}_{{\mathrm{\Sigma }}_t}$ by step 1.1; this field is orthogonal to ${\mathrm{\Sigma }}_t$ by [F8] and unit by [F7], and it is the outward normal because $d{r}_p(\nu )=g_q(\nu ,\nu )=1>0$ along the increasing direction of $r_p$. For $X\in T_q{\mathrm{\Sigma }}_t=T^{\perp }$ and $Y\in T_q{\mathrm{\Sigma }}_t$, the shape operator of [F31] is $S_{\nu }X=-({\mathrm{\nabla }}_X\nu {)}^{\mathrm{\top }}=-({\mathrm{\nabla }}_X\mathrm{grad}{r}_p{)}^{\mathrm{\top }}=-S(X)=-D_t{J}_X(t),$ because ${\mathrm{\nabla }}_X\mathrm{grad}{r}_p=S(X)$ is orthogonal to $T$ by step 3.2 and step 3.1 and $T_q{\mathrm{\Sigma }}_t=T^{\perp }$; and the value at $q$ depends only on $X_q$ and ${\nu }_q$ by [F31]. By [F32] and $\mathrm{\mid }\nu {\mathrm{\mid }}_g=1$, the second fundamental form is the normal vector $\mathrm{I}\mathrm{I}(X,Y)=g_q(S_{\nu }X,Y)\nu =-g_q(D_t{J}_X(t),Y)\nu ,$ which is assertion (c). Here $T_q{\mathrm{\Sigma }}_t=T^{\perp }$ by step 1.1 and $\mathrm{\mid }\nu {\mathrm{\mid }}_g=1$ by [F7].

Boundary, endpoint and choice audit. In dimension zero $T_{p}M=\{0_p\}$, so $S_{p}M=\mathrm{\varnothing }$ by [F5], there is no unit direction $v$, and the assertion is vacuous; the empty manifold carries no base point $p$, and the statement also covers every $n\ge 1$. The parameter range $0<t<c_p(v)$ is open and both endpoints are genuinely excluded: at $t=0$ one has $q=p$, while the regularity theorem [F6] gives the smoothness of $r_p$ only on the open set $U=M\setminus (\{p\}\cup \mathrm{Cut}(p))$, which does not contain $p$, so no Hessian at $p$ is asserted; for finite $t=c_p(v)$ the point $q$ lies in $\mathrm{Cut}(p)$ by [F5], hence outside $U$, and again no Hessian is asserted there; the case $c_p(v)=+\mathrm{\infty }$ is allowed and then every $t>0$ is covered. The zero vector is never used: $v$ is unit, so $tv\mathrm{\ne }0$ and $J_X$ is defined for every $X$, including the degenerate direction $X=0$, for which $W=0$, $J_X=0$ and both sides of (b) vanish, while (a) asserts existence and uniqueness of the zero field. The construction selects nothing arbitrary: $W$ is the value of a linear inverse at $X$ by [F4], $v(s)$ is an explicit normalized curve, and the inverse $J(t{)}^{-1}$ is the inverse of the linear isomorphism of step 2.1; the choice assumption [A1] is inherited exactly through the completeness, exponential, cut-time, Gauss and level-set suppliers [F3, F4, F9, F29, F34], and no further countable selection is made. The statement contains no biconditional: (a) is an existence and uniqueness assertion with the normal and endpoint properties, (b) and (c) are identities, and the equivalence used in step 2.1, namely $g_q(T,X)=t{g}_p(v,W)$ for the single pair at hand, comes from the cited Gauss lemma [F9]. Assertion (b) uses only unit directions $v$; the metric and its positive definiteness [F35] are used at the displayed pairings, and no compactness or positive injectivity radius is assumed.