[proofplan] The Bishop-Gromov argument compares the polar density $J_\theta(\rho)$ with the model density $\operatorname{sn}_k(\rho)^{n-1}$ and proves that their quotient is nonincreasing along every radial geodesic before the cut time. Equality of the ball-volume ratios at $r$ and $R$ forces the weighted averages of this nonincreasing quotient to be constant throughout the annulus. Since the quotient is monotone in each radial variable, constancy of these averages forces the radial derivative defect to vanish almost everywhere. Under the additional smooth equality assumptions and absence of cut points, the shape-operator equation integrates directly to the warped product polar metric. [/proofplan]

[step:Compare the polar density with the model density along each radial geodesic] The theorem assumes $n\geq2$, so the normal space $\dot\gamma_\theta(\rho)^\perp$ has positive dimension $n-1$ and the divisions by $n-1$ below are legitimate. Let $I_k$ denote the maximal positive interval on which the model sine function is positive. Define \begin{align*} \operatorname{sn}_k:I_k&\to(0,\infty) \end{align*} to be the solution of $\operatorname{sn}_k''+k\operatorname{sn}_k=0$ with $\operatorname{sn}_k(0)=0$ and $\operatorname{sn}_k'(0)=1$, interpreted by the right-hand limit at $0$. Define the model cotangent function \begin{align*} \operatorname{ct}_k:I_k&\to\mathbb{R}\\ \rho&\mapsto \frac{\operatorname{sn}_k'(\rho)}{\operatorname{sn}_k(\rho)}. \end{align*} Let $\sigma_p$ denote the Riemannian measure on $S_pM$, and define the model radial density \begin{align*} m_k:I_k&\to(0,\infty)\\ \rho&\mapsto \operatorname{sn}_k(\rho)^{n-1}. \end{align*} For $\theta\in S_pM$, let $\tau:S_pM\to(0,\infty]$ denote the ordinary cut-time function and define the truncated cut-time function $c:S_pM\to(0,\sup I_k]$ by $c(\theta):=\min\{\tau(\theta),\sup I_k\}$. Let $\gamma_\theta:[0,c(\theta))\to M$ denote the unit-speed radial geodesic $\gamma_\theta(\rho)=\exp_p(\rho\theta)$. Let $J_\theta:(0,c(\theta))\to(0,\infty)$ denote the polar Jacobian along $\gamma_\theta$, and let $H_\theta:(0,c(\theta))\to\mathbb{R}$ denote the radial mean curvature, so that $\partial_\rho J_\theta(\rho)=H_\theta(\rho)J_\theta(\rho)$ at every smooth polar point. Define the extended density quotient as the map \begin{align*} a_\theta:(0,\sup I_k)&\to[0,\infty)\\ \rho&\mapsto \begin{cases} \dfrac{J_\theta(\rho)}{m_k(\rho)},&0<\rho<c(\theta),\\ 0,&c(\theta)\leq\rho<\sup I_k. \end{cases} \end{align*} The radial Riccati comparison estimate used in the [Bishop-Gromov monotonicity theorem](/page/Bishop-Gromov%20Inequality) applies on $(0,c(\theta))$. Before the cut time the polar coordinates are smooth, $J_\theta(\rho)>0$, and the radial shape operator \begin{align*} S_\theta(\rho):\dot\gamma_\theta(\rho)^\perp&\to\dot\gamma_\theta(\rho)^\perp \end{align*} satisfies the matrix Riccati equation \begin{align*} \partial_\rho S_\theta(\rho)+S_\theta(\rho)^2+R_{\dot\gamma_\theta(\rho)}=0, \end{align*} where $R_{\dot\gamma_\theta(\rho)}$ is the curvature endomorphism $v\mapsto R(v,\dot\gamma_\theta(\rho))\dot\gamma_\theta(\rho)$ on $\dot\gamma_\theta(\rho)^\perp$. Taking traces gives \begin{align*} \partial_\rho H_\theta(\rho)+\operatorname{tr}(S_\theta(\rho)^2)+\operatorname{Ric}_g(\dot\gamma_\theta(\rho),\dot\gamma_\theta(\rho))=0. \end{align*} The [Cauchy-Schwarz inequality](/page/Cauchy-Schwarz%20Inequality) for the eigenvalues of the symmetric endomorphism $S_\theta(\rho)$ gives \begin{align*} \operatorname{tr}(S_\theta(\rho)^2)\geq \frac{H_\theta(\rho)^2}{n-1}, \end{align*} and the curvature hypothesis gives \begin{align*} \operatorname{Ric}_g(\dot\gamma_\theta(\rho),\dot\gamma_\theta(\rho))\geq(n-1)k. \end{align*} Therefore \begin{align*} \partial_\rho H_\theta(\rho)+\frac{H_\theta(\rho)^2}{n-1}+(n-1)k\leq0. \end{align*} The model function $h_k:(0,\sup I_k)\to\mathbb{R}$ defined by $h_k(\rho)=(n-1)\operatorname{ct}_k(\rho)$ satisfies \begin{align*} h_k'(\rho)+\frac{h_k(\rho)^2}{n-1}+(n-1)k=0, \end{align*} because $\operatorname{sn}_k''+k\operatorname{sn}_k=0$. The initial polar asymptotics are \begin{align*} J_\theta(\rho)&=\rho^{n-1}(1+O(\rho^2)),\\ H_\theta(\rho)&=\frac{n-1}{\rho}+O(\rho),\\ h_k(\rho)&=\frac{n-1}{\rho}+O(\rho) \end{align*} as $\rho\downarrow0$, so $H_\theta(\rho)-h_k(\rho)\to0$. We now apply the singular endpoint form of the [Riccati comparison theorem](/page/Riccati%20Equation). For completeness, here is the scalar reduction. Define \begin{align*} y_\theta:(0,c(\theta))&\to\mathbb{R}\\ \rho&\mapsto H_\theta(\rho)-h_k(\rho). \end{align*} Subtracting the model equality from the preceding differential inequality gives, at every smooth polar radius, \begin{align*} y_\theta'(\rho)+\frac{H_\theta(\rho)+h_k(\rho)}{n-1}y_\theta(\rho)\leq0. \end{align*} Fix $0<\rho_0<\rho<c(\theta)$. On the compact interval $[\rho_0,\rho]$, all polar quantities are smooth because the interval lies strictly before the cut time, and $H_\theta+h_k$ is continuous. Hence the positive integrating factor \begin{align*} E_{\rho_0,\rho}:=\exp\left(\int_{\rho_0}^{\rho}\frac{H_\theta(t)+h_k(t)}{n-1}\,d\mathcal{L}^1(t)\right) \end{align*} is finite. Multiplying the differential inequality by this integrating factor and integrating over $[\rho_0,\rho]$ gives \begin{align*} y_\theta(\rho)\leq y_\theta(\rho_0)E_{\rho_0,\rho}^{-1}. \end{align*} The asymptotics above imply that $H_\theta(t)+h_k(t)=2(n-1)t^{-1}+O(t)$ near $0$, hence $E_{\rho_0,\rho}^{-1}=O(\rho_0^2)$ as $\rho_0\downarrow0$ for fixed $\rho$. Since $y_\theta(\rho_0)=O(\rho_0)$, the right-hand side tends to $0$. Therefore \begin{align*} H_\theta(\rho)\leq (n-1)\operatorname{ct}_k(\rho) \end{align*} for every $\rho<c(\theta)$ at which the polar density is smooth. Differentiating $a_\theta$ on $(0,c(\theta))$ gives \begin{align*} \partial_\rho a_\theta(\rho) &= \frac{\partial_\rho J_\theta(\rho)}{m_k(\rho)} - \frac{J_\theta(\rho)m_k'(\rho)}{m_k(\rho)^2} \\ &= \frac{J_\theta(\rho)}{m_k(\rho)} \left( H_\theta(\rho)-(n-1)\operatorname{ct}_k(\rho) \right). \end{align*} Define the Bishop-Gromov radial density defect by \begin{align*} D:S_pM\times(0,\sup I_k)&\to[0,\infty)\\ (\theta,\rho)&\mapsto \begin{cases} \left((n-1)\operatorname{ct}_k(\rho)-H_\theta(\rho)\right)J_\theta(\rho),&0<\rho<c(\theta),\\ 0,&c(\theta)\leq\rho<\sup I_k. \end{cases} \end{align*} The comparison inequality says precisely that $D(\theta,\rho)\geq0$ before the cut time, and the preceding derivative identity is equivalently \begin{align*} \partial_\rho a_\theta(\rho)=-\frac{D(\theta,\rho)}{m_k(\rho)}. \end{align*} Hence $\partial_\rho a_\theta(\rho)\leq0$ before the cut time. With the extension by $0$ after $c(\theta)$, the function $a_\theta$ is nonincreasing on $(0,\sup I_k)$. [/step]

[step:Convert equality of volume ratios into pointwise vanishing of the radial defect]For $s\in I_k$, the [polar-coordinate area formula](/page/Polar%20Coordinates) for the exponential map at $p$ gives \begin{align*} \operatorname{vol}_g(B(p,s)) = \int_{S_pM}\int_0^s a_\theta(\rho)m_k(\rho)\,d\mathcal{L}^1(\rho)\,d\sigma_p(\theta), \end{align*} where the extension $a_\theta(\rho)=0$ for $\rho\geq c(\theta)$ incorporates the cut locus, which has zero Riemannian measure in polar coordinates. The model volume function is the map \begin{align*} V_k^n:I_k&\to(0,\infty)\\ s&\mapsto \sigma_p(S_pM)\int_0^s m_k(\rho)\,d\mathcal{L}^1(\rho). \end{align*} Define \begin{align*} F:I_k&\to\mathbb{R}\\ s&\mapsto \frac{\operatorname{vol}_g(B(p,s))}{V_k^n(s)}. \end{align*} The [Bishop-Gromov monotonicity theorem](/page/Bishop-Gromov%20Inequality) applies because $(M^n,g)$ is complete, $\operatorname{Ric}_g\geq(n-1)k g$, and $r,R\in I_k$ keep the comparison model density positive. Hence $F$ is nonincreasing. For any $s\in[r,R]$, monotonicity gives \begin{align*} F(r)\geq F(s)\geq F(R). \end{align*} Since $F(r)=F(R)$ and $r<R$, both inequalities are equalities, so $F$ is constant on $[r,R]$. Let this constant be $C$. For $\mathcal{L}^1$-almost every $s\in(r,R)$, differentiating the two absolutely continuous volume functions gives \begin{align*} \int_{S_pM}a_\theta(s)m_k(s)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM)m_k(s). \end{align*} Since $m_k(s)>0$, this is equivalent to \begin{align*} \int_{S_pM}a_\theta(s)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM) \end{align*} for $\mathcal{L}^1$-almost every $s\in(r,R)$. Also, \begin{align*} \int_{S_pM}\int_0^r a_\theta(t)m_k(t)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM)\int_0^r m_k(t)\,d\mathcal{L}^1(t). \end{align*} Let $G\subset(r,R)$ be the set of radii for which the differentiated identity \begin{align*} \int_{S_pM}a_\theta(s)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM) \end{align*} holds. Then $\mathcal{L}^1((r,R)\setminus G)=0$. For $s\in G$, since each $a_\theta$ is nonincreasing and $0<t<r<s$, we have \begin{align*} a_\theta(t)-a_\theta(s)\geq0 \end{align*} for every $\theta\in S_pM$ and every $t\in(0,r)$ at which the chosen monotone representative is evaluated. Each monotone representative has at most countably many discontinuities, and changing values on these countable one-dimensional sets does not affect any $\mathcal{L}^1$ or product-measure statement below. Define the nonnegative measurable function \begin{align*} N:S_pM\times(0,r)\times G&\to[0,\infty]\\ (\theta,t,s)&\mapsto \left(a_\theta(t)-a_\theta(s)\right)m_k(t). \end{align*} Measurability follows from measurability of the polar Jacobian, the cut-time function, and the extension by zero. For each fixed $s\in G$, [Tonelli's theorem](/page/Tonelli%20Theorem) applies because $N(\cdot,\cdot,s)$ is nonnegative, and the equalities above give \begin{align*} &\int_{S_pM}\int_0^r \left(a_\theta(t)-a_\theta(s)\right)m_k(t)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta)\\ &= \int_{S_pM}\int_0^r a_\theta(t)m_k(t)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta) - \left(\int_{S_pM}a_\theta(s)\,d\sigma_p(\theta)\right) \left(\int_0^r m_k(t)\,d\mathcal{L}^1(t)\right)\\ &= 0. \end{align*} Applying [Tonelli's theorem](/page/Tonelli%20Theorem) again on $S_pM\times(0,r)\times G$ gives \begin{align*} \int_G\int_{S_pM}\int_0^r N(\theta,t,s)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta)\,d\mathcal{L}^1(s)=0. \end{align*} Since $N\geq0$, it follows that $N=0$ for $(\sigma_p\otimes\mathcal{L}^1\otimes\mathcal{L}^1)$-almost every $(\theta,t,s)\in S_pM\times(0,r)\times G$. Because $m_k(t)>0$ on $(0,r)$, [Fubini's theorem](/page/Fubini%20Theorem) gives a full $\sigma_p$-measure set $E\subset S_pM$ such that for every $\theta\in E$, \begin{align*} a_\theta(t)=a_\theta(s) \end{align*} for $(\mathcal{L}^1\otimes\mathcal{L}^1)$-almost every $(t,s)\in(0,r)\times G$. Fix $\theta\in E$. By [Fubini's theorem](/page/Fubini%20Theorem), there exists $t_\theta\in(0,r)$ and a full-measure subset $G_\theta\subset G$ such that \begin{align*} a_\theta(s)=a_\theta(t_\theta) \end{align*} for every $s\in G_\theta$. Since $G_\theta$ has full measure in $(r,R)$, the monotone function $\rho\mapsto a_\theta(\rho)$ is constant for $\mathcal{L}^1$-almost every $\rho\in(r,R)$. Therefore its a.e. classical derivative vanishes on the annulus. With \begin{align*} \Omega_{r,R}:=\{(\theta,\rho)\in S_pM\times(r,R):\rho<c(\theta)\}, \end{align*} we obtain \begin{align*} \partial_\rho a_\theta(\rho)=0 \end{align*} for $(\sigma_p\otimes\mathcal{L}^1)$-almost every $(\theta,\rho)\in\Omega_{r,R}$.[/step]

[guided]We first recall all objects used in this step so that the argument is self-contained. The measure $\sigma_p$ is the Riemannian measure on $S_pM$, the model density is $m_k(\rho)=\operatorname{sn}_k(\rho)^{n-1}$ on $I_k$, and the extended density quotient is \begin{align*} a_\theta(\rho)= \begin{cases} \dfrac{J_\theta(\rho)}{m_k(\rho)},&0<\rho<c(\theta),\\ 0,&c(\theta)\leq\rho<\sup I_k. \end{cases} \end{align*} The previous comparison step proved that $a_\theta$ is nonincreasing in $\rho$ for each fixed direction $\theta$ by showing that, before the cut time, \begin{align*} \partial_\rho a_\theta(\rho) = -\frac{D(\theta,\rho)}{m_k(\rho)}\leq0, \end{align*} where $D$ is the nonnegative Bishop-Gromov radial density defect. After the cut time the extension by zero preserves monotonicity. The key point is that we are not merely using that one averaged quantity is equal at two endpoints. We use monotonicity in two ways: first, Bishop-Gromov says the ball-volume ratio is nonincreasing in the radius, so equality at $r$ and $R$ forces equality at every intermediate radius; second, the density quotient $a_\theta(\rho)$ is nonincreasing along each fixed radial geodesic. The polar-coordinate [area formula](/theorems/3075) applies to the exponential map because, away from the cut locus, $\exp_p$ is a smooth local diffeomorphism in polar variables, the cut-time function is measurable, and the cut locus has zero Riemannian measure. It writes the actual ball volume as \begin{align*} \operatorname{vol}_g(B(p,s)) = \int_{S_pM}\int_0^s a_\theta(\rho)m_k(\rho)\,d\mathcal{L}^1(\rho)\,d\sigma_p(\theta). \end{align*} The model volume is \begin{align*} V_k^n(s) = \sigma_p(S_pM)\int_0^s m_k(\rho)\,d\mathcal{L}^1(\rho). \end{align*} Thus the volume ratio is a weighted average of the radial density quotient $a_\theta$ over all directions and all radii up to $s$. Let $C$ be the common value of the ratio on $[r,R]$. Since the volume functions are absolutely continuous in $s$, differentiating for almost every $s\in(r,R)$ gives \begin{align*} \int_{S_pM}a_\theta(s)m_k(s)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM)m_k(s). \end{align*} After dividing by $m_k(s)>0$, this says that the spherical average of $a_\theta(s)$ is exactly $C$ for almost every $s\in(r,R)$: \begin{align*} \int_{S_pM}a_\theta(s)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM). \end{align*} The equality at radius $r$ says \begin{align*} \int_{S_pM}\int_0^r a_\theta(t)m_k(t)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta) = C\,\sigma_p(S_pM)\int_0^r m_k(t)\,d\mathcal{L}^1(t). \end{align*} Let $G\subset(r,R)$ be the full-measure set of radii for which the differentiated identity holds. We must be careful about quantifiers: proving equality for one radius $s$ is not enough; we need a statement that holds simultaneously for almost every direction and almost every annular radius. This is exactly where [Tonelli's theorem](/page/Tonelli%20Theorem) and [Fubini's theorem](/page/Fubini%20Theorem) enter. For $s\in G$ and $0<t<r<s$, monotonicity along the same radial geodesic gives \begin{align*} a_\theta(t)\geq a_\theta(s). \end{align*} Define \begin{align*} N:S_pM\times(0,r)\times G&\to[0,\infty]\\ (\theta,t,s)&\mapsto \left(a_\theta(t)-a_\theta(s)\right)m_k(t). \end{align*} The function is measurable because $J_\theta(\rho)$ is measurable in polar coordinates, $c(\theta)$ is measurable, and $a_\theta$ was extended by zero after the cut time. It is nonnegative, so [Tonelli's theorem](/page/Tonelli%20Theorem) applies without a prior integrability assumption. For each $s\in G$, Tonelli's theorem permits separating the two nonnegative integrals, and the identities above give \begin{align*} &\int_{S_pM}\int_0^r \left(a_\theta(t)-a_\theta(s)\right)m_k(t)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta)\\ &= C\,\sigma_p(S_pM)\int_0^r m_k(t)\,d\mathcal{L}^1(t) - C\,\sigma_p(S_pM)\int_0^r m_k(t)\,d\mathcal{L}^1(t)\\ &= 0. \end{align*} Integrating this identity over $s\in G$ and applying [Tonelli's theorem](/page/Tonelli%20Theorem) on the triple product gives \begin{align*} \int_G\int_{S_pM}\int_0^r N(\theta,t,s)\,d\mathcal{L}^1(t)\,d\sigma_p(\theta)\,d\mathcal{L}^1(s)=0. \end{align*} Since $N\geq0$, the vanishing of the integral forces $N=0$ for almost every triple $(\theta,t,s)$. Because $m_k(t)>0$ on $(0,r)$, this means \begin{align*} a_\theta(t)=a_\theta(s) \end{align*} for almost every triple $(\theta,t,s)\in S_pM\times(0,r)\times G$. Now [Fubini's theorem](/page/Fubini%20Theorem) lets us fix the direction correctly. There is a full $\sigma_p$-measure set $E\subset S_pM$ such that, for every $\theta\in E$, the equality above holds for almost every pair $(t,s)\in(0,r)\times G$. For such a $\theta$, [Fubini's theorem](/page/Fubini%20Theorem) again gives a radius $t_\theta\in(0,r)$ and a full-measure set $G_\theta\subset G$ such that \begin{align*} a_\theta(s)=a_\theta(t_\theta) \end{align*} for every $s\in G_\theta$. Since $G_\theta$ has full measure in $(r,R)$, the monotone function $\rho\mapsto a_\theta(\rho)$ is constant for almost every $\rho\in(r,R)$. A monotone real-valued function is differentiable almost everywhere, and the derivative of a function that is almost everywhere constant is zero at almost every differentiability point. Therefore, with \begin{align*} \Omega_{r,R}:=\{(\theta,\rho)\in S_pM\times(r,R):\rho<c(\theta)\}, \end{align*} we have \begin{align*} \partial_\rho a_\theta(\rho)=0 \end{align*} for $(\sigma_p\otimes\mathcal{L}^1)$-almost every $(\theta,\rho)\in\Omega_{r,R}$.[/guided]

[step:Identify the vanishing derivative with equality of mean curvature] On $\Omega_{r,R}$, the polar Jacobian satisfies $J_\theta(\rho)>0$, and \begin{align*} a_\theta(\rho)=\frac{J_\theta(\rho)}{m_k(\rho)}. \end{align*} For almost every point of $\Omega_{r,R}$, \begin{align*} 0 = \partial_\rho a_\theta(\rho) = \frac{J_\theta(\rho)}{m_k(\rho)} \left( H_\theta(\rho)-(n-1)\operatorname{ct}_k(\rho) \right). \end{align*} Since $J_\theta(\rho)>0$ and $m_k(\rho)>0$, this implies \begin{align*} H_\theta(\rho)=(n-1)\operatorname{ct}_k(\rho) \end{align*} for $(\sigma_p\otimes\mathcal{L}^1)$-almost every $(\theta,\rho)\in\Omega_{r,R}$. Equivalently, for the Bishop-Gromov radial density defect $D$ defined in the comparison step, \begin{align*} D(\theta,\rho)=0 \end{align*} almost everywhere on $\Omega_{r,R}$. [/step]

[step:Derive the equality shape-operator equation and integrate it to obtain the radial warping] Assume now that $\tau(\theta)>R$ for every $\theta\in S_pM$. Then the polar map \begin{align*} \Phi:S_pM\times(r,R)&\to B(p,R)\setminus\overline{B(p,r)}\\ (\theta,\rho)&\mapsto \exp_p(\rho\theta) \end{align*} is a smooth polar parametrization of the annulus, because no radial geodesic meets the cut locus before radius $R$. The pulled-back metric has the form \begin{align*} \Phi^*g=d\rho^2+g_\rho, \end{align*} where $g_\rho$ is the Riemannian metric induced on the angular slice $S_pM\times\{\rho\}$. We next explain precisely how the smooth equality assumptions imply the shape-operator equation in the present no-cut annulus. Suppose the Cauchy-Schwarz equality case in the trace Riccati estimate and the radial Ricci equality hold smoothly along a radial geodesic. The trace Riccati equation is \begin{align*} \partial_\rho H_\theta(\rho)+\operatorname{tr}(S_\theta(\rho)^2)+\operatorname{Ric}_g(\dot\gamma_\theta(\rho),\dot\gamma_\theta(\rho))=0. \end{align*} The equality case in the [Cauchy-Schwarz inequality](/page/Cauchy-Schwarz%20Inequality) for the eigenvalues of the symmetric endomorphism $S_\theta(\rho)$ says precisely that all eigenvalues are equal, hence \begin{align*} S_\theta(\rho)=\frac{H_\theta(\rho)}{n-1}\operatorname{Id}_{\dot\gamma_\theta(\rho)^\perp}. \end{align*} The previous step gives $H_\theta(\rho)=(n-1)\operatorname{ct}_k(\rho)$ for $(\sigma_p\otimes\mathcal{L}^1)$-almost every $(\theta,\rho)$ in the no-cut annulus. Under the no-cut hypothesis, $H_\theta(\rho)$ and $(n-1)\operatorname{ct}_k(\rho)$ are smooth functions of $(\theta,\rho)$ on $S_pM\times(r,R)$, so this almost-everywhere equality extends to every $(\theta,\rho)\in S_pM\times(r,R)$ by continuity. Therefore \begin{align*} S_\theta(\rho)=\operatorname{ct}_k(\rho)\operatorname{Id}_{\dot\gamma_\theta(\rho)^\perp}. \end{align*} Thus the smooth Cauchy-Schwarz and radial Ricci equalities, together with the volume-ratio equality already used to obtain the almost-everywhere mean-curvature equality, imply the shape-operator equation. Conversely, the shape-operator equation intrinsically implies both smooth equality conditions: if $S_\theta(\rho)=\operatorname{ct}_k(\rho)\operatorname{Id}_{\dot\gamma_\theta(\rho)^\perp}$, then the Cauchy-Schwarz equality case holds because all eigenvalues are equal. Taking the trace of the matrix Riccati equation gives \begin{align*} (n-1)\operatorname{ct}_k'(\rho)+(n-1)\operatorname{ct}_k(\rho)^2+\operatorname{Ric}_g(\dot\gamma_\theta(\rho),\dot\gamma_\theta(\rho))=0. \end{align*} Since $\operatorname{ct}_k'(\rho)+\operatorname{ct}_k(\rho)^2+k=0$, we obtain \begin{align*} \operatorname{Ric}_g(\dot\gamma_\theta(\rho),\dot\gamma_\theta(\rho))=(n-1)k. \end{align*} Let $T_\theta S_pM$ denote the tangent space at $\theta$ to the unit sphere $S_pM\subset T_pM$, and let $v,w\in T_\theta S_pM$. For each $\rho\in(r,R)$, define the differential of the angular exponential map \begin{align*} A_{\theta,\rho}:T_\theta S_pM&\to\dot\gamma_\theta(\rho)^\perp\\ v&\mapsto d\exp_p|_{\rho\theta}(\rho v). \end{align*} Because the annulus is free of cut points, $A_{\theta,\rho}$ is an isomorphism onto the tangent space of the geodesic sphere at $\gamma_\theta(\rho)$. In the expression $S_\theta(\rho)v$ below, $S_\theta(\rho)$ acts on the vector $A_{\theta,\rho}v$, and the result is pulled back to $T_\theta S_pM$ through this identification. The [first variation formula](/theorems/2728) for the angular metric along the radial geodesic gives \begin{align*} \partial_\rho g_\rho(v,w) = 2g_\rho(S_\theta(\rho)v,w). \end{align*} Using the assumed shape-operator equality \begin{align*} S_\theta(\rho)=\operatorname{ct}_k(\rho)\operatorname{Id}_{\dot{\gamma}_\theta(\rho)^\perp}, \end{align*} we obtain \begin{align*} \partial_\rho g_\rho(v,w) = 2\operatorname{ct}_k(\rho)g_\rho(v,w). \end{align*} Now define the angular tensor \begin{align*} q_\rho:=\operatorname{sn}_k(\rho)^{-2}g_\rho. \end{align*} For all $v,w\in T_\theta S_pM$, \begin{align*} \partial_\rho q_\rho(v,w) &= -2\operatorname{sn}_k(\rho)^{-3}\operatorname{sn}_k'(\rho)g_\rho(v,w) + \operatorname{sn}_k(\rho)^{-2}\partial_\rho g_\rho(v,w)\\ &= -2\operatorname{ct}_k(\rho)\operatorname{sn}_k(\rho)^{-2}g_\rho(v,w) + 2\operatorname{ct}_k(\rho)\operatorname{sn}_k(\rho)^{-2}g_\rho(v,w)\\ &= 0. \end{align*} Thus $q_\rho$ is independent of $\rho$. Setting \begin{align*} h:=q_r=\operatorname{sn}_k(r)^{-2}g_r, \end{align*} we obtain \begin{align*} g_\rho=\operatorname{sn}_k(\rho)^2h \end{align*} for every $\rho\in(r,R)$. Therefore \begin{align*} \Phi^*g=d\rho^2+\operatorname{sn}_k(\rho)^2h. \end{align*} Let $g_{S_pM}^{\mathrm{round}}$ denote the standard round metric on the unit sphere $S_pM\subset T_pM$ induced by the [inner product](/page/Inner%20Product) $g_p$ on $T_pM$. If $g_r=\operatorname{sn}_k(r)^2g_{S_pM}^{\mathrm{round}}$, then $h=g_{S_pM}^{\mathrm{round}}$, so the pulled-back polar metric is \begin{align*} \Phi^*g=d\rho^2+\operatorname{sn}_k(\rho)^2g_{S_pM}^{\mathrm{round}}. \end{align*} Thus the metric has the round model form in these polar coordinates. If, in addition, the polar parametrization $\Phi:S_pM\times(r,R)\to B(p,R)\setminus\overline{B(p,r)}$ is a global diffeomorphism onto the annulus, this pulled-back metric identity descends to an isometry of the annulus with the corresponding constant-curvature model annulus. Conversely, any such round model-annulus isometry expressed in the polar parametrization forces the angular metric at radius $r$ to be $\operatorname{sn}_k(r)^2g_{S_pM}^{\mathrm{round}}$. This proves the stated pulled-back polar rigidity and the necessity of the angular initial data condition for a full round model-annulus isometry. [/step]