A concave cometric and convex collars

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson constructs the metric on the manifold \(M\) of The manifold and its fundamental group. On the neck \([0,1]\times T\) it is \[ g=dt^2+dx^{\mathsf T}P(t)^{-1}dx, \] where \(P(t)\) is a smooth path of positive definite \(2\times2\) matrices, the cometric, chosen concave and nearly constant, with a small shear in the middle that realizes the twisted gluing. Near each boundary torus the metric is a product of a flat circle and a convex warped surface, which continues into a flat torus with a hole. Outside the open neck every sectional curvature is nonpositive; on the neck the curvature is not controlled, and the next two lessons show that the concavity of \(P\) is enough to exclude conjugate points.

Matrix norms are Euclidean operator norms, \({}^{\mathsf T}\) denotes transpose, and for symmetric matrices \(R\preceq S\) means that \(S-R\) is positive semidefinite.

1. The cutoff and the matrix path

Define \(a\colon\mathbb R\to[0,1)\) by \(a(u)=\exp\bigl(-1/(\tfrac23-u)\bigr)\) for \(u<\tfrac23\) and \(a(u)=0\) for \(u\ge\tfrac23\), and its primitive \(A\) by \[ A(u)=\int_0^ua(y)\,dy . \tag{1.1} \] The function \(a\) is smooth: every derivative of \(\exp(-1/w)\) is a polynomial in \(1/w\) times \(\exp(-1/w)\), which tends to \(0\) as \(w\to0^+\). For \(u<\frac23\), \[ a'(u)=-\frac{a(u)}{(\frac23-u)^2}<0 . \tag{1.2} \] So \(a\) is nonincreasing, \(A\) is concave, and \(A\) is constant on \([\frac23,\infty)\). Fix once and for all a smooth nondecreasing function \(\chi\colon\mathbb R\to[0,1]\) equal to \(0\) on \((-\infty,\frac5{12}]\) and to \(1\) on \([\frac7{12},\infty)\).

For \(0<\delta\le\frac12\) put \(\epsilon=\delta^2\) and, for \(0\le t\le1\), \[ \begin{aligned} H(t)&=e_2+\epsilon\chi(t)e_1,\\ d_1(t)&=1+\delta A(t),\\ d_2(t)&=1+\delta A(1-t),\\ P(t)&=d_1(t)\,e_1e_1^{\mathsf T}\\ &\quad+d_2(t)\,H(t)H(t)^{\mathsf T} . \end{aligned} \tag{1.3} \] Put \(Q=P^{-1}\), the matrix of the torus metric, and \[ \begin{aligned} \kappa&=\delta a(0)=\delta e^{-3/2}>0,\\ b&=1+\delta A(1) . \end{aligned} \tag{1.4} \] Since \(A(1)=A(\frac23)\le\frac23e^{-3/2}<\frac16\), we have \(1\le d_i\le\frac{13}{12}\) on \([0,1]\).

The two end frames are the matrices with columns \[ \begin{aligned} T_1&=(e_1,\ e_2),\\ T_2&=(e_2+\epsilon e_1,\ e_1), \end{aligned} \tag{1.5} \] those of (1.1) in The manifold and its fundamental group with \(\epsilon=B_1/B_2\). A linear change of coordinates \(x=T_iy\) changes the cometric to \(T_i^{-1}PT_i^{-\mathsf T}\). The inward coordinate is \(\tau=t\) at the first end and \(\tau=1-t\) at the second.

Lemma 1.1 (the ends of the path). For \(0\le\tau\le\frac14\), the cometric in the end frame \(T_i\) and inward coordinate \(\tau\) is \[ P_i(\tau)=\begin{pmatrix}1+\delta A(\tau)&0\\0&b\end{pmatrix} . \tag{1.6} \] In particular \(P_i(0)e_1=e_1\) and \(P_i'(0)=\kappa\,e_1e_1^{\mathsf T}\), primes denoting derivatives in \(\tau\).

Proof. For \(t\in[0,\frac14]\), \(\chi(t)=0\), so \(H=e_2\); and \(1-t\ge\frac23\), so \(d_2=b\). Thus \(P=\operatorname{diag}(1+\delta A(t),b)\), and \(T_1\) is the identity. For \(t\in[\frac34,1]\), \(\chi(t)=1\), so \(H=e_2+\epsilon e_1\) is the first column of \(T_2\); \(t\ge\frac23\) gives \(d_1=b\), and \(d_2=1+\delta A(\tau)\). Hence \(P=T_2\operatorname{diag}(1+\delta A(\tau),b)T_2^{\mathsf T}\), which is (1.6) after the change of frame. Finally \(A(0)=0\) and \(A'=a\). \(\square\)

2. Bounds on the cometric

Lemma 2.1 ([OpenAI-NC, Lemma 3.1]). There are \(\delta_0\in(0,\frac12]\) and \(C_0\ge18\), depending only on \(\chi\), such that for \(0<\delta\le\delta_0\) the following hold on \([0,1]\), in the original coordinates and in either end frame with its inward coordinate: \[ \begin{aligned} &\tfrac19\operatorname{Id}\preceq P\preceq18\operatorname{Id},\\ &\tfrac1{18}\operatorname{Id}\preceq Q\preceq9\operatorname{Id},\\ &\|P'\|\le C_0\delta, \end{aligned} \tag{2.1} \] \[ \begin{aligned} &-P''\succeq0,\\ &|P'p|^2\le C_0\delta\,p^{\mathsf T}(-P'')p \end{aligned} \tag{2.2} \] for every \(p\in\mathbb R^2\).

Proof. Size. Let \(R=(e_1,H)\) be the matrix with columns \(e_1\) and \(H\). Then \(P=R\operatorname{diag}(d_1,d_2)R^{\mathsf T}\), and \(R\) and \(R^{-1}=(e_1,\ e_2-\epsilon\chi e_1)\) have squared norms at most their squared Frobenius norms \(2+\epsilon^2\chi^2\le3\). With \(1\le d_i\le2\) this gives \(\frac13\operatorname{Id}\preceq P\preceq6\operatorname{Id}\). The frames \(T_i\) and their inverses also have squared norms at most \(3\), which gives the bounds (2.1) for \(P\) in the end frames, and those for \(Q=P^{-1}\) follow.

Two estimates for \(a\). On \([0,1]\), with \(w=\frac23-u\in(0,\frac23]\) where \(a\ne0\), (1.2) gives \(-a'=e^{-1/w}/w^2\). Since \(w^{-2}\ge\frac94\ge e^{-1/w}\), we get \(a^2=e^{-1/w}\cdot e^{-1/w}\le e^{-1/w}w^{-2}=-a'\); and \(x^2e^{-x}\le4e^{-2}<1\) for \(x=1/w\) gives \(|a'|\le1\). So on \([0,1]\) \[ a^2\le-a',\qquad|a'|\le1 . \tag{2.3} \]

Outside \(I_0=[\frac5{12},\frac7{12}]\). There \(\chi'=0\), so \(H\) is constant and, with \(q=R^{\mathsf T}p\), \[ \begin{aligned} &P'=\delta a(t)\,e_1e_1^{\mathsf T}-\delta a(1-t)\,HH^{\mathsf T},\\ &p^{\mathsf T}(-P'')p\\ &\quad=\delta\bigl(-a'(t)q_1^2-a'(1-t)q_2^2\bigr)\ge0 . \end{aligned} \] Since \(|H|^2\le\frac{17}{16}\), \[ \begin{aligned} |P'p|^2&\le2\delta^2\bigl(a(t)^2q_1^2+\tfrac{17}{16}a(1-t)^2q_2^2\bigr)\\ &\le3\delta\,p^{\mathsf T}(-P'')p \end{aligned} \] by (2.3), and \(\|P'\|\le3\delta\).

On \(I_0\). There \(t\) and \(1-t\) lie in \([\frac5{12},\frac7{12}]\), so \(w\in[\frac1{12},\frac14]\), where \(e^{-1/w}/w^2\) is increasing; hence \(-a'(t),-a'(1-t)\ge m:=144e^{-12}\). Compare \(P\) with \(P_0=\operatorname{diag}(d_1,d_2)\): the difference \[ \begin{aligned} E=P-P_0&=d_2\bigl(\epsilon\chi\,(e_1e_2^{\mathsf T}+e_2e_1^{\mathsf T})\\ &\qquad+\epsilon^2\chi^2e_1e_1^{\mathsf T}\bigr) \end{aligned} \] and its first two derivatives are sums of terms each containing a factor \(\epsilon\) or \(\epsilon^2\), a factor \(d_2\), \(d_2'=-\delta a(1-t)\) or \(d_2''=\delta a'(1-t)\) (all bounded by \(2\)), and factors \(\chi,\chi',\chi''\). So \(\|E'\|\le C_1\epsilon\) and \(\|E''\|\le C_2\epsilon\) with \(C_1,C_2\) depending only on \(\sup|\chi'|\) and \(\sup|\chi''|\). Now \(-P_0''\succeq m\delta\operatorname{Id}\), because \(-P_0''=\delta\operatorname{diag}(-a'(t),-a'(1-t))\), and \(\epsilon=\delta^2\); for \(\delta\le m/(2C_2)\), \[ \begin{aligned} &-P''=-P_0''-E''\\ &\quad\succeq\bigl(m\delta-C_2\delta^2\bigr)\operatorname{Id}\succeq\tfrac12m\delta\operatorname{Id},\\ &\|P'\|\le\delta+C_1\delta^2\le(1+C_1)\delta, \end{aligned} \] so \[ \begin{aligned} |P'p|^2&\le(1+C_1)^2\delta^2|p|^2\\ &\le\frac{2(1+C_1)^2}m\,\delta\,p^{\mathsf T}(-P'')p . \end{aligned} \]

End frames. For \(\widetilde P=T_i^{-1}PT_i^{-\mathsf T}\) and \(q=T_i^{-\mathsf T}p\), \(\widetilde P''=T_i^{-1}P''T_i^{-\mathsf T}\), so \(p^{\mathsf T}(-\widetilde P'')p=q^{\mathsf T}(-P'')q\ge0\), and \(|\widetilde P'p|^2\le3|P'q|^2\le3C\delta\,p^{\mathsf T}(-\widetilde P'')p\) with \(C\) the constant found above; also \(\|\widetilde P'\|\le3\|P'\|\). Replacing \(t\) by \(1-t\) changes the sign of the first derivative and leaves the second unchanged. Take \(\delta_0=\min\{\frac12,\,m/(2C_2)\}\), and let \(C_0\) be the largest of the numbers \(18\), \(9\), \(3(1+C_1)\) and \(6(1+C_1)^2/m\). \(\square\)

The middle of the path carries the shear: \(H\) turns from \(e_2\) to \(e_2+\epsilon e_1\) while both diagonal coefficients are strictly concave, and since the shear has size \(\epsilon=\delta^2\), it does not destroy the concavity of size \(\delta\). The periods \(B_1=\epsilon B_2\) turn this small shear into the fixed integral gluing \(h_2=h_1+f_1\), \(f_2=h_1\).

3. Warped products and the ends of the neck

Lemma 3.1 (warped surfaces and products with a circle). Let \(\ell>0\) be smooth on an interval \(J\). The metric \(d\tau^2+\ell(\tau)^2dy^2\) on \(J\times\mathbb R/B\mathbb Z\) has Gaussian curvature \(-\ell''/\ell\). If a surface has Gaussian curvature \(K\le0\), every sectional curvature of its Riemannian product with a circle lies in \([K,0]\) at each point.

Proof. In the coordinates \((\tau,y)\), the only nonzero Christoffel symbols are \(\Gamma^\tau_{yy}=-\ell\ell'\) and \(\Gamma^y_{\tau y}=\Gamma^y_{y\tau}=\ell'/\ell\). Hence \(\nabla_{\partial_\tau}\partial_\tau=0\) and \(\nabla_{\partial_\tau}\partial_y=\nabla_{\partial_y}\partial_\tau=(\ell'/\ell)\partial_y\), and \[ \begin{aligned} &R(\partial_y,\partial_\tau)\partial_\tau\\ &\quad=\nabla_{\partial_y}\nabla_{\partial_\tau}\partial_\tau-\nabla_{\partial_\tau}\nabla_{\partial_y}\partial_\tau\\ &\quad=-\Bigl(\bigl(\tfrac{\ell'}\ell\bigr)'+\bigl(\tfrac{\ell'}\ell\bigr)^2\Bigr)\partial_y\\ &\quad=-\frac{\ell''}\ell\,\partial_y . \end{aligned} \] So \(K=\langle R(\partial_y,\partial_\tau)\partial_\tau,\partial_y\rangle/|\partial_y|^2=-\ell''/\ell\). For the product, the curvature tensor is that of the surface on surface vectors and vanishes as soon as one argument is tangent to the flat circle. For orthonormal \(X=X_1+X_2\), \(Y=Y_1+Y_2\), split into surface and circle components, the sectional curvature is \(K\bigl(|X_1|^2|Y_1|^2-\langle X_1,Y_1\rangle^2\bigr)\), and the bracket lies in \([0,1]\). \(\square\)

In the end frame \(T_i\) with inward coordinate \(\tau\), the neck metric for \(0\le\tau\le\frac14\) is, by Lemma 1.1, \[ \begin{aligned} g&=d\tau^2+\ell_0(\tau)^2dy_1^2\\ &\quad+b^{-1}dy_2^2,\\ \ell_0(\tau)&=(1+\delta A(\tau))^{-1/2}, \end{aligned} \tag{3.1} \] the product of a warped surface and a flat circle. The function \(\ell_0\) is defined for \(\tau\ge-1\), where \(A(\tau)\ge-1\) and \(1+\delta A\ge\frac12\). Writing \(w=1+\delta A\), \(w'=\delta a\), \(w''=\delta a'\), \[ \begin{aligned} &\frac{\ell_0''}{\ell_0}=\frac{3\delta^2a^2}{4(1+\delta A)^2}\\ &\qquad-\frac{\delta a'}{2(1+\delta A)}\ge0 \end{aligned} \tag{3.2} \] on \([-1,\frac14]\), because \(a'\le0\) there. By Lemma 3.1, the neck has nonpositive sectional curvature for \(0\le t\le\frac14\) and for \(\frac34\le t\le1\). Also \(\ell_0'(\tau)=-\delta a(\tau)/(2w^{3/2})<0\), with \(\ell_0(0)=1\) and \(\ell_0'(0)=-\kappa/2\).

4. Convex collars, flat cores and the metric on \(M\)

Fix \(\delta\in(0,\delta_0]\) and a period \(B\in(0,1]\). We build a holed torus \(\Sigma\) with a collar of period \(B\) whose metric has curvature at most \(0\) and is given near its boundary by \(d\tau^2+\ell_0(\tau)^2dy_1^2\).

The warping function. Fix a smooth \(\tilde\chi\colon\mathbb R\to[0,1]\) equal to \(0\) on \((-\infty,-1]\) and to \(1\) on \([-\frac12,\infty)\). Since \(\ell_0''\ge0\) on \([-1,0]\), \(0\le\int_{-1}^0\tilde\chi\ell_0''\le\ell_0'(0)-\ell_0'(-1)\), which is at most \(|\ell_0'(-1)|<1\), and \(\kappa/2<1\). As \(2\pi/B\ge2\pi\), we may choose a smooth function \(\beta\ge0\) supported in \((-2,-1)\) with \[ \int\beta=\frac{2\pi}B-\frac\kappa2-\int_{-1}^0\tilde\chi\,\ell_0''>0 . \] Put \(g_\ell=\tilde\chi\ell_0''+\beta\ge0\) on \([-3,0]\), and let \(\ell\) be the solution of \(\ell''=g_\ell\) with \(\ell(0)=1\), \(\ell'(0)=-\kappa/2\). Then:

  1. \(\ell=\ell_0\) on \([-\frac12,0]\), since both solve the same second-order equation there with the same data at \(0\); so \(\ell\) continues smoothly by \(\ell_0\) on \([0,\frac14]\).
  2. \(\ell'(\tau)=-\frac\kappa2-\int_\tau^0g_\ell<0\), so \(\ell\) is decreasing and \(\ell\ge1\) on \([-3,0]\); and \(\ell''\ge0\).
  3. For \(\tau\le-2\), \(\ell'(\tau)=-2\pi/B\), so \(\ell(\tau)=\frac{2\pi}B\,(r_1-2-\tau)\) on \([-3,-2]\), where \(r_1=\ell(-2)B/(2\pi)\).

The collar. Let \(C=[-3,0]\times\mathbb R/B\mathbb Z\) with the metric \(d\tau^2+\ell(\tau)^2dy_1^2\). By Lemma 3.1 its curvature is \(-\ell''/\ell\le0\). On \([-3,-2]\times\mathbb R/B\mathbb Z\), the substitution \(r=r_1-2-\tau\), \(\theta=2\pi y_1/B\) turns the metric into \(dr^2+r^2d\theta^2\), the Euclidean metric in polar coordinates, on the annulus \(r_1\le r\le r_1+1\).

The flat part. Choose \(c>2(r_1+1)\) and let \(F_c\), \(\pi\) and \(D_\rho\) be as in Section 1 of The manifold and its fundamental group. The closed annulus \(r_1\le|x|\le r_1+1\) about the origin embeds isometrically in \(F_c\) under \(\pi\), and the substitution above identifies it with \([-3,-2]\times\mathbb R/B\mathbb Z\subseteq C\). Let \(\Sigma\) be the union of \(F_c\setminus\pi(D_{r_1})\) and \(C\) glued along this isometry. Each point of \(\Sigma\) has a neighbourhood contained in one of the two pieces: points with \(r=r_1+1\) lie in the interior of the flat piece and points with \(\tau=-2\) in the interior of \(C\). So \(\Sigma\) is a smooth compact surface with boundary \(\{\tau=0\}\) and a smooth metric, flat on the first piece and of curvature \(-\ell''/\ell\le0\) on the second. Every sectional curvature of \(\Sigma\) is nonpositive.

\(\Sigma\) is a holed torus with a collar of period \(B\): the collar is \((-1,0]\times\mathbb R/B\mathbb Z\subseteq C\), and a diffeomorphism of \([-3,0]\) onto \([\frac{r_1}2,r_1+1]\) that is decreasing and equals \(r_1-2-\tau\) near \([-3,-2]\) defines, together with \(\theta=2\pi y_1/B\), a diffeomorphism of \(\Sigma\) onto \(F_c\setminus\pi(D_{r_1/2})\).

The metric on \(M\). Fix \(\delta\in(0,\delta_0]\) and \(B_2\in(0,1]\), and put \(B_1=\delta^2B_2\), so that \(B_1/B_2=\epsilon\). Build \(\Sigma_1\) with period \(B_1\) and \(\Sigma_2\) with period \(B_2\) as above, and form \(M\) as in Section 1 of The manifold and its fundamental group. Give the neck the metric \(dt^2+dx^{\mathsf T}Q(t)dx\), which is well defined on \([0,1]\times T\) because it does not depend on \(x\), and give \(N_i=\Sigma_i\times\mathbb R/B_{3-i}\mathbb Z\) the product of the metric of \(\Sigma_i\) with \(b^{-1}dy_2^2\).

Proposition 4.1 (the metric). These metrics fit together to a smooth Riemannian metric on the closed manifold \(M\). Every sectional curvature at every point outside the open neck \(U=(0,1)\times T\) is nonpositive, and so is every sectional curvature at points of the neck with \(t\le\frac14\) or \(t\ge\frac34\).

Proof. Near the first boundary torus, in the chart \((\tau,y)\mapsto(t,x)=(\tau,T_1y)\), the metric of \(N_1\) for \(-\frac12<\tau\le0\) is \(d\tau^2+\ell_0(\tau)^2dy_1^2+b^{-1}dy_2^2\), and by (3.1) the neck metric for \(0\le\tau\le\frac14\) is given by the same formula. So on \((-\frac12,\frac14)\times T\) the metric is given by one smooth formula. The second boundary torus is the same with \(T_2\) and \(\tau=1-t\). The curvature statement on \(N_i\) is Lemma 3.1 applied to \(\Sigma_i\), and on the end portions of the neck it is (3.2). \(\square\)

The neck metric, lifted to \([0,1]\times\mathbb R^2\), depends only on \(\delta\) and the fixed function \(\chi\), not on \(B_1\), \(B_2\), \(\tilde\chi\), \(\beta\) or \(c\). All estimates of the following lessons concern this lifted neck metric; \(\delta\) is fixed by them, and the periods may be chosen afterwards. In [OpenAI-NC, Section 3 and Appendix A] the cores are hyperbolic punctured tori truncated along a horocycle, whose cusp has the warped form \(dr^2+c^2e^{-2r}d\theta^2\); a convex collar joins that cusp to (3.1) in the same way, and any compact nonpositively curved holed torus whose boundary germ matches (3.1) would serve.

5. Exercises

5.1. Show that \(a(\frac18)/a(0)=e^{-9/26}\).

5.2. Show that \(T_2\operatorname{diag}(\alpha,\beta)T_2^{\mathsf T}=\alpha HH^{\mathsf T}+\beta e_1e_1^{\mathsf T}\) for \(H=e_2+\epsilon e_1\), and deduce Lemma 1.1 at the second end directly.

5.3. Compute the curvature of \(d\tau^2+e^{-2\tau}dy^2\) and of \(d\tau^2+(T-\tau)^2dy^2\) with Lemma 3.1, and identify the two surfaces.

5.4. Let \(\ell\) be smooth and positive with \(\ell''(\tau_0)<0\) at some point. Show that the warped surface has positive curvature near \(\tau_0\); explain why the construction of Section 4 needs \(g_\ell\ge0\).

5.5. Show that \(H(t)H(t)^{\mathsf T}-e_2e_2^{\mathsf T}\) equals \(\epsilon\chi(e_1e_2^{\mathsf T}+e_2e_1^{\mathsf T})+\epsilon^2\chi^2e_1e_1^{\mathsf T}\), and that its norm is at most \(3\epsilon\).

6. Solutions

5.1. By the definition of \(a\), \[ \begin{aligned} &\log a(\tfrac18)-\log a(0)\\ &\quad=-\frac1{2/3-1/8}+\frac1{2/3}=-\frac{24}{13}+\frac32\\ &\quad=-\frac9{26}. \end{aligned} \]

5.2. Writing \(\operatorname{diag}(\alpha,\beta)=\alpha\,e_1e_1^{\mathsf T}+\beta\,e_2e_2^{\mathsf T}\), \[ \begin{aligned} &T_2\operatorname{diag}(\alpha,\beta)T_2^{\mathsf T}\\ &\quad=\alpha\,T_2e_1(T_2e_1)^{\mathsf T}+\beta\,T_2e_2(T_2e_2)^{\mathsf T} \end{aligned} \] with \(T_2e_1=H\), \(T_2e_2=e_1\). For \(t\ge\frac34\) and \(\tau=1-t\), \[ \begin{aligned} P&=b\,e_1e_1^{\mathsf T}+(1+\delta A(\tau))HH^{\mathsf T}\\ &=T_2\operatorname{diag}(1+\delta A(\tau),b)T_2^{\mathsf T}, \end{aligned} \] which is (1.6) in the frame \(T_2\).

5.3. For \(\ell=e^{-\tau}\), \(-\ell''/\ell=-1\): a hyperbolic cusp. For \(\ell=T-\tau\), \(\ell''=0\): the flat metric in polar coordinates with \(r=T-\tau\).

5.4. \(K=-\ell''/\ell>0\) where \(\ell''<0\), by continuity near \(\tau_0\). If \(g_\ell\) were negative somewhere, the collar would have positive curvature there, and Proposition 4.1 would fail.

5.5. Expand \((e_2+\epsilon\chi e_1)(e_2+\epsilon\chi e_1)^{\mathsf T}\). The norm of \(e_1e_2^{\mathsf T}+e_2e_1^{\mathsf T}\) is \(1\) and that of \(e_1e_1^{\mathsf T}\) is \(1\), so the norm is at most \(\epsilon+\epsilon^2\le3\epsilon\).

References