Analytic double fibrations and Bolker visibility — Examples and Counterexamples

The defining function $F(\theta ,s,x)=s-x\cdot \theta$ is real analytic and has ${\mathrm{\partial }}_{s}F=1$. Thus $dF$ is surjective at every point of $Z_{\mathrm{\Theta }}$, and [F6] makes $Z_{\mathrm{\Theta }}$ an analytic hypersurface of dimension $3$.

Use a local angular coordinate $\alpha$ on $S^1$, so ${\mathrm{\partial }}_{\alpha }\theta ={\theta }^{\perp }$. The defining function $F(\theta ,s,x)=s-x\cdot \theta$ has differential $dF=ds-(x\cdot {\theta }^{\perp })d\alpha -\theta \cdot dx\mathrm{.}$ By [F4], a conormal covector is $\tau dF$ for a nonzero scalar $\tau$. Matching its data components and object component with the sign convention [F3] gives ${\eta }_s=\tau ,{\eta }_{\alpha }=-\tau x\cdot {\theta }^{\perp },\xi =\tau \theta \mathrm{.}$ Thus any object covector in the relation is parallel to the measured normal $\theta$.

A tangent vector satisfies $\dot{s}-\dot{x}\cdot \theta -x\cdot \dot{\theta }=0$. Given any data tangent pair $(\dot{\theta },\dot{s})$, choose $\dot{x}=(\dot{s}-x\cdot \dot{\theta })\theta$; then ${\pi }_G$ has a tangent lift and is a submersion. Given any object tangent vector $\dot{x}$, choose $\dot{\theta }=0$, $\dot{s}=\dot{x}\cdot \theta$; hence ${\pi }_X$ is a submersion. Both maps are analytic restrictions of product projections and $4>3>2\ge 2$. The displayed analytic total density is nowhere zero in these incidence coordinates. For compact $K\subset {\mathbb{R}}^2$ and compact $L\subset \mathrm{\Theta }\times \mathbb{R}$, the preimage of $L$ under ${\pi }_G{\mathrm{\mid }}_{{\pi }_X^{-1}(K)}$ is closed in ${\mathrm{pr}}_1(L)\times K$, which is compact because ${\mathrm{pr}}_1(L)⋐\mathrm{\Theta }$. Thus the restricted projection is proper and [F5] applies.

If $(\theta ,s,\eta ;x_0,{\xi }_0)\in C_{Z_{\mathrm{\Theta }}}$, step 1.2 implies ${\xi }_0\mathrm{/}\mathrm{\mid }{\xi }_0\mathrm{\mid }=\theta$ or $-\theta$. Since $\mathrm{\Theta }=-\mathrm{\Theta }$, this would place ${\xi }_0\mathrm{/}\mathrm{\mid }{\xi }_0\mathrm{\mid }$ in $\mathrm{\Theta }$, contradicting the hypothesis. No such data covector exists.

Put $F(n,x)=n\cdot x$ on $S^2\times S^2$. At a point with $F=0$, the vector $x$ lies in $T_n{S}^2$ by [F5], and the tangent variation $(\dot{n},\dot{x})=(x,0)$ satisfies $dF(\dot{n},\dot{x})=x\cdot x=1$. Thus $dF\mathrm{\ne }0$; [F6] makes $Z=F^{-1}(0)$ an analytic hypersurface of dimension $3$ in the $4$-dimensional product.

A tangent vector to $Z$ satisfies $x\cdot \dot{n}+n\cdot \dot{x}=0$. Given any $\dot{n}\in T_n{S}^2$, set $\dot{x}=-(x\cdot \dot{n})n$; then $\dot{x}\in T_x{S}^2$ by [F5], and the tangent equation holds. Hence ${\pi }_G$ is a submersion. Given any $\dot{x}\in T_x{S}^2$, set $\dot{n}=-(n\cdot \dot{x})x$; the same calculation proves ${\pi }_X$ is a submersion. Both maps are analytic restrictions of product projections. Since $Z$ is compact, ${\pi }_G$ is proper on ${\pi }_X^{-1}(K)$ for every compact $K\subset X$. The density induced by the positive-definite analytic product round metric has strictly positive analytic coefficient in every local incidence chart. The dimensions satisfy $4>3>2\ge 2$, so [F1] applies.

The defining differential on tangent vectors to the product is $dF=(x^{\mathrm{♭}},n^{\mathrm{♭}})$ by the metric identification [F4]. Therefore by [F3] the relation at $(n,x)$ consists of $(n,\eta ;x,\xi )=(n,\tau x^{\mathrm{♭}};x,-\tau n^{\mathrm{♭}}),\tau \mathrm{\ne }0.$ At the antipodal point $-x$, choose multiplier $-\tau$. Its data covector is $(-\tau )(-x{)}^{\mathrm{♭}}=\tau x^{\mathrm{♭}}$, the same as at $x$, while the object point changes from $x$ to $-x$. Both are nonzero relation points. Hence the left projection is not injective, and [F2] implies failure of global Bolker.

Let $v=n\times x$, a unit tangent vector at $x$. For $s$ near zero put $n_s=\cos (s)n+\sin (s)v$, $v_s=n_s\times x$, and ${\gamma }_s(t)=\cos (t)x+\sin (t)v_s,0\le t\le \pi \mathrm{.}$ Each $n_s$ is a unit vector orthogonal to $x$, and $v_s$ is a unit vector orthogonal to $x$; hence ${\gamma }_s$ parametrizes a great circle at unit speed and is a geodesic by [F10]. Thus [F7] makes this a geodesic variation. Its endpoints are fixed: ${\gamma }_s(0)=x$ and ${\gamma }_s(\pi )=-x$. Differentiating in $s$ at zero gives $J(t)={{\mathrm{\partial }}_s{\gamma }_s(t)\mid }_{s=0}=\sin (t)(v\times x)=-\sin (t)n,$ because $(n\times x)\times x=-n$ when $n\cdot x=0$ and $\mathrm{\mid }x\mathrm{\mid }=1$. By [F11] this field is Jacobi and hence satisfies the Jacobi equation in [F8]. It is nonzero for $0<t<\pi$ and vanishes at both endpoints, so [F9] makes the endpoints conjugate. The proof uses no choice axiom; the sphere has no boundary, so this example lies outside the simple-surface hypotheses.

The defining function $F(\theta ,s,x)=s-x\cdot \theta$ is analytic and has ${\mathrm{\partial }}_{s}F=1$. Thus [F6] makes $Z=F^{-1}(0)$ an analytic hypersurface of dimension $3$.

In a local angle coordinate $\alpha$ with ${\mathrm{\partial }}_{\alpha }\theta ={\theta }^{\perp }$, the defining function has differential $dF=ds-(x\cdot {\theta }^{\perp })d\alpha -\theta \cdot dx\mathrm{.}$ By [F2], the canonical relation consists of $\xi =\tau \theta ,{\eta }_s=\tau ,{\eta }_{\theta }=-\tau x\cdot {\theta }^{\perp },\tau \mathrm{\ne }0.$ These signs follow by matching $(\eta ,-\xi )=\tau dF$.

A tangent vector satisfies $\dot{s}-\dot{x}\cdot \theta -x\cdot \dot{\theta }=0$. Given any data tangent pair $(\dot{\theta },\dot{s})$, choose $\dot{x}=(\dot{s}-x\cdot \dot{\theta })\theta$; this lifts it to $TZ$, so ${\pi }_G$ is a submersion. Given any object tangent vector $\dot{x}$, take $\dot{\theta }=0$ and $\dot{s}=\dot{x}\cdot \theta$, so ${\pi }_X$ is a submersion. Since ${\pi }_X^{-1}(K)$ is parameterized by the compact set $S^1\times K$ for every compact $K\subset {\mathbb{R}}^2$, the restriction to $G$ is proper. The specified analytic total density is nowhere zero. The dimensions satisfy $4>3>2\ge 2$, so [F5] applies.

For fixed $(\theta ,s,{\eta }_s,{\eta }_{\theta })$ in the image, $\tau ={\eta }_s\mathrm{\ne }0$ and $x=s\theta -\frac{{\eta }_{\theta }}{\tau }{\theta }^{\perp },\xi =\tau \theta \mathrm{.}$ Thus the left fiber is a singleton. To check the differential, let a tangent variation be in the kernel of the left projection. Then $\dot{\theta }=0$, $\dot{s}=0$, $\dot{\tau }={\dot{\eta }}_s=0$, and $0=\dot{s}=\dot{x}\cdot \theta ,0={\dot{\eta }}_{\theta }=-\tau \dot{x}\cdot {\theta }^{\perp }\mathrm{.}$ Since $\tau \mathrm{\ne }0$ and $(\theta ,{\theta }^{\perp })$ is a basis, $\dot{x}=0$ and $\dot{\xi }=0$. Hence the left projection is an injective immersion, so [F3] gives global Bolker.

For the specified object covector, $\theta =\xi \mathrm{/}\mathrm{\mid }\xi \mathrm{\mid }$ and $\tau =\mathrm{\mid }\xi \mathrm{\mid }$ satisfy $\xi =\tau \theta$. With $s$ and $\eta$ chosen in the statement, step 1.2 supplies the remaining relation equations. Therefore the data-side canonical pair exists for every nonzero $\xi$.

The contrapositive of [F4] shows that $(x,\xi )\in {\mathrm{WF}}_A(f)$ forces $(\theta ,s;{\eta }_{\theta },{\eta }_s)\in {\mathrm{WF}}_A(Rf)$. Its hypotheses hold by the analytic nonvanishing density, properness and full Bolker calculations in steps 1.1–2.2.