1. Properly and strictly convex

This section and the next review some well-known results concerning properly and strictly convex projective manifolds that are needed to prove the theorems. The results needed subsequently are (1.13)–(1.11) at the end of this section. The only mild innovation is that, to avoid appealing to results about word-hyperbolic groups, a more direct approach was taken to the proof of (1.11). The early sections of [Reference Cooper, Long and TillmannCLT15] greatly expand on the background in this section.

If $M$ is a manifold, the universal cover is $\pi _{M}:\widetilde M \to M$ , and if $g\in \pi _1M$ , then $\tau _g:\widetilde M\to \widetilde M$ is the covering transformation corresponding to $g$ . A geometry is a pair $(G,X)$ , where $G$ is a group that acts analytically and transitively on a manifold $X$ . A $(G,X)$ -structure on a manifold $M$ is determined by a development pair $(\textrm {dev},\rho )$ that consists of the holonomy $\rho \in {\textrm {Hom}}(\pi _1M,G)$ and the developing map $\textrm {dev}:\widetilde M\to X$ which is a local homeomorphism. The pair satisfies for all $x\in \widetilde M$ and $g\in \pi _1M$ that $\textrm {dev}(\tau _{g}x)=(\rho g)\textrm {dev} x$ .

In what follows $V={\mathbb{R}}^{n+1}$ , its dual vector space is $V^*={\textrm {Hom}}(V,{\mathbb{R}})$ , and ${\mathbb{R}}^{n+1}_0=V_0=V\setminus \{0\}$ . Projective space is ${\mathbb{P}} V=V_0/{\mathbb{R}}_0$ , and $[A]\in {\textrm {Aut}}({\mathbb{P}} V)=\textrm {PGL}(V)$ acts on ${\mathbb{P}} V$ by $[A][x]=[Ax]$ . Projective geometry is $({\textrm {Aut}}({\mathbb{P}} V),{\mathbb{P}} V)$ and is also written $({\textrm {Aut}}(\mathbb{RP}^n),\mathbb{RP}^n)$ . Radiant-affine geometry is $(\textrm {GL}(V),V_0)$ .

We use the notation ${\mathbb{R}}^+=(0,\infty )$ . Positive projective space is $\mathbb{RP}^n_+={\mathbb{P}}_+V=V_0/{\mathbb{R}}^+$ and $[x]_+=\{\lambda x:\lambda \gt 0\}$ for $x\in V_0$ . Sometimes we identify ${\mathbb{P}}_+V$ with the unit sphere $S^n\subset {\mathbb{R}}^{n+1}$ via $[x]_+\equiv x/\|x\|$ . The group ${\textrm {Aut}}({\mathbb{P}}_+V):=\textrm {SL}_{\pm }(V)\subset \textrm {GL}(V)$ is the subgroup with $\det =\pm 1$ . Positive projective geometry $({\textrm {Aut}}({\mathbb{P}}_+V),{\mathbb{P}}_+V)$ is the double cover of projective geometry. We will pass back and forth between projective geometry and positive projective geometry without mention, often omitting the term positive.

If $U$ is a vector subspace of $V$ , then ${\mathbb{P}} U$ is a projective subspace of ${\mathbb{P}} V$ . This is a (projective) line if $\dim U=2$ and a (projective) hyperplane if $\dim U=\dim V-1$ . The (projective) dual of $U$ is the projective subspace ${\mathbb{P}} U^0\subset {\mathbb{P}} V^*$ where $U^0=\{\phi \in V^* :\phi (U)=0\}$ . We use the same terminology in positive projective geometry. By lifting developing maps one obtains the following.

The frontier of a subset $X\subset Y$ is $\textrm {Fr} X=\textrm {cl}(X)\setminus \textrm {int}(X)$ , and the boundary is $\partial X=X\cap \textrm {Fr} X$ . A segment is a connected, proper subset of a projective line that contains more than one point. In what follows $\Omega \subset \mathbb{RP}^n$ . If $H$ is a hyperplane and $x\in H\cap \textrm {Fr}\Omega$ and $H\cap \textrm {int}\Omega =\emptyset$ , then $H$ is called a supporting hyperplane (to $\Omega$ ) at $x$ . The set $\Omega$ :

• • is convex if every pair of points in $\Omega$ is contained in a segment in $\Omega$ ;

• • is properly convex if it is convex, and $\textrm {cl}\Omega$ does not contain a projective line;

• • is strictly convex if it is properly convex and $\textrm {Fr}\Omega$ does not contain a segment;

• • is flat if it is convex and $\dim \Omega \lt n$ ;

• • is a convex domain if it is a properly convex open set;

• • is $C^1$ if for each $x\in \textrm {Fr}\Omega$ there is a unique supporting hyperplane at $x$ .

The same terms will be applied to a lift of $\Omega$ to ${\mathbb{R}\textrm {P}_+}^n$ . If $V=U\oplus W$ , then projection from ${\mathbb{P}} W$ onto ${\mathbb{P}} U$ is $\pi :{\mathbb{P}} V\setminus {\mathbb{P}} W\longrightarrow {\mathbb{P}} U$ given by $\pi [u+w]=[u]$ . Duality is the map that sends each point $\theta =[\phi ]\in {\mathbb{P}} V^*$ to the hyperplane $H_{\theta }={\mathbb{P}}\ker \phi \subset {\mathbb{P}} V$ . If $L$ is a line in ${\mathbb{P}} V^*$ the hyperplanes $H_{\theta }$ dual to the points $\theta \in L$ are called a pencil of hyperplanes. Then $Q=\cap H_{\theta }$ is the projective dual of $L$ and is called the core of the pencil. It is a codimension $2$ projective subspace.

In what follows $\Omega \subset \mathbb{RP}^n$ is a properly convex domain, and ${\textrm {Aut}}(\Omega )\subset {\textrm {Aut}}(\mathbb{RP}^n)$ is the subgroup that preserves $\Omega$ . The Hilbert metric $d_{\Omega }$ on $\Omega$ is defined as follows. If $\ell \subset \mathbb{RP}^n$ is a line and $\alpha =\Omega \cap \ell \ne \emptyset$ , then $\alpha$ is a proper segment in $\Omega$ and $\alpha =(a_-,a_+)$ with $a_{\pm }\in \textrm {Fr}\Omega$ . There is a projective isomorphism $f:\alpha \rightarrow {\mathbb{R}}^+$ and

\begin{equation*} d_{\Omega }(x,y)=\frac {1}{2}\left |\;\log \frac {f(x)}{f(y)}\;\right | \end{equation*}

for $x,y\in \alpha$ . It follows from the next result that $d_{\Omega }$ satisfies the triangle inequality, and thus is a metric. It is a Finsler metric: given by a norm on the tangent space. The factor of $1/2$ ensures that this gives the hyperbolic metric when $\Omega$ is a round disc. A geodesic in $\Omega$ is a curve whose length is the distance between its endpoints. It is immediate that ${\textrm {Aut}}(\Omega )$ acts by isometries of  $d_{\Omega }$ .

The interior of a triangle in the projective plane is a properly convex domain that is projectively equivalent to the positive quadrant in an affine patch ${\mathbb{R}}^2$ . A metric ball in this domain with center $p$ is a hexagon. The vertices of the hexagon lie on the lines through $p$ that contain a vertex of the triangle. A smooth curve in the interior of a triangle is a geodesic for the Hilbert metric if, and only if, the tangent line at each point on the curve intersects the same pair of sides of the triangle. Thus, in a properly convex domain, there may be many geodesics with the same endpoints.

Let $H_{\pm }$ be supporting hyperplanes for $\Omega$ at $a_{\pm }\in \textrm {Fr}\Omega$ , and $P=H_+\cap H_-$ . Projection from $P$ onto $\alpha$ is the map $\pi :\Omega \to \alpha =(a_-,a_+)$ given by $\pi [u+v]=[v]$ where $[u]\in P$ and $[v]\in \alpha$ . The choice of $H_{\pm }$ is unique only if $a_{\pm }$ are $C^1$ points. If $\Omega$ is strictly convex, then for each $x\in \Omega$ by (1.3)(iii) there is a unique point $\Pi (x)\in \alpha$ closest to $x$ , and $\Pi :\Omega \rightarrow \alpha$ is continuous and called the nearest point retraction. In general $\pi \ne \Pi$ .

A line in a projective manifold is flat connected $1$ -manifold. A projective manifold $M$ is convex if every pair of points in $M$ are contained in a line. The developing map might not be injective on a lift of a line. For example, every covering space of $\mathbb{RP}^1$ is convex. It is easy to see (cf. Figure 2) the following.

Figure 2. Distance between two lines near infinity.

A subset $\mathcal{C}\subset V_0$ is a cone if $t\cdot \mathcal{C}=\mathcal{C}$ for all $t\gt 0$ . The dual cone $\mathcal{C}^*=\textrm {int}\left (\{\phi \in V^*:\ \phi (\mathcal{C})\geqslant 0\}\right )$ is convex. If $\Omega \subset {\mathbb{P}}_+ V$ , then $\mathcal{C}\Omega$ is the cone $\{v\in V_0:\ [v]_+\in \Omega \}$ . If $\Omega$ is open and properly convex, then the dual domain $\Omega ^*={\mathbb{P}}(\mathcal{C}\Omega ^*)$ is open, and properly convex. Using the natural identification $V\cong V^{**}$ it is immediate that $(\Omega ^*)^*=\Omega$ .

If $\Gamma \subset {\textrm {Aut}}(\Omega )$ is a discrete and torsion-free subgroup, then $M=\Omega /\Gamma$ is a properly (respectively strictly) convex manifold if $\Omega$ is properly (respectively strictly) convex. Then $\widetilde M=\Omega$ and there is a development pair $(\textrm {dev},\rho )$ , where $\textrm {dev}$ is the identity map, and the holonomy $\rho :\pi _1M\to \Gamma$ is an isomorphism.

The tautological line bundle is the quotient map $\pi :V_0\rightarrow {\mathbb{P}} V$ . If $M$ is a projective manifold modelled on ${\mathbb{P}} V$ , then the pull-back of this line bundle by the developing map $\textrm {dev}:\widetilde M\rightarrow {\mathbb{P}} V$ is a line bundle over $\widetilde M$ . Covering transformations induce bundle isomorphisms, and the quotient is a line bundle over $M$ called the tautological line bundle. It is a trivial bundle, since the lines can be oriented to point away from $0$ .

This bundle has a natural structure as a radiant-affine manifold. In the case that $M=\Omega /\Gamma$ is properly convex, the holonomy lifts to $\widetilde \Gamma \subset \textrm {SL}_{\pm } V$ . Then this bundle over $\widetilde M$ is naturally identified with $\mathcal{C}\Omega$ and the quotient $\mathcal{C}\Omega /\widetilde \Gamma$ is a radiant-affine manifold (3.3) that can be naturally identified with the tautological line bundle over $M$ . This plays a key role in the proof of the Open Theorem. If $X$ is a subset of a metric space $Y$ , the $r$ -neighborhood of $X$ is

\begin{equation*}N_r(X)=N(X,r)=\{y\in Y:\ d(y,X)\leqslant r\}.\end{equation*}

We now use a compactness argument to show that if $\Omega$ is the universal cover of a closed strictly convex manifold, then projection from a projective subspace onto a geodesic in $\Omega$ is uniformly distance decreasing in the following sense.

The following is due to Benoist [Reference BenoistBen04]. A triangle is a disc $\Delta$ in a projective plane bounded by three segments. If $\Delta \subset \textrm {cl}\Omega$ and $\Delta \cap \textrm {Fr}\Omega =\partial \Delta$ , then $\Delta$ is called a properly embedded triangle, or PET. A properly convex set $\Omega$ has thin triangles if there is $\delta \gt 0$ such that for every triangle $T$ in $\Omega$ , each side of $T$ is contained in a $\delta$ -neighborhood of the union of the other two sides with respect to the Hilbert metric.

It follows that in dimension $2$ either $\Omega$ is a triangle (and $M$ is a torus or Klein bottle) or else is strictly convex. Using a basic fact from geometric group theory, it follows that a properly convex manifold (of any dimension) is strictly convex if and only if $\pi _1M$ has thin triangles. This is an algebraic characterization of strictly convex. However, we will give a direct geometric argument.

Given $K\geqslant 1$ and $L\geqslant 0$ , a $(K,L)$ -quasi-isometric embedding is a map $f:X\to Y$ between metric spaces $(X,d_X)$ and $(Y,d_Y)$ such that

\begin{equation*}K^{-1}(d_X(x,x^{\prime})-L)\;\leqslant \; d_Y(fx,fx^{\prime})\;\leqslant \; K\;d_X(x,x^{\prime})+L \end{equation*}

for all $x,x^{\prime}\in X.$ If $X=[a,b]\subset {\mathbb{R}}$ , then $f$ is a quasi-geodesic. The map $f$ is a quasi-isometry, or QI, if also $Y\subset N(fX,L)$ .

If $g:Y\to Z$ is a $(K^{\prime},L^{\prime})$ -QI, then $g\circ f$ is a $(KK^{\prime},K^{\prime}L+L^{\prime})$ -QI. In particular, if $g$ is a QI and $f$ is a quasi-geodesic, then $g\circ f$ is a quasi-geodesic with QI constants that only depend on those of $f$ and $g$ .

If $G$ is a finitely generated group, then a choice of finite generating set gives a word metric on  $G$ . A different choice of generating set gives a bi-Lipschitz equivalent metric. The Švarc–Milnor lemma in this setting is

A metric space $(Y,d_Y)$ is ML if it satisfies the Morse Lemma, i.e. for all $(K,L)$ there is $S=S(K,L)\gt 0$ , called a tracking constant, such that if $\alpha$ and $\beta$ are $(K,L)$ -quasi-geodesics in $Y$ with the same endpoints, then $\alpha \subset N_{S}(\beta )$ and $\beta \subset N_{S}(\alpha )$ . Since quasi-geodesics are sent to quasi-geodesics by quasi-isometries, it follows that the property of ML is preserved by QI.

Clearly ${\mathbb{R}}^2$ is not ML, and any norm on ${\mathbb{R}}^2$ is QI to the standard norm, so ${\mathbb{R}}^2$ with any norm is not ML. A geodesic in the Cayley graph of $G$ picks out a sequence in $G$ that is a quasi-geodesic. It follows from (1.8) that whether or not a closed properly convex manifold is ML only depends on $\pi _1M$ .

Proof. If $\Omega$ is not strictly convex, then by (1.7) it contains a PET $\Delta$ . The metric $d_{\Omega }$ restricted to $\Delta$ is isometric to a norm on ${\mathbb{R}}^2$ . Hence, $(\Omega ,d_{\Omega })$ is not ML.

Figure 3. Quasi-geodesics.

Now suppose $\Omega$ is strictly convex. A quasi-geodesic is nice if the image consists of finitely many line segments and each line segment is parameterized by arc length. Every $(K,L)$ -quasi-geodesic $\alpha$ in $\Omega$ is approximated by a nice $(K^{\prime},L^{\prime})$ -quasi-geodesic $\overline {\alpha }$ in the sense that $\alpha \subset N(\overline {\alpha },D)$ and $\overline {\alpha }\subset N(\alpha ,D)$ and $(D, K^{\prime},L^{\prime})$ only depends on $(K,L)$ . Thus it suffices to prove there is a tracking constant for nice quasi-geodesics.

Refer to Figure 3. Let $\alpha ^{\prime}$ be a nice $(K,L)$ -quasi-geodesic with endpoints $p$ and $q$ . Let $\beta ^{\prime}$ be the line segment with the same endpoints as $\alpha ^{\prime}$ . Let $R=R(2K)$ be given by (1.6) and set $S^{\prime}=R+4RK+K+L$ . We show that $S=2KS^{\prime}+2L+2S^{\prime}$ is a tracking constant. Let $\beta ^+$ be the geodesic in $\Omega$ that contains $\beta ^{\prime}$ and $\pi :\Omega \rightarrow \beta ^+$ be a projection.

Claim 1: $\alpha ^{\prime}\subset N(\beta ^+,S^{\prime})$ .

Claim 2: $\alpha ^{\prime}\subset N(\beta ^{\prime},S/2)$ and $\beta ^{\prime}\subset N(\alpha ^{\prime},S/2)$ .

Assuming the claims, if $\gamma ^{\prime}$ is another nice $(K,L)$ -quasi-geodesic with endpoints $p$ and $q$ , then

\begin{equation*}\beta ^{\prime}\subset N(\gamma ^{\prime},S/2)\quad \textrm {so}\quad \alpha ^{\prime}\subset N(\beta ^{\prime},S/2)\subset N(N(\gamma ^{\prime},S/2),S/2)=N(\gamma ^{\prime},S),\end{equation*}

which proves that $S$ is a tracking constant. Hence, $(\Omega ,d_{\Omega })$ is ML.

For Claim 1, suppose that $\alpha$ is the closure of a component of $\alpha ^{\prime}\setminus N(\beta ^+,R)$ . The endpoints $x$ and $y$ of $\alpha$ satisfy $d(x,\beta ^+)=d(y,\beta ^+)=R$ . Let $x^{\prime},y^{\prime}\in \beta ^+$ be chosen so that $d(x,x^{\prime})=R=d(y,y^{\prime})$ and let $\beta =[\pi x,\pi y]\subset \beta ^+$ . By (1.6),

\begin{equation*}{\textrm {length}}(\beta )\leqslant 1+{\textrm {length}}(\alpha )/2K\end{equation*}

by definition of $R$ . Since $\pi$ is distance non-increasing $d(x^{\prime},\pi x)=d(\pi x^{\prime},\pi x)\leqslant d(x^{\prime}, x)=R$ , and similarly $d(y^{\prime},\pi y)\leqslant R$ . Thus

\begin{equation*}d(x,y)\leqslant d(x,x^{\prime})+d(x^{\prime},\pi x)+d(\pi x,\pi y)+d(\pi y,y^{\prime})+d(y^{\prime},y)\leqslant 4R+{\textrm {length}}(\beta ).\end{equation*}

Since $\alpha$ is parametrized by arc length, the domain of $\alpha$ is ${\textrm {length}}(\alpha )$ , Since $\alpha$ is a $(K,L)$ -quasi-geodesic, we have

\begin{equation*}\begin{array}{lrcl} & K^{-1}({\textrm {length}}(\alpha )-L)& \leqslant & d(x,y)\\ \Rightarrow & {\textrm {length}}(\alpha ) &\leqslant & K\cdot d(x,y)+L \\ & & \leqslant & K(4R +{\textrm {length}}(\beta ))+L\\ & & \leqslant & K(4R+1+{\textrm {length}}(\alpha )/2K)+L\\ & & = &4RK+K+L+{\textrm {length}}(\alpha )/2\\ \Rightarrow & (1/2){\textrm {length}}(\alpha )&\leqslant & 4RK+K+L. \end{array}\end{equation*}

Since the endpoints of $\alpha$ are in $N(\beta ^+,R)$ , it follows that

\begin{equation*}\alpha \subset N(\ N(\beta ^+,R)\ ,\ (1/2){\textrm {length}}(\alpha ))=N(\beta ^+,R+(1/2){\textrm {length}}(\alpha )).\end{equation*}

Now

\begin{equation*}R+(1/2){\textrm {length}}(\alpha )\leqslant R+4RK+K+L=S^{\prime}.\end{equation*}

This proves Claim 1.

For Claim 2, let $\Pi :\alpha ^{\prime}\rightarrow \beta ^+$ be the nearest point retraction. This map is continuous, and $p,q$ are both in the image, so the image of $\Pi \circ \alpha ^{\prime}$ contains $\beta ^{\prime}$ . If $x^{\prime}\in \beta ^{\prime}$ there is $y^{\prime}\in \alpha ^{\prime}$ with $\Pi y^{\prime}=x^{\prime}$ . Then $d(\beta ^+,y^{\prime})=d(x^{\prime},y^{\prime})$ , and by Claim 1 $d(\beta ^+,y^{\prime})\leqslant S^{\prime}$ so $d(x^{\prime},y^{\prime})\leqslant S^{\prime}$ thus

\begin{equation*} \beta ^{\prime}\subset N(\alpha ^{\prime},S^{\prime})\subset N(\alpha ^{\prime},S/2)\quad {\rm and} \quad \eta =\Pi ^{-1}(\beta ^{\prime})\subset N(\beta ^{\prime},S^{\prime})\subset N(\beta ^{\prime},S/2).\end{equation*}

Let $\delta \subset \alpha ^{\prime}$ be the closure of a component of $\alpha ^{\prime}\setminus \eta$ , thus $\delta$ is an arc in the interior of $\alpha ^{\prime}$ . Let $r,s$ be the endpoints of $\delta$ . Then $\Pi (r)=\Pi (s)\in \{p,q\}$ . By Claim 1,

\begin{equation*}d(r,s)\leqslant d(r,\Pi (r))+d(\Pi (r),\Pi (s))+d(\Pi (s),s)\leqslant S^{\prime}+0+S^{\prime}=2S^{\prime}.\end{equation*}

Since $\delta$ is a nice $(K,L)$ -quasi-geodesic,

\begin{equation*}{\textrm {length}}(\delta )\leqslant K\cdot d(r,s) +L\leqslant K\cdot 2S^{\prime}+L.\end{equation*}

By Claim 1, the endpoints of $\delta$ are distance at most $S^{\prime}$ from $\beta ^{\prime}$ so

\begin{equation*}\delta \subset N(N(\beta ^{\prime},S^{\prime}),(1/2){\textrm {length}}(\delta ))\leqslant N(\beta ^{\prime},S^{\prime}+(1/2)(K\cdot 2S^{\prime}+L))\subset N(\beta ^{\prime},S/2).\end{equation*}

Since $\delta \cup \eta \subset N(\beta ^{\prime},S/2)$ it follows that $\alpha ^{\prime}\subset N(\beta ^{\prime},S/2)$ , proving Claim 2.

In the proof of the Closed Theorem, we construct a properly convex manifold with the correct holonomy. In order to know this manifold is closed and strictly convex we use the pair of results (1.10) and (1.11). A properly convex manifold has contractible universal cover. Whitehead’s theorem (that a weak homotopy equivalence between CW complexes is a homotopy equivalence) implies that if $M$ and $M^{\prime}$ are properly convex and $\pi _1M\cong \pi _1M^{\prime}$ , then $M$ and $M^{\prime}$ are homotopy equivalent. This is key in the following result.

The remaining discussion in this sections comes logically after Section 2 since it depends on (2.6), but it fits better into the narrative here. We let the reader choose their preferred order.

The displacement distance of $\gamma \in {\textrm {Aut}}(\Omega )$ is $t(\gamma )=\inf \{d_{\Omega }(x,\gamma x): x\in \Omega \}$ . The element $\gamma$ is called hyperbolic if $t(\gamma )\gt 0$ . It is elliptic if it fixes a point in $\Omega$ and parabolic if it has no fixed point in $\Omega$ but $t(\gamma )=0$ . By [Reference Cooper, Long and TillmannCLT15, Proposition 2.1] we have that $t(\gamma )=\log |\lambda /\mu |$ , where $\lambda$ and $\mu$ are the eigenvalues of $t(\gamma )$ of largest and smallest modulus. We will not make use of elliptics or parabolics in this paper.

Lemma 1.12 (Hyperbolics). Suppose $\Omega \subset {\mathbb{P}} V$ and $M=\Omega /\Gamma$ is a strictly convex closed manifold and $1\ne \gamma \in \Gamma$ . Then $\gamma$ is hyperbolic and there are $a_{\pm }\in \textrm {Fr}(\Omega )$ such that for all $x\in {\mathbb{P}} V\setminus (H_+\cup H_-)$ we have

\begin{equation*}\lim _{n\to \pm \infty }\gamma ^nx=a_{\pm },\end{equation*}

where $H_{\pm }$ is the supporting hyperplane to $\Omega$ at $a_{\pm }$ .

The point $a_+$ is called the attracting, and $a_-$ is the repelling, fixed point of $\gamma$ , and $(a_-,a_+)\subset \Omega$ is called the axis of $\gamma$ . This axis is the only proper segment in $\Omega$ preserved by $\gamma$ . The attracting fixed point of $\gamma ^{-1}$ is the repelling fixed point of $\gamma$ . The following result is used in our proof of Chuckrow’s theorem, that is needed for the Closed Theorem.

Two projective structures on the same manifold are equivalent if there is a projective isomorphism of one onto the other that is homotopic to the identity (rather than isotopic to the identity). For closed hyperbolic manifolds of dimension at most $3$ , homotopy implies isotopy [Reference EpsteinEps66, Reference GabaiGab01]. The next two results imply that the holonomy of a strictly convex manifold determines the domain $\Omega$ , and hence determines the projective structure, up to equivalence. However, given the holonomy, there may be two developing maps that are homotopic but not isotopic. Thus the space of equivalence classes of strictly convex projective structures on a closed manifold can be identified with a subset of the space of representations modulo conjugacy.

2. Convex cones

This section is based on work of Vinberg, as simplified by Goldman. Write $V={\mathbb{R}}^{n+1}$ . If $\Omega$ is properly convex there is a nice convex hypersurface in the cone $\mathcal{C}\Omega$ that is preserved by every projective automorphism of $\mathcal{C}\Omega$ . This is a generalization of the hyperboloid model of hyperbolic space that sits inside the lightcone. In general there is a choice of such surfaces. One is Vinberg’s characteristic surface, and another is an affine sphere. The characteristic surface is defined below using a simple geometric condition. One consequence of this is (2.8), which provides an analogue of the centroid for properly convex subsets of the sphere. The purpose of this section is to prove the following and derive some consequences.

In the special case that $\mathcal{C}$ is the cone on a round ball, then $\partial \mathcal{D}$ is the hyperboloid model of hyperbolic space. Fix an inner product $\langle \;\cdot \;,\;\cdot \;\rangle$ on $V$ . This determines a norm on $V$ , and induces a Riemannian metric and associated volume form on every smooth submanifold of $V$ .

The centroid of a bounded convex set $K$ in $V$ is the point $\mu (K)$ in $K$ given by

\begin{equation*}\mu (K)=\left .\int _K x\ \textrm {dvol}_{K_x}\right /\int _K \textrm {dvol}_{K_x}.\end{equation*}

Here, $\dim K\leqslant \dim V$ and $\textrm {dvol}_K$ is the induced volume form on $K$ . The centroid is independent of the inner product.

Figure 4. Here, $T\cap \mathcal{D}$ is the centroid of $T\cap \mathcal{C}$ .

Let $S^n=\{x\in V:\ \|x\|^2=1\}$ . Suppose $\Omega$ is an open, properly convex subset of $S^n$ and $\mathcal{C}=\mathcal{C}\Omega$ is the corresponding cone in $V$ . Given $0\ne \phi \in V^*$ , the set $\phi ^{-1}(1)$ is an affine hyperplane in $V$ . If $\phi \in \mathcal{C}^*$ , then $\phi (\mathcal{C})\gt 0$ , so $\mathcal{C}_{\phi }=\{x\in \mathcal{C}: \phi (x)=1\}$ is a bounded subset of this hyperplane that separates $\mathcal{C}$ . The subset of $\mathcal{C}$ below this hyperplane is $\mathcal{C}\cap \phi ^{-1}(0,1]$ , and has finite volume and boundary $\mathcal{C}_{\phi }$ . The centroid of $\mathcal{C}_{\phi }$ is a point $\mu (\mathcal{C}_{\phi })\in \mathcal{C}$ . Define $\Theta (\phi )=\mu (\mathcal{C}_{\phi })$ . We will show this is a homeomorphism $\Theta :\mathcal{C}^*\rightarrow \mathcal{C}$ .

The volume function $\mathcal{V}:\mathcal{C}^*\to {\mathbb{R}}$ is defined by

(1) \begin{align} \mathcal{V}(\phi )=\textrm {vol}(\mathcal{C}\cap \phi ^{-1}(0,1])=\int _{\mathcal{C}\cap \phi ^{-1}(0,1]}\textrm {dvol} .\end{align}

Let $\pi :\mathcal{C}_{\phi }\to \Omega =S^n\cap \mathcal{C}$ be the radial projection $\pi (x)=x/\|x\|$ . Let $\textrm {dB}$ be the induced volume form on $S^n$ . We compute this integral in polar coordinates, so $\textrm {dvol}=r^{n} \textrm {dr}\wedge \pi ^*\textrm {dB}$ .

Given $\phi \in \mathcal{C}^*$ , there is $v\in V$ such that $\phi (x)=\langle x,v\rangle$ for all $x\in V$ . Let $y=\pi (x)$ . If $\phi (x)=1$ , then

(2) \begin{align} r=r(x)=\|x\|=\|x\|/\|\pi x\|=\phi (x)/\phi (\pi x)=\langle y,v\rangle ^{-1}\quad \textrm {and}\quad x=\langle y,v\rangle ^{-1}y .\end{align}

Using polar coordinates

(3) \begin{align} \mathcal{V}(\phi )=\int _{C_{\phi }\cap \phi ^{-1}(0,1]}r^n\textrm {dr}\wedge \pi ^*\textrm {dB}= \int _{\Omega }\left (\int _0^{\langle y,v\rangle ^{-1}} r^n \textrm {dr}\right )\textrm {dB}_y =(n+1)^{-1}\int _{\Omega }\langle y,v\rangle ^{-n-1}\; \textrm {dB}_y. \end{align}

For $q\in \mathcal{C}$ , the set $\mathcal{C}^*_q=\{\phi \in \mathcal{C}^*:\ \phi (q)=1\ \}$ is the intersection of a hyperplane in $V^*$ with $\mathcal{C}^*$ , and has compact closure. Elements of $\mathcal{C}^*_q$ correspond to hyperplanes in $V_0$ that contain $q$ and separate $\mathcal{C}$ .

Proof. We use the inner product to identify $V$ with $V^*$ so that $\mathcal{C}^*$ is a subset of $V$ and

\begin{equation*}\mathcal{C}_q^*=\{x\in \mathcal{C}^*:\langle x,q\rangle=1\}.\end{equation*}

We also regard $\mathcal{V}(\phi )$ as the function $\mathcal{W}(v)$ given by the right-hand side of (3) and prove corresponding statements for $\mathcal{W}$ .

For fixed $y$ , the integrand $f(v)=\langle y,v\rangle ^{-n-1}$ in (3) is a smooth and convex function of $v$ . It follows that $\mathcal{W}(v)$ is a smooth, strictly convex function of $v$ . This proves (i).

If $0\ne \psi \in \textrm {Fr}\mathcal{C}^*$ , then there is $0\ne x\in \textrm {Fr}\mathcal{C}$ with $\psi (x)=0$ . Thus ${\mathbb{R}}^+\cdot x\subset \textrm {cl}\mathcal{C}_{\psi }$ , so $\mathcal{C}_{\psi }$ is not compact. Moreover $\mathcal{C}_{\psi }$ is convex and has non-empty interior, so $\textrm {vol}(\mathcal{C}_{\psi })=\infty$ . It easily follows that $\mathcal{V}(\phi )\to \infty$ as $\phi \to \psi$ . This proves (ii), and (iii) follows from (3). It follows from convexity and (ii) that $\mathcal{V}|\mathcal{C}^*_q$ has a unique critical point, and it is a minimum.

The gradient of $\mathcal{W}(v)$ is

(4) \begin{align} \nabla \mathcal{W} =-\int _{\Omega } \langle y,v\rangle ^{-n-2} y\;\textrm {dB}_y. \end{align}

From the definition of $\mathcal{C}^*_q$ it follows that the condition for a critical point is that $\nabla \mathcal{W}\in {\mathbb{R}}\cdot q$ . We use radial projection $\pi$ to change variables in (4) to obtain (6).

Refer to Figure 5. Let $v\in V$ be dual to $\phi$ , so $\phi (x)=\langle x,v\rangle$ . Thus $\phi (v\|v\|^{-2})=1$ and $v\|v\|^{-2}$ is the point on $\phi ^{-1}(1)$ closest to $0$ . Thus the distance of $\phi ^{-1}(1)$ from $0$ is $\|v\|^{-1}$ . Given $x\in \mathcal{C}_{\phi }$ , then $1=\phi (x)=\langle x,v\rangle$ . Set $y=\pi (x)$ , then $\|y\|=1$ , and define $\cos \theta =\langle y,v\rangle /\|v\|$ . Since $\langle x,v\rangle=1$ it follows that $x=y/\langle y,v\rangle$ and so $\|x\|=1/\langle y,v\rangle$ .

Let $S_x\subset {\mathbb{R}}^{n+1}$ denote the sphere with center $0$ and radius $\|x\|$ . The volume form on $S_x$ is $d_{S_x}=\|x\|^{n}\pi _1^*dB_y$ , where $\pi _1:S_x\rightarrow S^n$ is radial projection. Let $d A_x$ be the volume element on $\mathcal{C}_{\phi }$ . Then $dA_x=(\cos \theta )^{-1}\pi _2^*d_{S_x}$ , where $\pi _2:\mathcal{C}_{\phi }\rightarrow S_x$ is radial projection. Then $\pi =\pi _1\circ \pi _2:\mathcal{C}_{\phi }\rightarrow S^n$ is radial projection, and combining gives

(5) \begin{align} d A_x=(\cos \theta )^{-1}\|x\|^{n}\; \pi ^*d B_y= \|v\|\langle y,v\rangle ^{-n-1}\; \pi ^*d B_y .\end{align}

It follows from (4) and (5) that

(6) \begin{align} \nabla \mathcal{W} =-\|v\|^{-1}\int _{\mathcal{C}_{\phi }}x \; \textrm {dA}_x=-\|v\|^{-1}\mu (\mathcal{C}_{\phi })\int _{\mathcal{C}_{\phi }} \textrm {dA}_x .\end{align}

Since the condition for a critical point is $\nabla \mathcal{W}\in {\mathbb{R}}\cdot q$ , this happens when

\begin{equation*}\mu (\mathcal{C}_{\phi })\in {\mathbb{R}}\cdot q.\end{equation*}

The centroid $\mu (\mathcal{C}_{\phi })$ is in $\mathcal{C}_{\phi }$ , and $q\in \mathcal{C}_{\phi }$ . Hence at the minimum we have $\mu (\mathcal{C}_{\phi })=q$ , proving (iv).

Figure 5. Volume forms on $\Omega \subset S^n$ and $\mathcal{C}_{\phi }$ .

Proof of Theorem 2.1. Clearly $\mathcal{D}$ is a closed convex subset of $V$ . Given $q\in \mathcal{C}$ , there is $\phi \in \mathcal{C}^*$ given by (2.2)(iv). Then by (2.2)(iii) there is $t\gt 0$ such that $\mathcal{V}(t\cdot \phi )=1$ . Replace $\phi$ by $t\cdot \phi$ and replace $q$ by $t^{-1}q$ . Then $\mathcal{V}(\phi )=1$ and the hyperplane $T_{\phi }=\phi ^{-1}(1)$ contains $q$ . The half-space $H={\phi }^{-1}[1,\infty )$ has the property that $\textrm {vol}(\mathcal{C}\setminus H)=1$ , so $\mathcal{D}\subset H$ .

Suppose that $\mathcal{V}(\psi )=1$ and $\psi \ne \phi$ , then $\lambda =\psi (q)\gt 1$ since otherwise if $\psi ^{\prime}=\lambda ^{-1}\psi \in \mathcal{C}^*_q$ and $\mathcal{V}(\psi ^{\prime})=\lambda ^{n+1}\leqslant 1$ . This contradicts that $\phi$ is the unique minimum of $\mathcal{V}|\mathcal{C}^*$ . Hence ${\psi }(q)\gt 1$ so $q\in \partial \mathcal{D}$ and $T_{\phi }$ is a supporting hyperplane to $\mathcal{D}$ at $q$ . Also $q=T_{\phi }\cap \mathcal{D}$ is the centroid of $\mathcal{C}_{\phi }$ by (iv). This proves (2) and (5). It also follows that $\mathcal{D}\cap ({\mathbb{R}}^+\cdot q)=[1,\infty )\cdot q$ , which proves (3).

Figure 6. Approaching the frontier.

Suppose that $T$ is any supporting hyperplane at $q$ . Then there is $\psi \in V^*$ with $T=\psi ^{-1}(1)$ and $\psi (\mathcal{D})\geqslant 1$ . The same reasoning shows that $\phi =\psi$ , thus $T_{\phi }$ is the unique supporting hyperplane at $q$ , which proves (6). Now (4) follows from the fact that $\textrm {SL}(\mathcal{C})$ preserves volume.

It only remains to prove (1). Refer to Figure 6. Since $\mathcal{D}$ is convex, it suffices to prove $\mathcal{D}$ contains no point in the frontier of $\mathcal{C}$ . Suppose $x\in \mathcal{D}\cap \textrm {Fr}\mathcal{C}$ . Then $x\ne 0$ by definition of $\mathcal{D}$ . Choose $y\in \mathcal{C}$ and let $U\subset V$ be the two-dimensional vector subspace that contains both $x$ and $y$ . Let $x_n$ be a sequence in $(x,y)$ that converges to $x$ . When $x_n$ is close to $x$ , there is a half-space $H^{\prime}_n$ with $x_n\in \partial H^{\prime}_n$ and $\textrm {vol}(\mathcal{C}\setminus H^{\prime}_n)$ is very small. Thus if $\phi _n\in \mathcal{C}^*_{x_n}$ is as given by (iv), then $\epsilon _n=\mathcal{V}(\phi _n)\to 0$ as $n\to \infty$ . Since $x$ is above $\partial H_n=\phi _n^{-1}(1)$ it follows that $y$ is below $y_n=({\mathbb{R}}\cdot y)\cap \partial H_n$ .

Let $\psi _n=\epsilon _n^{1/(n+1)}\phi _n$ , then by (2.2)(iii), we have $\mathcal{V}(\psi _n)=1$ . Then $\mathcal{D}\subset \psi _n^{-1}[1,\infty )$ . The hyperplane $T_n=\psi _n^{-1}(1)$ intersects ${\mathbb{R}}\cdot y$ at $z_n=\epsilon _n^{-1/(n+1)}y_n$ . Since $\|y_n\|\gt \|y\|$ it follows that $z_n\to \infty$ as $n\to \infty$ . Let $r_n$ be the distance of $z_n$ from $\partial \mathcal{C}$ . Then $r_n\to \infty$ . Half the ball of radius $r_n$ center at $z_n$ is below $T_n$ and in $\mathcal{C}$ . But this volume is at most $1$ . This contradicts the existence of $x$ , proving (1).

The dual action of $A\in \textrm {SL} _{\pm }V$ on ${\mathbb{P}} V^*$ is given by $A[\phi ]=[\phi \circ A^{-1}]$ . If $\Gamma \subset \textrm {SL}_{\pm } V$ and preserves $\Omega$ , then the dual action of $\Gamma$ preserves $\Omega ^*$ . It is clear that $\Theta$ is equivariant with respect to these actions. The following directly follows from (1.5) and (2.3).

If $M=\Omega /\Gamma$ , then $M$ is called $C^1$ if $\Omega$ is $C^1$ . From (1.5) we have the following.

The centroid of a bounded open convex set $\Omega \subset {\mathbb R}^n$ is a distinguished point in $\Omega$ . We wish to define something similar for certain subsets of the sphere $S^n=\{x\in {\mathbb{R}}^{n+1}:\|x\|=1\}$ . Imagine a room that contains a transparent globe close to one wall, with a light source at the center of the globe. The shadow of Belgium appears on the wall. You can rotate the globe so that the centroid of this shadow is the point $p$ on the wall closest to the center of the globe. The point on the globe that projects to $p$ is called the (spherical) center of Belgium.

The open hemisphere that is the $\pi /2$ neighborhood of $y\in S^n$ is $U_y=\{x\in S^n: \langle x,y\rangle \gt 0\}$ . Radial projection $\pi _y:U_y\longrightarrow {\textrm {T}_y}S^n$ from the origin onto the tangent space to ${\mathbb{R}\textrm {P}_+}^n$ at $y$ is given by

\begin{equation*}\pi _y(x)=\frac {x-\langle x,y\rangle y}{\langle x,y\rangle },\end{equation*}

where we identify $S^n$ with ${\mathbb{R}\textrm {P}_+}^n$ .

If $A\in {\textrm {O}}(n+1),$ then $A$ is an isometry of the inner product and so $(A\Omega )^*=A(\Omega ^*)$ , and $Ay$ is a center of $A\Omega$ if $y$ is a center of $\Omega$ . The following property of $\partial \mathcal{D}$ is important for the proof of the Closed Theorem.

3. Open

If $M$ is a closed $(n-1)$ -manifold, then $\pi _1M$ is finitely generated by some set of elements $g_1,\cdots ,g_k$ . Now, $\rho \in \textrm {Rep}(M)$ is uniquely determined by the $k$ elements $\rho (g_i)\in \textrm {PGL}(n,{\mathbb{R}})$ . The Veronese embedding embeds real-projective space (and thus $\textrm {PGL}(n,{\mathbb{R}})$ ) into a Euclidean space. This collection of elements can then be thought of as one point in a finite-dimensional real vector space $V$ . This gives an injective map of $\textrm {Rep}(M)$ into $V$ . The Euclidean topology on $\textrm {Rep}(M)$ is the subspace topology (of the standard topology on $V$ ) on the image of this map. In this topology $\rho$ and $\rho ^{\prime}$ are close if there are matrix representatives of $\rho (g_i)$ and $\rho ^{\prime}(g_i)$ that are close.

Proof. Let $\widetilde {M}$ be the universal cover of $M$ . We identify $\pi _1M$ with the group of covering transformations of $\widetilde M$ . Choose a triangulation $\mathcal{T}_M$ of $M$ . Let $\mathcal{T}$ be the lifted triangulation on  $\widetilde M$ .

Suppose that $\rho \in \textrm {Rep}(M)$ and there is a collection of maps

\begin{equation*}f_{\sigma }:\sigma \rightarrow \mathbb{RP}^n,\end{equation*}

one for each simplex $\sigma$ in $\mathcal{T}$ , satisfying the following properties.

1. (i) Equivariance. If $\sigma ^{\prime}=g\cdot \sigma$ for some $g\in \pi _1M$ , then

   \begin{equation*}f_{\sigma ^{\prime}}=\rho (g)\circ f_{\sigma }.\end{equation*}

2. (ii) Compatibility. If $\tau$ is a face of $\sigma$ , then

   \begin{equation*}f_{\tau }=f_{\sigma }|\tau .\end{equation*}

Compatibility implies there is a map $F:\widetilde {M}\longrightarrow \mathbb{RP}^n$ such that $F|\sigma =f_{\sigma }$ . Equivariance implies

\begin{equation*}F(g\cdot x)=\rho (g)(x).\end{equation*}

Suppose that $\textrm {dev}_0:\widetilde {M}\rightarrow \mathbb{RP}^n$ is the developing map, and $\rho _0\in \textrm {Rep}(M)$ is the holonomy of some real-projective structure on $M$ . If $\rho =\rho _0$ , then the collection $f_{\sigma }=\textrm {dev}_0|\sigma$ satisfies the required properties and $F=\textrm {dev}_0$ .

Choose the triangulation $\mathcal{T}_M$ of $M$ to be by projective simplices. We show below that if $\rho$ is close enough to $\rho _0$ , then the maps $f_{\sigma }$ may be chosen to be close to $\textrm {dev}_0|\sigma$ on any finite set of simplices. If this finite collection contains a neighborhood, $U$ , of a fundamental domain $D$ , this will ensure $F|U$ is a local homeomorphism. By equivariance $F$ is a local homeomorphism everywhere. Then $F$ is a developing map for $\rho$ , and this defines a projective structure on $M$ with holonomy $\rho$ ,

It suffices to define $f_{\sigma }$ on one simplex in each $\pi _1M$ orbit. Then the equivariance property defines the maps on the rest of the orbit. In order to ensure compatibility, we define the maps in order of increasing dimension of simplices. At each stage we have a well-defined equivariant map $F_k:\mathcal{T}^k\rightarrow \mathbb{RP}^n$ defined on the $k$ -skeleton $\mathcal{T}^k$ of $\mathcal{T}$ . The extension process is to choose one $(k+1)$ -simplex $\sigma$ in each $\pi _1M$ orbit, and define $f_{\sigma }$ to be a homeomorphism onto a projective simplex that has boundary $F_k(\partial \sigma )$ . By choosing $\rho$ close enough to $\rho _0$ we can arrange that the projective simplex $f_{\sigma }(\sigma )$ is non-degenerate and close to $\textrm {dev}_0(\sigma )$ . The induction starts by choosing one vertex $v$ in each $\pi _1M$ orbit and defining $f_v(v)=\textrm {dev}_0(v)$ .

The next goal is to characterize properly convex manifolds in such a way that the above proof applies. A set is locally convex if every point has a convex neighborhood. There is a basic local-to-global principle for convexity.

Local convexity only needs to be checked at points in $\partial K$ . Suppose that $K$ is contained in the upper half-space $x_1\geqslant 0$ in ${\mathbb{R}}^{n+1}$ . Then $K\cap (0\times {\mathbb{R}}^{n})$ is a convex subset of ${\mathbb{R}}^n$ if the subset $S\subset \partial K$ where $x_1\gt 0$ is a locally convex hypersurface in ${\mathbb{R}}^{n+1}$ . Informally: a mountain with a convex surface has a convex base. We now extend this to show that a manifold is properly convex.

A projective manifold is convex if every pair of points is contained in a segment. If $M$ is a closed projective manifold, the argument in (3.2), that the union of the segments starting at a point is closed, fails. For example, consider the projective torus $T$ that is the quotient of ${\mathbb{R}}^2_0$ by a cyclic group generated by a homothety. Then $T$ is not convex. Let $\pi :{\mathbb{R}}_0^2\rightarrow T$ be the projection. If $x\in {\mathbb{R}}^2_0$ and $y_n\in {\mathbb{R}}^2_0$ is a sequence that converge to $-x$ then the Hausdorff limit of the segments $\pi [x,y_n]$ in $T$ is not connected. It is the pair of rays, $\pi [x,0)\cup \pi (0,-x]$ . Each ray maps into in one of the pair of circles $\pi ({\mathbb{R}}_0\cdot x)\subset T$ .

A smooth hypersurface $S\subset {\mathbb{R}}^n$ is Hessian convex if the surface is locally the zero-set of a smooth, real-valued function with positive–definite Hessian. Suppose $\Omega \subset {\mathbb{RP}}^n_+$ is properly convex and $\mathcal{C}\Omega =\{v\in {\mathbb{R}}^{n+1}_0:[v]\in \Omega \}$ is the corresponding convex cone. Suppose $M=\Omega /\Gamma$ is a compact, and properly convex, $n$ -manifold. Then $\widetilde W=\mathcal{C}\Omega /\Gamma \cong M\times (0,\infty )$ is a properly convex affine $(n+1)$ -manifold. We may quotient out by a homothety to obtain a compact affine $(n+1)$ -manifold $W\cong M\times S^1$ .

By (2.1) there is a hypersurface $\partial \mathcal{D}\subset \mathcal{C}\Omega$ that is $\Gamma$ -invariant and Hessian convex away from $0$ . Then $Q=\partial \mathcal{D}/\Gamma$ is a compact, Hessian-convex, codimension-1 submanifold of $W$ . Let $K\subset \mathbb{RP}^{n+1}$ be the closure of $\mathcal{D}$ and regard $\mathbb{RP}^{n+1}={\mathbb{R}}^{n+1}\sqcup \mathbb{RP}^n_{\infty }$ . The interior of $K\cap \mathbb{RP}^n_{\infty }$ can be identified with $\Omega$ . The existence of the hypersurface $Q$ in $W$ implies the existence of $\mathcal{D}\subset {\mathbb{R}}^{n+1}$ , which implies $\Omega$ is properly convex, by the reasoning above.

By the Ehresmann–Thurston principle, deforming $\Gamma$ a small amount to $\Gamma ^{\prime}$ gives a new projective $n$ -manifold $M^{\prime}\cong M$ , and a new affine $(n+1)$ -manifold $W^{\prime}\cong M\times S^1$ . Now $W^{\prime}$ contains a hypersurface $Q^{\prime}$ that is Hessian convex, provided the deformation of the developing map is small enough in $C^2$ . It then follows that $M^{\prime}$ is properly convex. This is the approach taken in [Reference Cooper, Long and TillmannCLT18] for non-compact manifolds.

Here, we choose to work with a piecewise linear submanifold $Q$ in place of a smooth one. Hessian convexity is replaced by the condition that at each vertex of the triangulation the determinants, of certain matrices formed by the relative positions of vertices, are strictly positive. This ensures local convexity near the vertex. This version of convexity is preserved by small $C^0$ -deformations of the developing map. Thus it is easier to apply. We start by reviewing these ideas for hyperbolic manifolds.

The hyperboloid model of the hyperbolic plane is the action of ${\textrm {O}}(2,1)$ on the surface $S\subset {\mathbb{R}}^3$ given by $x^2+y^2-z^2=-1$ . If we identify ${\mathbb{R}}^3$ with $\mathbb{RP}^3\setminus \mathbb{RP}^2_{\infty }$ , then we can regard this as an affine action on $\mathbb{RP}^3$ by using the affine subgroup ${\textrm {O}}(2,1)\oplus (1)\subset \textrm {SL}(4,{\mathbb{R}})$ . The open disc

\begin{equation*}D=\{[x:y:z:0]:\ x^2+y^2\lt z^2\}\subset \mathbb{RP}^2_{\infty }\end{equation*}

has the same frontier as $S$ , and $({\textrm {O}}(2,1),D)$ is the projective (Klein) model of the hyperbolic plane. See Figure 7.

Figure 7. The hyperboloid and Klein models of ${\mathbb{H}}^2$ .

The surface $\textrm {cl}(S\cup D)\subset \mathbb{RP}^3$ bounds a closed $3$ -ball $B\subset \mathbb{RP}^3$ . Let $\Omega =B\setminus \textrm {Fr} D\cong D\times I$ . The fact that $S$ is a strictly convex surface implies $\textrm {cl}(S\cup D)$ is a convex surface in some affine patch which implies $\Omega$ is properly convex, and this implies $D$ is properly convex. If one thinks of $B$ as an upside down mountain, then the convexity of the surface $S$ implies the convexity of $D$ .

The action of ${\textrm {O}}(2,1)\oplus (1)$ preserves $\Omega$ . Suppose $\Gamma \subset {\textrm {O}}(2,1)$ and $\Sigma =D/\Gamma$ is a hyperbolic surface. Let $\Gamma ^{\prime}=\Gamma \oplus (1)$ , then $N=\Omega /\Gamma ^{\prime}\cong \Sigma \times [0,1]$ is a properly convex 3-manifold with one flat boundary component $\Sigma =D/\Gamma$ and one strictly convex boundary component $M=S/\Gamma$ . The fact that $M$ is a strictly convex surface implies $\Sigma$ is properly convex. We will generalize this construction to arbitrary properly convex manifolds in place of $\Sigma$ . But first we divide out by a homothety.

Consider the cone $\mathcal{C}=\{\lambda \cdot x:x\in S,\ \ \lambda \gt 0\}$ . There is a product structure $\widetilde \phi :S\times {\mathbb{R}}^+ \to \mathcal{C}$ given by $\widetilde \phi (x,\lambda )=\lambda \cdot x$ on $\mathcal{C}$ that is preserved by $\Gamma ^{\prime}$ . Let $H\subset \textrm {GL}(4,{\mathbb{R}})$ be the cyclic group generated by $\textrm {Diag}(2,2,2,1)$ . Then $\Gamma ^{\prime}$ centralizes $H$ and the group $\Gamma ^+\subset \textrm {GL}(4,{\mathbb{R}})$ generated by $\Gamma ^{\prime}$ and $H$ preserves $\mathcal{C}$ and $W:=\mathcal{C}/\Gamma ^+$ is a closed $3$ -manifold homeomorphic to $\Sigma \times S^1$ .

In what follows ${\mathbb{R}}^+=\{x\in {\mathbb{R}}:x\gt 0\}$ , and $S^1={\mathbb{R}}^+/\exp ({\mathbb{Z}})$ , has universal cover ${\mathbb{R}}^+$ . There is a product structure $\phi :\Sigma \times S^1\to W$ covered by $\widetilde \phi$ , and the surfaces $\phi (\Sigma ,\theta )$ are convex. The reader might contemplate all this in the case of one dimension lower, where $\Gamma \subset {\textrm {SO}}(1,1)$ is generated by a hyperbolic and $\mathcal{C}/\Gamma ^+$ is an affine structure on $S^1\times S^1$ .

We now return to the general setting. The radiant-affine group is the subgroup of the affine group acting on ${\mathbb{R}}^n$ that fixes the origin

\begin{equation*}\textrm {RA}(n)=\begin{pmatrix}\textrm {GL}(n,{\mathbb{R}}) & & & 0\\ 0& & &1\end{pmatrix}\subset \textrm {Aff}({\mathbb{R}}^n).\end{equation*}

Radiant-affine geometry is $(\textrm {RA}(n),{\mathbb{R}}^n_0)$ and is a subgeometry of affine geometry, so we may use affine notions. It is also isomorphic to a subgeometry of projective geometry under the embedding

\begin{equation*}{\mathbb{R}}^n_0\hookrightarrow \{[x:1]: x\in {\mathbb{R}}^n_0\}\subset \mathbb{RP}^n.\end{equation*}

The subgroup $t {\textrm {I}}_n\oplus 1$ with $t\gt 0$ is called the homothety flow. This subgroup acts on ${\mathbb{R}}^n_0$ by homotheties: $x\mapsto t\cdot x$ . The quotient of the tautological line bundle by a cyclic group of homotheties is the tautological circle bundle over $M$ .

Let $\pi :{\mathbb{R}}^n_0\rightarrow S^{n-1}$ be radial projection $\pi (x)=x/\|x\|$ . The second condition implies that

\begin{equation*}(\pi \circ \textrm {dev}|(\widetilde {M}\times 1),\rho |\pi _1(M\times 1))\end{equation*}

is a positive projective structure on $M$ . Conversely a positive projective structure on $M$ determines a radiant-affine geometry structure on $M\times S^1$ . Multiplication by $S^1$ on the right factor of $W=M\times S^1$ corresponds by the developing map to homotheties. We call all of these actions homothety or the radial flow.

Suppose $W$ is an affine $n$ -manifold, and $S$ is a hypersurface in $W$ . Then $S$ is a convex hypersurface, and it is locally convex, if for every point $p\in S$ there is an $n$ -dimensional submanifold $Q\subset W$ with $Q\cong \partial Q\times [0,1)$ and $\partial Q\subset S$ , and $\partial Q$ is a neighborhood of $p$ in $S$ and $\textrm {dev}(U)\subset {\mathbb{R}}^n$ is convex. If $W=M\times S^1$ is a tautological circle bundle, then such $S$ is outwards convex if, in addition, $t\cdot \partial Q\cap Q\ne \emptyset$ for some $t\gt 1$ . In other words $\textrm {dev}\widetilde S$ is locally convex away from $0$ in ${\mathbb{R}}^{n+1}$ .

A locally convex hypersurface $S$ is strictly convex if every connected flat subset of $S$ is a single point. If, instead, every connected flat subset of $\widetilde S$ is contained in a compact set, then $S$ is called strongly convex. We will make use of strongly convex when $S$ has a triangulation by flat simplices, no two of which are in the same hyperplane. In this case $S$ is called simplicial convex.