Let \(M\) be a closed, connected, smooth \(n\)-manifold.

Our goal is to describe all strictly convex projective structures on \(M\). To this end, we seek a strictly convex domain \(\Omega \subseteq \mathbb R \mathrm P^n\) such that \(M\) is homeomorphic to the orbit space \(\Omega / \Gamma\) for some subgroup \(\Gamma\) of \(\mathrm{PGL}(\Omega)\). Recall that \(\Omega\) is properly convex if its closure lies in an affine patch, and that \(\Omega\) is strictly convex if it is properly convex and if \(\partial \overline \Omega\) contains no line segment of positive length.

A marking \(h : M \to \Omega / \Gamma\) gives a holonomy representation

\[\rho = h_* : \pi_1 M \stackrel{\cong}{\longrightarrow} \Gamma.\]

Proposition. \(\Gamma\) determines \(\Omega\); hence, \(\rho\) determines \(\Omega\).

Proof. This follows from the fact that \(\partial \overline \Omega\) is the limit set of \(\Gamma\). \(\square\)

Thus describing strictly convex projective structures on \(M\) amounts to describing holonomy representations \(\pi_1 M \to \Gamma \leq \mathrm{PGL}(n + 1)\).

Let \(SC(M)\) denote the set of all holonomy representations \(\pi_1 M \to \mathrm{PGL}(n + 1)\) of strictly convex projective structures on \(M\). We have

\[SC(M) \leq \mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1)) \cong \mathrm{Hom}(\pi_1 M, \mathrm{SL}_\pm (n + 1)).\]

Theorem (Benoist 2000). \(SC(M)\) is both open and closed in \(\mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1))\).

We will prove the “open” part of this theorem using a general result of Thurston and Ehresmann.

\((G, X)\)-MANIFOLDS, THE HOLONOMY THEOREM

Let \(X\) be a manifold, and let \(G\) be a Lie group that acts continuously (and hence analytically) and transitively on \(X\) by diffeomorphisms.

Recall that a manifold \(M\) is a \((G, X)\)-manifold if there exists an atlas \(\{\varphi_i : U_i \subseteq M \to \varphi_i(U_i) \subseteq X\}\) of \(M\) such that the transition map

\[g_{ij} = \varphi_j \circ \varphi_i^{-1} : \varphi_i^{-1}(U_i \cap U_j) \longrightarrow \varphi_j^{-1}(U_i \cap U_j)\]

is the restriction of an element of \(G\) whenever \(U_i \cap U_j\) is nonempty, and such that the following cocycle condition is satisfied:

\[g_{ik} = g_{jk} \circ g_{ij} \quad\text{whenever}\quad U_i \cap U_j \cap U_k \neq \emptyset.\]

Alternatively, \(M\) is a \((G, X)\)-manifold if and only if it is homeomorphic to a quotient \(X / \Gamma\) for some discrete torsion-free subgroup of \(G\). Then the action of \(\Gamma\) on \(X\) is properly discontinuous and free, and the diagram

\[\begin{matrix} \pi_1 M & \smash{\stackrel{\mathrm{hol}}{\longrightarrow}} & \Gamma \\ \curvearrowright & & \curvearrowright \\ \widetilde M & \smash{\stackrel{\mathrm{dev}}{\longrightarrow}} & X \\ \end{matrix}\]

commutes. Here \(\pi_1 M\) acts on the universal cover \(\widetilde M\) of \(M\) by deck transformations, \(\mathrm{dev}\) is the developing map, and \(\mathrm{hol}\) is the holonomy of \(M\).

Theorem (Thurston–Ehresmann). Let \(M\) be a closed, connected, smooth \((G, X)\)-manifold. Then the set of all holonomies \(\pi_1 M \to G\) of \((G, X)\)-structures on \(M\) is open in \(\mathrm{Hom}(\pi_1 M, G)\).

Sketch of proof (Thurston 1978). Let \(\pi : \widetilde M \to M\) be the universal cover, and let \(D \subseteq \widetilde M\) be a fundamental domain. There exist finitely many charts fo \(\varphi_: U_i \to X\) covering \(D\). For each index \(i\), we choose \(U_i\) small enough that \(\pi\) is one-to-one on \(U_i\), and we use \(\pi\) to identify \(U_i\) with a subset of \(M\). We have

\[M = \bigsqcup U_i \Big / \sim,\]

where the equivalence relation \(\sim\) is given by

\[x_i \sim x_j \Longleftrightarrow g_{ij} \circ \varphi_i(x_i) = \varphi_j(x_j).\]

Here is the idea of the proof: If \(\rho'\) is close to a holonomy representation \(\rho : \pi_1 M \to G\), then there are maps \(g'_{ij}\) close to the maps \(g_{ij}\) satisfying the cocycle condition. Then define

\[M' = \bigsqcup U_i \Big / \sim',\]

where the relation \(\sim'\) is defined by replacing \(g_{ij}\) with \(g'_{ij}\) above.

Why can we do this? Take the nerve of the cover, i.e., a graph \(H\) with one vertex for each set \(U_i\) and one edge between corresponding vertices whenever \(U_i \cap U_j\) is nonempty. We may assume without loss of generality that the sets \(U_i\) are round balls, so that \(H\) is connected. Now let \(T\) be a maximal tree with a chosen basepoint. Define \(g'_{ij}\) to be \(g_{ij}\) if the edge for \(g_{ij}\) is in \(T\); otherwise, choose \(g'_{ij}\) so that the based loop \(\alpha_{ij}\) in \(H\) containing the edge for \(g_{ij}\) is \(\rho(\alpha_{ij})\), a product of maps \(g_{kl}\) around the loop. \(\square\)

Theorem (Kozul 1962). \(SC(M)\) is open in \(\mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1))\).

Proof. Suppose we are given a strictly convex projective structure, and hence a holonomy representation \(\rho : \pi_1 M \to \mathrm{PGL}(n + 1)\) by the Proposition, and a nearby representation \(\rho'\). By the holonomy theorem, there exists a nearby projective structure, and this structure is a strictly convex structure (why?). \(\square\)

HESSIAN METRICS

Let \(U \subseteq \mathbb R^n\) be open, and let \(c : U \to \mathbb R^n\) be a smooth function that is strictly convex (i.e., whose Hessian matrix is positive definite). The map \(c\) induces a Riemannian metric on \(U\) given by

\[\langle u, v \rangle_c = u^\top A v \quad\text{for all }v, w \in T_x U \text{ and all } x \in U,\]

where

\[A = D^2 c = {\left(\displaystyle{\frac{\partial^2 c}{\partial x_i \partial x_j}}\right)}\]

is the Hessian matrix of \(c\).

For example, the map \(c : \mathbb R^2 \to \mathbb R\) given by \(c(x_1, x_2) = x_1^2 + x_2^2\) induces the Euclidean metric on \(\mathbb R^2\).

Recall that a nonzero tangent vector \(v \in T_x U\) is specified by a smooth curve \(\gamma : (-\epsilon, \epsilon) \to U\) with \(\gamma(0) = x\) and \(\gamma'(0) = v\), where \(x \in U\) and \(\epsilon > 0\). Then \(\|v\|_c^2 = F''(0) > 0\) by strict convexity, where \(F = c \circ \gamma\).

Let \(M\) be an affine \(n\)-manifold. Then we have charts \(\varphi_i : U_i \subseteq M \to \mathbb R^n\) and transition maps in the affine group \(\mathrm{Aff}(n)\).

Let \(c : M \to \mathbb R\) be smooth. Then

\[\begin{align*} c \text{ is strictly convex} &\Longleftrightarrow {} \text{all maps } c \circ \varphi_i^{-1} \text{ are strictly convex} \\ &\Longleftrightarrow {} (c \circ \gamma)''(t) > 0 \text{ for all } t \in (-\epsilon, \epsilon) \text{ for all curves } \gamma \text{ defined as above.} \end{align*}\]

Thus \(c\) determines a Hessian metric on \(M\).

Remark. The Vinberg character functions on cones (April 14) are examples of this phenomenon.

Theorem. Suppose \(M\) is a simply connected boundaryless affine \(n\)-manifold with a complete Hessian metric \(d\). Then the developing map \(M \to \mathbb R^n\) is injective and has image a convex set.

Proof. Since \(M\) is simply connected, the developing map is a diffeomorphism, so we identify \(M\) with its image in \(\mathbb R^n\).

Let \(a, b \in M\) be distinct; we must show that the line segment joining them is contained in \(M\). Consider the triangle formed by \(a\), \(b\), and an arbitrary point \(p \in M\). Let \(\alpha, \beta : [0, 1] \to M\) be the straight-line paths paths from \(p\) to \(a\) and from \(p\) to \(b\), respectively. For each \(t \in [0, 1]\), let \(\gamma_t : [0, 1] \to M\) be the straight-line path from \(\alpha(t)\) to \(\beta(t)\). We claim that

\[\mathrm{length}(\gamma_t) \leq K < \infty\quad \text{for some } K > 0 \text{ for all } t.\]

Then \(d(x, p) < L\) for some \(L > 0\) for all \(x \in M\), and we are done by completeness.

Let \(F_t = c \circ \gamma_t\) for each \(t\). We have

\[\mathrm{length}(\gamma_t) = \int^1_0 \sqrt{F''_t(s)}\,ds \leq {\left(\int^1_0 F''_t(s)\,ds\right)}^{1/2} {\left(\int^1_0 ds\right)}^{1/2} \leq (F'_t(1) + F'_t(0))^{1/2} \leq K.\]

The first inequality follows from the Cauchy–Schwarz ineqality, and the last inequality (for some \(K > 0\)) follows from compactness. This proves the claim. \(\square\)

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