Convex and Concave Decompositions of Affine 3-Manifolds

Abstract

A (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space ${{\mathbb {R}}}^3$ with transition maps in the affine transformation group ${\mathbf {Aff}}({{\mathbb {R}}}^3)$. We will show that a connected closed affine 3-manifold is either an affine Hopf 3-manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral $\pi$-submanifolds and 2-convex affine manifolds, each of which is an irreducible 3-manifold. It follows that if there is no toral $\pi$-submanifold, then M is prime. Finally, we prove that if a closed affine manifold is covered by a connected open set in ${{\mathbb {R}}}^{3}$, then M is irreducible or is an affine Hopf manifold.

Appendices

Appendix A: Contraction Maps

Here, we will discuss contraction maps in ${{\mathbb {R}}}^{n}$. A contracting map$f:X \rightarrow X$ for a metric space X with metric d is a map so that $d(f(x), f(y)) < d(x, y)$ for $x, y \in X$.

Lemma 18

A linear map L has the property that all the norms of the eigenvalues are $< 1$. if and only if L is a contracting map for the distance induced by a norm.

Proof

See Corollary 1.2.3 of Katok and Hasselblatt (1995). $\square$

Proposition 7

$\langle g\rangle$ acts on ${{\mathbb {R}}}^{n} -\{O\}$ (resp. $U-\{ O\}$ for the upper half space $U \subset {{\mathbb {R}}}^{n}$) properly if and only if the all the norms of the eigenvalues of g are $> 1$ or $< 1$.

Proof

Suppose that $\langle g\rangle$ acts on ${{\mathbb {R}}}^{n} -\{O\}$ properly. Fix a standard norm on ${{\mathbb {R}}}^n$. For a unit sphere S with center 0, the image $g^{i}(S)$ is inside a unit ball B for some integer i by the properness of the action. This implies that $g^{i}(B) \subset B$, and the norms of the eigenvalues of $g^{i}$ are $< 1$ by Lemma 18. Hence the conclusion follows for g. The case of the half space U is similar.

For the converse, by replacing g with $g^{-1}$ if necessary, we assume that all norms of eigenvalues $< 1$. Lemma 18 shows that $g(B) \subset B$ for a unit ball corresponding to a norm. This implies the result. $\square$

Given two subsets A, B in an affine subspace ${{\mathbb {R}}}^n$, we denote by $A*B$ the union of all segments with end points in A and B respectively.

Proposition 8

Let D be a connected open set in ${{\mathbb {S}}}^{n}$ in an affine patch ${{\mathbb {R}}}^n$. Let g be a projective automorphism acting on D and an affine patch ${{\mathbb {R}}}^{n}$. We assume the following :

• S is a compact connected smoothly embedded submanifold of codimension-one of D so that $D-S$ has two components $D_{1}$ and $D_{2}$ where $D_{1}$ is bounded in an affine patch ${{\mathbb {R}}}^n$ in ${{\mathbb {S}}}^n$.

• g acts with a fixed point $x \in {{\mathbb {R}}}^{n}$ in the closure of $D_{1}$.

• $g(\mathrm{Cl}(D_1)) \subset D_{1}$.

• Every complete affine line containing x meets S at at least one point.

• $D_{1} \subset \{x\}*S$ where $\{x\}*S$ is the union of all segments from x ending at S.

Then x is the global attracting fixed point of g in ${{\mathbb {R}}}^{n}$.

Proof

Choose the coordinate system on ${{\mathbb {R}}}^n$ so that x is the origin. Let L(g) denote the linear part of the g in this coordinate system. Suppose that there is a norm of the eigenvalue of L(g) greater than or equal to 1. Then there is a subspace V of dimension 1 or 2 so that $V \otimes {{\mathbb {C}}}$ is an eigenspace in ${{\mathbb {C}}}^n$ associated with an eigenvalue of norm $\ge 1$. We obtain $S_{V}:= V \cap S \ne \varnothing$ by the above paragraph. Let ${\varTheta }(S_{V})$ denote the set of directions of $S_{V}$ from x. L(g) acts on the space of directions from x. Since $\{x\} *g(S) \subset \{x\} *S$, we obtain $L(g)({\varTheta }(S_{V})) \subset {\varTheta }(S_{V})$. Hence, ${\varTheta }(S_{V})$ is either the set of a point, the set of a pair of antipodal points, or a subset of a circle where every point or its antipode are in it. Now, V has a Riemannian metric where g acts as a rotation times a scalar map. There is a point t of $S_{V}$ where a maximal radius of $S_{V}$ takes place under this metric. Then $g(t) \in g(S_{V})$ must meet $D_{2} \cup S_{V}$, a contradiction.

Thus, the norms of eigenvalues of L(g) are $< 1$. By Lemma 18, L(g) has a fixed point x as an attracting fixed point. $\square$

Now, we prove without a g-invariant affine subspace.

Proposition 9

Let D be a connected open set in ${{\mathbb {S}}}^{n}$ in an affine subspace ${{\mathbb {R}}}^n$. Let g be a projective automorphism of ${{\mathbb {S}}}^{n}$ acting on D. We assume the following :

• S is a compact connected smoothly embedded $(n-1)$-sphere of D so that $D-S$ has two components $D_{1}$ and $D_{2}$ where $D_{1}$ is bounded in an affine path ${{\mathbb {R}}}^n$ in ${{\mathbb {S}}}^n$.

• $g(\mathrm{Cl}(D_1)) \subset D_{1}$.

Then g acts on an open affine subspace ${{\mathbb {R}}}^n$ containing $D_1$, and g has the global attracting fixed point x in ${{\mathbb {R}}}^{n}$, and

Proof

By the Schoenflies theorem, a component of ${{\mathbb {S}}}^3 - S$ is a 3-cell $D'_1$ bounded in ${{\mathbb {R}}}^n$. So, $D'_1$ is in a cell in ${{\mathbb {R}}}^n$. Then $g(D'_1) \subset D'_1$ since $g(S) \subset D'_1$ and the external component of ${{\mathbb {S}}}^2 - g(S)$ is not contained in a properly convex domain. By the Brouwer fixed-point theorem, g fixes a point in the interior of $D'_1$.

The convex hull $C'$ of $D_1$ is still in an affine patch and is a properly convex domain. An easy argument shows that $g(C')$ is a compact subset of the interior of $C'$ since every pair of points of $g(D_1)$ has a pair of convex open neighborhoods in the interior of $D_1$ useful for taking segments.

Then $C'':=\bigcup _{i=1}^\infty g^{-n}(C')$ is a convex open subset of ${{\mathbb {S}}}^n$, and hence is in an open hemisphere by Proposition 2.3 of Choi (1999) since $C''$ cannot be a great sphere. We claim that $C''$ is an open n-hemisphere. Suppose that $C''$ is not an open n-hemisphere. $C''$ has a family of open i-hemispheres foliating $C''$ for $1\le i < n$ or is properly convex by Proposition 2.4 of Choi (1999). The space of i-hemispheres forms a properly convex open domain K of dimension $n-i < n$ as shown in Section 1.4 of Chae et al. (1993). Let ${\varPi }_K$ denote the projection $C'' \rightarrow K$. When $C''$ is properly convex, we let ${\varPi }_K$ be the identity map. Again g acts on K with a fixed point $x'$ in the interior of ${\varPi }_K(C')$ so that $g({\varPi }_K(C'))$ into the interior of ${\varPi }_K(C')$. By the existence of Hilbert metric Kobayashi (1984) for a properly convex domain, if g fixes an interior point, then $g({\varPi }_K(C'))$ cannot go into the interior of ${\varPi }_K(C')$ by the existence of the points of maximal distance from x at ${\varPi }_K(C')$. Hence, $C''$ is an affine patch where g acts on.

Since S is a separating sphere, every complete affine ray starting from x meets S at at least one point. Since every point of $D_1$ is on a complete affine ray starting from x, $D_{1} \subset \{x\}*S$ where $\{x\}*S$ is the union of all segments from x ending at S. Since g acts on an affine patch $C''$, and $D_1 \subset C''$, Proposition 8 in Appendix 1 implies that x is an attracting fixed point of g on $C''$. $\square$

Appendix B: The Boundary of a Concave Affine Manifolds is Not Strictly Concave

The following is the easy generalization of the maximum property in Section 6.2 of Choi (1994). Let N be an affine manifold with boundary. Hence, each boundary point has a chart going to an affine space where the boundary subset of the open set where the chart is defined maps to a submanifold of codimension-one. A strictly concave boundary point of an affine manifold N is a boundary point y where a totally geodesic open disk D contains y, $y\in D^{o}$, and $D -\{y\} \subset N^{o}$.

Theorem 11

Let N be a concave affine 3-manifold of type II in a compact real projective manifold M with convex boundary. Then $\partial N$ has no strictly concave point.

Proof

Let $M_{h}$ be a cover as in the main part of the paper. Let $N_h$ be a component of the inverse image of N in $M_h$.

Suppose that the conclusion does not hold. Then there is a boundary point y of $N_h$ with a disk D as above. Then if y is a boundary point of $M_{h}$, then D must be in $\partial M_{h}$ since $\partial M_h$ is convex: we can use a chart of a point of $\partial M_h$ to a convex subset of ${{\mathbb {S}}}^n$ mapping into an affine patch and deduce by looking these as graphs of convex functions. This contradicts the premise since $D -\{y\} \subset N^{o}$.

Now, N is covered by ${\varLambda }(R) \cap M_{h}$ for bihedral 3-crescent R in $\check{M}_h$. Since y is not a boundary point of $M_{h}$, we take a convex compact neighborhood B(y) of the convex point y so that $\mathbf{dev}_{h}(B(y))$ is an $\epsilon$-${\mathbf {d}}$-ball for some $\epsilon > 0$. Then $B(y) - {\varLambda }(R)$ is a properly convex domain with the image $\mathbf{dev}_{h}(B(y) - {\varLambda }(R))$ is properly convex. For each point $z \in \mathrm{bd}{\varLambda }(R)\cap B(y)$, let $S_{z}$, $S_{z} \sim R$, be a bihedral 3-crescent containing z. Since ${\varLambda }(R)$ is maximal, $\mathbf{dev}_{h}(I_{S_{z}})$ is a supporting plane at $\mathbf{dev}_{h}(z)$ of $\mathbf{dev}_{h}(B(y) - {\varLambda }(R))$.

We perturb a small convex disk $D \subset I_{S_{y}}$ containing y away from y, so that the perturbed convex disk $D'$ is such that the closure of $D' \cap B(y) - {\varLambda }(R)$ is a small compact disk $D''$ with

$$\begin{aligned} \partial D'' \subset \mathrm{bd}{\varLambda }(R) \cap M_{h} \hbox { and } D^{\prime \prime o} \cap {\varLambda }(R) = \varnothing . \end{aligned}$$

Moreover, $\partial D''$ bounds a compact disk $B'$ in $\mathrm{bd}{\varLambda }(R) \cap B(y)$. Choose a point $z_{0}$ in the interior of $D''$. For each point $z \in B'$, $I_{S_z}^o$ is transversal to $\overline{z_0z}$ since $z_0 \not \in S_z$. Since $S_z^o$ is further away from $z_0$ than z, we can choose a maximal segment $s_{z} \subset S_{z}$ starting from $z_{0}$ passing z ending at a point $\delta _{+} s_{z}$ of $\alpha _{S_{z}}$. We obtain a compact 3-ball $B_{z_{0}} = \bigcup _{z\in B'} s_{z}$ with its boundary in $\delta _{\infty }{\varLambda }(R)$. The boundary is the union of $D_{z_{0}} := \bigcup _{z\in \partial D''} s_{z}$, a compact disk, and an open disk

$$\begin{aligned} \alpha _{z_{0}} := \bigcup _{z\in B^{o}_{z_{0}}} \delta _{+}s_{z} \subset \delta _{\infty } {\varLambda }(R). \end{aligned}$$

The injectivity of $\mathbf{dev}_h| B_{z_0}$ is clear since we are using maps $\mathbf{dev}_h|s_z$, $z \in B'$ which are radiant from $\mathbf{dev}_h(z_0)$. Hence, $B_{z_{0}}$ is a bihedral 3-crescent.

Since $B_{z_0}$ is a union of segments from $z_0$ passing points of $B'$ containing a segment passing y and transversal to $I_{S_y}^o$, it overlaps with $S_y$. Since $B_{z_{0}}$ is a bihedral 3-crescent $\sim S_{y}, S_{y} \sim R$, we obtain $B_{z_{0}} \subset {\varLambda }(R)$. This contradicts our choice of y and $D''$. $\square$