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Convex and Concave Decompositions of Affine 3-Manifolds

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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.

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Correspondence to Suhyoung Choi.

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This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 2013R1A1A2056698).


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.


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 \(< 1\).


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 AB 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 \({{\mathbb {R}}}^{n}\).


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


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.


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 \(D''\). \(\square \)

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Choi, S. Convex and Concave Decompositions of Affine 3-Manifolds. Bull Braz Math Soc, New Series 51, 243–291 (2020). https://doi.org/10.1007/s00574-019-00152-1

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  • Geometric structures
  • Flat affine structure
  • 3-manifolds

Mathematics Subject Classification

  • Primary 57M50
  • Secondary 53A15
  • 53A20