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Convex and Concave Decompositions of Affine 3-Manifolds

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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.

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References

  1. Apanasov, B.: Conformal Geometry of Discrete Groups and Manifolds, Volume 32 of De Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (2000)

  2. Apanasov, B., Bradlow, S., Rodrigues Jr., W., Uhlenbeck, K.: Geometry, Topology, and Physic. de Gruyter, Berlin (1997)

  3. Barbot, T.: Variétés affines radiales de dimension 3. Bull. Soc. Math. France 128(3), 347–389 (2000)

  4. Barbot, T., Choi, S.: Radiant affine \(3\)-manifolds with boundary, and certain radiant affine \(3\)-manifolds. Mem. Am. Math. Soc. 730, 98–120 (2001). (Appendix C of “The decomposition and classification of radian affine \(3\)-manifolds”)

  5. Benoist, Y.: Nilvariétés projectives. Comment. Math. Helv. 69(3), 447–473 (1994)

  6. Benoist, Y.: Tores affines. Crystallographic Groups and Their Generalizations (Kortrijk, 1999), Volume 262 of Contemporary Mathematics, pp. 1–37. American Mathematical Society, Providence (2000)

  7. Berger, M.: Geometry I. Universitext. Springer, Berlin (2009). (Translated from the 1977 French original by M. Cole and S. Levy, Fourth printing of the 1987 English translation [MR0882541])

  8. Carrière, Y.: Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math. 95(3), 615–628 (1989)

  9. Chae, Y., Choi, S., Park, C.: Real projective manifolds developing into an affine space. Int. J. Math. 4(2), 179–191 (1993)

  10. Choi, S.: Convex decompositions of real projective surfaces. I. \(\pi \)-annuli and convexity. J. Differ. Geom. 40(1), 165–208 (1994)

  11. Choi, S.: The Convex and Concave Decomposition of Manifolds with Real Projective Structures, Volume 78 of Mém. Soc. Math. Fr. (N.S.). Société Mathématique de France, Paris (1999)

  12. Choi, S.: The universal cover of an affine three-manifold with holonomy of shrinkable dimension \(\le \) two. Int. J. Math. 11(3), 305–365 (2000)

  13. Choi, S.: The decomposition and classification of radiant affine 3-manifolds. Mem. Am. Math. Soc. 154(730), viii+122 (2001)

  14. Cooper, D., Goldman, W.: A 3-manifold with no real projective structure. Ann. Fac. Sci. Toulouse Math. (6) 24(5), 1219–1238 (2015)

  15. Fried, D.: An informal note

  16. Fried, D., Goldman, W., Hirsch, M.: Affine manifolds with nilpotent holonomy. Comment. Math. Helv. 56(4), 487–523 (1981)

  17. Goldman, W.: Geometric structures on manifolds. In: Preparations, Formerly Titled “Projective Geometry on Manifolds”. http://www.math.umd.edu/~wmg/gstom.pdf

  18. Gordon, C.: Dehn surgery on knots. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pp. 631–642. Mathematical Society of Japan, Tokyo (1991)

  19. Hempel, J.: 3-Manifolds. AMS Chelsea Publishing, Providence (2004). (Reprint of the 1976 original)

  20. Hopf, H.: Zur Topologie der komplexen Mannigfaltigkeiten. Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 167–185. Interscience Publishers, Inc., New York (1948)

  21. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Volume 54 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1995). (With a supplementary chapter by Katok and Mendoza (1995))

  22. Kobayashi, S.: Projectively invariant distances for affine and projective structures. Differential Geometry (Warsaw, 1979), Volume 12 of Banach Center Publ., pp. 127–152. PWN, Warsaw (1984)

  23. Kulkarni, R.: Some topological aspects of Kleinian groups. Am. J. Math. 100(5), 897–911 (1978)

  24. Marden, A.: Outer Circles. Cambridge University Press, Cambridge (2007)

  25. Nagano, T., Yagi, K.: The affine structures on the real two-torus. I. Osaka J. Math. 11, 181–210 (1974)

  26. Simha, R.: The uniformisation theorem for planar Riemann surfaces. Arch. Math. (Basel) 53(6), 599–603 (1989)

  27. Smale, S.: Diffeomorphisms of the \(2\)-sphere. Proc. Am. Math. Soc. 10, 621–626 (1959)

  28. Smillie, J.: Affinely flat manifolds. PhD thesis, University of Chicago, Chicago, USA (1977)

  29. Sullivan, D., Thurston, W.: Manifolds with canonical coordinate charts: some examples. Enseign. Math. (2) 29(1–2), 15–25 (1983)

  30. Wu, W.: On embedded spheres in affine manifolds. PhD thesis, University of Maryland, College Park (2012). arXiv:1110.3541

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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).

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

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

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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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Keywords

  • Geometric structures
  • Flat affine structure
  • 3-manifolds

Mathematics Subject Classification

  • Primary 57M50
  • Secondary 53A15
  • 53A20