# American Institute of Mathematical Sciences

January 2018, 38(1): 1-41. doi: 10.3934/dcds.2018001

## Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds

Received  April 2016 Revised  July 2017 Published  September 2017

Fund Project: The author was partially supported by research grant ERC 301179, and by INEM

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemannian manifolds contains a residual set of the metrics on a given smooth manifold of dimension $3$.

Citation: Pablo Angulo. Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 1-41. doi: 10.3934/dcds.2018001
##### References:
 [1] W. Ambrose, Parallel translation of riemannian curvature, Ann. of Math(2), 64 (1956), 337-363. doi: 10.2307/1969978. [2] P. Angulo, Cut and Conjugate Points of the Exponential Map, with Applications Ph. D. Dissertation at Universidad Autónoma de Madrid, 2014, arXiv: 1411.3933 [3] P. Angulo and L. Guijarro, Balanced split sets and Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 40 (2011), 223-252. doi: 10.1007/s00526-010-0338-y. [4] R. A. Blumenthal and J. J. Hebda, The generalized Cartan-Ambrose-Hicks theorem, C. R. Acad. Sci. Paris Sér. I Math, 305 (1987), 647-651. doi: 10.1007/BF00182117. [5] M. A. Buchner, Stability of the cut locus in dimensions less than or equal to 6, Invent. Math., 43 (1977), 199-231. [6] É. Cartan, Leçons sur la Géométrie des Espaces de Riemann (French) 2d ed. Gauthier-Villars, Paris, 1951. [7] M. Castelpietra and L. Rifford, Regularity Properties of the Distance Functions to Conjugate and Cut Loci for Viscosity Solutions of Hamilton-Jacobi Equations and Applications in Riemannian Geometry, ESAIM Control Optim. Calc. Var., 16 (2010), 695–718. arXiv: 0812.4107 (2008). doi: 10.1051/cocv/2009020. [8] J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. [9] P. Griffiths and J. Wolf, Complete maps and differentiable coverings, Michigan Math. J., 10 (1963), 253-255. doi: 10.1307/mmj/1028998907. [10] B. Hambly and T. Lyons, Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math.(2), 171 (2010), 109-167. doi: 10.4007/annals.2010.171.109. [11] J. J. Hebda, Conjugate and cut loci and the Cartan-Ambrose-Hicks theorem, Indiana Univ. Math. J., 31 (1982), 17-26. doi: 10.1512/iumj.1982.31.31003. [12] J. J. Hebda, Parallel translation of curvature along geodesics, Trans. Amer. Math. Soc., 299 (1987), 559-572. doi: 10.1090/S0002-9947-1987-0869221-6. [13] J. J. Hebda, Metric structure of cut loci in surfaces and Ambrose's problem, J. Differential Geom., 40 (1994), 621-642. doi: 10.4310/jdg/1214455780. [14] J. J. Hebda, Heterogeneous Riemannian manifolds, Int. J. Math. Math. Sci. (2010), Article ID 187232, 7 pp. [15] N. Hicks, A theorem on affine connexions, Illinois J. Math., 3 (1959), 242-254. [16] M. Hirsch, Differential Topology Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976. [17] M. V. de Hoop, S. F. Holman, E. Iversen, M. Lassas and B. Ursin, Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times, J. Math. Pures Appl., 103 (2015), 830-848. doi: 10.1016/j.matpur.2014.09.003. [18] J. Itoh, The length of a cut locus on a surface and Ambrose's problem, J. Differential Geom., 43 (1996), 642-651. doi: 10.4310/jdg/1214458326. [19] J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc., 353 (2001), 21-40. doi: 10.1090/S0002-9947-00-02564-2. [20] S. Janeczko and T. Mostowski, Relative generic singularities of the exponential map, Compositio Mathematica, 96 (1995), 345-370. [21] F. Klok, Generic singularities of the exponential map on Riemannian manifolds, Geom. Dedicata, 14 (1983), 317-342. doi: 10.1007/BF00181572. [22] Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [23] S. Kurilev, M. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, American Journal of Mathematics, 132 (2010), 529-562. doi: 10.1353/ajm.0.0103. [24] B. O'Neill, Construction of Riemannian coverings, Proc. Amer. Math. Soc., 19 (1968), 1278-1282. doi: 10.1090/S0002-9939-1968-0232313-2. [25] V. Ozols, Cut loci in Riemannian manifolds, Tôhoku Math. J.(2), 26 (1974), 219-227. doi: 10.2748/tmj/1178241180. [26] K. Pawel and H. Reckziegel, Affine submanifolds and the theorem of Cartan-Ambrose-Hicks, Kodai Math. J., 25 (2002), 341-356. doi: 10.2996/kmj/1071674466. [27] A. Weinstein, The generic conjugate locus, In Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math, Soc., Providence, R. I., (1970), 299-301. [28] A. Weinstein, The cut locus and conjugate locus of a riemannian manifold, Ann. of Math.(2), 87 (1968), 29-41. doi: 10.2307/1970592.

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##### References:
 [1] W. Ambrose, Parallel translation of riemannian curvature, Ann. of Math(2), 64 (1956), 337-363. doi: 10.2307/1969978. [2] P. Angulo, Cut and Conjugate Points of the Exponential Map, with Applications Ph. D. Dissertation at Universidad Autónoma de Madrid, 2014, arXiv: 1411.3933 [3] P. Angulo and L. Guijarro, Balanced split sets and Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 40 (2011), 223-252. doi: 10.1007/s00526-010-0338-y. [4] R. A. Blumenthal and J. J. Hebda, The generalized Cartan-Ambrose-Hicks theorem, C. R. Acad. Sci. Paris Sér. I Math, 305 (1987), 647-651. doi: 10.1007/BF00182117. [5] M. A. Buchner, Stability of the cut locus in dimensions less than or equal to 6, Invent. Math., 43 (1977), 199-231. [6] É. Cartan, Leçons sur la Géométrie des Espaces de Riemann (French) 2d ed. Gauthier-Villars, Paris, 1951. [7] M. Castelpietra and L. Rifford, Regularity Properties of the Distance Functions to Conjugate and Cut Loci for Viscosity Solutions of Hamilton-Jacobi Equations and Applications in Riemannian Geometry, ESAIM Control Optim. Calc. Var., 16 (2010), 695–718. arXiv: 0812.4107 (2008). doi: 10.1051/cocv/2009020. [8] J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. [9] P. Griffiths and J. Wolf, Complete maps and differentiable coverings, Michigan Math. J., 10 (1963), 253-255. doi: 10.1307/mmj/1028998907. [10] B. Hambly and T. Lyons, Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math.(2), 171 (2010), 109-167. doi: 10.4007/annals.2010.171.109. [11] J. J. Hebda, Conjugate and cut loci and the Cartan-Ambrose-Hicks theorem, Indiana Univ. Math. J., 31 (1982), 17-26. doi: 10.1512/iumj.1982.31.31003. [12] J. J. Hebda, Parallel translation of curvature along geodesics, Trans. Amer. Math. Soc., 299 (1987), 559-572. doi: 10.1090/S0002-9947-1987-0869221-6. [13] J. J. Hebda, Metric structure of cut loci in surfaces and Ambrose's problem, J. Differential Geom., 40 (1994), 621-642. doi: 10.4310/jdg/1214455780. [14] J. J. Hebda, Heterogeneous Riemannian manifolds, Int. J. Math. Math. Sci. (2010), Article ID 187232, 7 pp. [15] N. Hicks, A theorem on affine connexions, Illinois J. Math., 3 (1959), 242-254. [16] M. Hirsch, Differential Topology Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976. [17] M. V. de Hoop, S. F. Holman, E. Iversen, M. Lassas and B. Ursin, Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times, J. Math. Pures Appl., 103 (2015), 830-848. doi: 10.1016/j.matpur.2014.09.003. [18] J. Itoh, The length of a cut locus on a surface and Ambrose's problem, J. Differential Geom., 43 (1996), 642-651. doi: 10.4310/jdg/1214458326. [19] J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc., 353 (2001), 21-40. doi: 10.1090/S0002-9947-00-02564-2. [20] S. Janeczko and T. Mostowski, Relative generic singularities of the exponential map, Compositio Mathematica, 96 (1995), 345-370. [21] F. Klok, Generic singularities of the exponential map on Riemannian manifolds, Geom. Dedicata, 14 (1983), 317-342. doi: 10.1007/BF00181572. [22] Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [23] S. Kurilev, M. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, American Journal of Mathematics, 132 (2010), 529-562. doi: 10.1353/ajm.0.0103. [24] B. O'Neill, Construction of Riemannian coverings, Proc. Amer. Math. Soc., 19 (1968), 1278-1282. doi: 10.1090/S0002-9939-1968-0232313-2. [25] V. Ozols, Cut loci in Riemannian manifolds, Tôhoku Math. J.(2), 26 (1974), 219-227. doi: 10.2748/tmj/1178241180. [26] K. Pawel and H. Reckziegel, Affine submanifolds and the theorem of Cartan-Ambrose-Hicks, Kodai Math. J., 25 (2002), 341-356. doi: 10.2996/kmj/1071674466. [27] A. Weinstein, The generic conjugate locus, In Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math, Soc., Providence, R. I., (1970), 299-301. [28] A. Weinstein, The cut locus and conjugate locus of a riemannian manifold, Ann. of Math.(2), 87 (1968), 29-41. doi: 10.2307/1970592.
A Standard T: The left hand side displays a curve $\alpha$ in ${T_p}M$, while the right hand side displays ${\exp _p} \circ \alpha$. Ⅰ, Ⅱ and Ⅳ are ACDCs, Ⅲ is the retort of Ⅱ, Ⅴ is the retort of Ⅳ, and Ⅵ is the retort of Ⅰ. Vertices 2 and 4 are $A_{3}$ joins, vertex 1 is a splitter, vertex 3 is a hit and vertex 5 is a reprise. There can be more than two segments between a splitter and its matching hit, and between a hit and its matching reprise.
Flow diagram for the linking curve algorithm
The distribution $D$ and the CDCs at the conjugate points near an $A_{4}$ point.
CDCs in the half-cone of first conjugate points near an elliptic umbilic point, using the chart $(x_{1} ,x_{2} ) \rightarrow (x_{1} ,x_{2} ,- \sqrt{x_{1}^{2} +x_{2}^{2}} )$, for $r_{0} =(0,0,1)$. The distribution $D$ makes half turn as we make a full turn around $x_{1}^{2} +x_{2}^{2} =1$, spinning in the opposite direction.
A hyperbolic umbilic point.
This picture shows a neighborhood of an $A_{4}$ point in ${T_p}M$, together with the linking curves that start at $x$ and $y$ (to the left) and the image of the whole sketch by ${\exp _p}$ (to the right).
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