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

Pablo Angulo

doi:10.3934/dcds.2018001

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2018, Volume 38, Issue 1: 1-41. Doi: 10.3934/dcds.2018001

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

Pablo Angulo

Universidad Politécnica de Madrid, ETSI Navales, Avd. Arco de la Victoria 4, 28040 Madrid

Received: April 2016

Revised: July 2017

Published: October 2017

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

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Abstract

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 $3$.

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Mathematics Subject Classification: Primary:53C20, 53C22;Secondary:49N45.

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Figure 1. 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.

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Figure 2. Flow diagram for the linking curve algorithm

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Figure 3. The distribution $D$ and the CDCs at the conjugate points near an $A_{4}$ point.

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Figure 4. 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.

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Figure 5. A hyperbolic umbilic point.

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Figure 6. 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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