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SUMMARY:Alexey Bolsinov (Loughborough University)
DTSTART:20220531T140000Z
DTEND:20220531T150000Z
DTSTAMP:20260423T021411Z
UID:DGSTO/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/DGSTO/33/">O
 n integrability of geodesic flows on 3-dimensional manifolds</a>\nby Alexe
 y Bolsinov (Loughborough University) as part of Differential Geometry Semi
 nar Torino\n\n\nAbstract\nThe goal of the talk is to discuss the behaviour
  of geodesics on 3-manifolds $M$ with $SL(2\,\\mathbb R)$ geometry\,  one 
 of the eight natural geometries according to Thurston\, appearing on three
 -dimensional manifolds.   It has been known that the corresponding geodesi
 c flows cannot be integrable\, however\, this particular case has not been
  studied in detail.   The situation turned out quite interesting:  we have
  observed (joint work with  Alexander Veselov and Yiru Ye) that the phase 
 space $T^*M$ contains to two open domains\, complementary to each other an
 d having common boundary\, with integrable and chaotic behaviour  of geode
 sics.  In the integrable domain\, we have integrability in the class of re
 al-analytic integrals\,  whereas in the chaotic domain the geodesic flow h
 as positive topological entropy.  As a specific example\, we study in more
  detail the geodesic flow on the modular  3-manifold $M=SL(2\,\\R)/ SL(2\,
 \\mathbb Z)$ homeomorphic to the complement of a trefoil knot $\\mathcal K
 $ in 3-sphere.\n\nI will try to talk about these results in the context of
  a more general problem on topological obstructions to integrability of ge
 odesic flows on smooth manifolds following papers by V. V. Kozlov\, I. A. 
 Taimanov and L. Butler.\n
LOCATION:https://researchseminars.org/talk/DGSTO/33/
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