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SUMMARY:Carlos Olmos (Universidad Nacional de Córdoba)
DTSTART:20230628T160000Z
DTEND:20230628T170000Z
DTSTAMP:20260423T052831Z
UID:VSGS/75
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VSGS/75/">Ho
 pf fibrations and totally geodesic submanifolds</a>\nby Carlos Olmos (Univ
 ersidad Nacional de Córdoba) as part of Virtual seminar on geometry with 
 symmetries\n\n\nAbstract\nA Hopf-Berger sphere of factor $\\tau$  is  the 
 total space of a Hopf fibration  such that the Riemannian metric is rescal
 ed by a factor $\\tau\\neq 1$ in the directions of the fibers.  If the Hop
 f fibration is the complex one\, a Hopf-Berger sphere of $\\tau <1$ is the
  usual Berger sphere. Any  Hopf-Berger sphere may be regarded as a geodesi
 c sphere $\\mathsf{S}_t^m(o)\\subset\\bar M$ of radius $t$ of a rank one s
 ymmetric space of non-constant curvature ($\\bar M$ is compact if and only
  if $\\tau <1$).  A Hopf-Berger sphere has positive curvature if and only 
 if $\\tau <4/3$. A standard totally geodesic submanifold of $\\mathsf{S}_t
 ^m(o)$ is obtained as the intersection of the geodesic sphere with a total
 ly geodesic submanifold of $\\bar M$ that contains the center $o$. In this
  talk we will refer to our recent classification of totally geodesic subma
 nifolds of Hopf-Berger spheres. In particular\,  for quaternionic and octo
 nionic fibrations\, non-standard totally geodesic spheres with the same di
 mension of the fiber appear\, for $\\tau <1/2$. Moreover\,  there are  tot
 ally geodesic $\\mathbb RP^2$\, and $\\mathbb RP^3$  (under some restricti
 ons on $\\tau$\,  the  dimension\, and the type of the fibration). On the 
 one hand\, as a consequence of the connectedness principle of Wilking\,  t
 here does not exist a  totally geodesic $\\mathbb RP^4$ in a  space of  po
 sitive curvature which  diffeomorphic to the sphere $S^7$.  On the other h
 and\, we construct an example of a totally geodesic $\\mathbb RP^2$ in a H
 opf-Berger sphere of dimension $7$ and positive curvature. Could there exi
 st a totally geodesic $\\mathbb RP^3$ in a space of positive curvature whi
 ch  diffeomorphic to $S^7$?.\n\nThis talk is based on a joint work with Al
 berto Rodríguez-Vázquez.\n
LOCATION:https://researchseminars.org/talk/VSGS/75/
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