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SUMMARY:Micro Squares\,  Connections and the Lie Bracket of Vector Fields 
 - Filip Bár (University of Cambridge)
DTSTART:20120312T170000Z
DTEND:20120312T183000Z
UID:TALK36973@talks.cam.ac.uk
CONTACT:Filip Bár
DESCRIPTION:Using the exponential laws in the cartesian closed category of
  microlinear spaces we obtain that the iterated tangent bundle is the bund
 le of micro squares. There are various ways to structure it as a bundle ov
 er the tangent bundle. One way is to see it as a bundle over ordered pairs
  of tangents by mapping a micro square to its pair of principal axes. This
  bundle turns out to be an affine bundle over the tangent bundle and plays
  an important (unifying) role for connections and Lie brackets of vector f
 ields.\n\nGeometrically\, a (linear) connection on a tangent bundle is a s
 tructure that 'connects' infinitesimally neighbouring tangent spaces (resp
 ecting their R-linear structures). Any section of the affine bundle of mic
 ro squares is considered a connection\, since passing from a pair of tange
 nt vectors to the respective micro square amounts to thickening the pair o
 f tangent vectors to an infinitesimal grid\, allowing infinitesimal 'paral
 lel' transport of one tangent along the other. If the section is homogeneo
 us in its arguments\, then this yields a linear connection.\n\nUsing expon
 ential adjunction there are three ways to consider vector fields. Defining
  vector fields as sections of the tangent bundle of M one can consider the
 m equivalently either as infinitesimal flows\, or tangents at the identity
  map of the microlinear space of diffeomorphisms\, i.e.\, as elements of t
 he Lie algebra of the Lie group Diff(M). The Lie bracket of vector fields 
 can be defined by using this Lie algebra representation and translating th
 e intended geometric meaning directly into algebra. Moreover\, using the a
 ffine bundle structure of micro squares the Lie bracket can be written in 
 a way resembling the canonical Lie bracket from ring theory. This is becau
 se any pair of vector fields determines a micro square in a natural way.  
         
LOCATION:Centre for Mathematical Sciences\, MR9
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