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Well-defined map: transvections in symmetric space

  1. Jun 25, 2012 #1

    I'm trying to understand a proof for the fact that the isometry group of a symmetric space is a Lie group. The proof uses a lemma and I don't see how the lemma works. Here is the statement in question:

    (Let me give you the definition for [itex]\tau_v[/itex]: Let M be a symmetric space and [itex]c:\mathbb R \rightarrow M[/itex] a geodesic with [itex]p = c(0), v = \dot c(0).[/itex] Then for [itex]t \in \mathbb R[/itex] the isometries

    [itex]\tau_{tv} = s_{c(t/2)}\circ s_{c(0)}[/itex]

    are called transvections.)

    I don't see how the map [itex](p, q) \mapsto \tau_p(q)[/itex] is even well-defined. There can be more than one transvection mapping [itex]p_0[/itex] to [itex]p[/itex], and different transvections will in general give different values [itex]\tau_p(q)[/itex].
    Last edited: Jun 25, 2012
  2. jcsd
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