# Well-defined map: transvections in symmetric space

1. Jun 25, 2012

### Sajet

Hi!

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 $\tau_v$: Let M be a symmetric space and $c:\mathbb R \rightarrow M$ a geodesic with $p = c(0), v = \dot c(0).$ Then for $t \in \mathbb R$ the isometries

$\tau_{tv} = s_{c(t/2)}\circ s_{c(0)}$

are called transvections.)

I don't see how the map $(p, q) \mapsto \tau_p(q)$ is even well-defined. There can be more than one transvection mapping $p_0$ to $p$, and different transvections will in general give different values $\tau_p(q)$.

Last edited: Jun 25, 2012