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Helgason’s formula II November 7, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, recall our goal was to prove Helgason’s formula,

\displaystyle \boxed{ (d \exp)_{tX}(Y) = \left( \frac{ 1 - e^{\theta( - tX^* )}}{\theta(tX^*)} (Y^*) \right)_{\exp(tX)}.}  

and that we have already shown

\displaystyle {(d \exp)_{tX}(Y) f = \sum_{n=0}^{\infty} \frac{t^n}{(n+1)!} ( X^{*n} Y^* + X^{*(n-1)} Y^* X^* + \dots + Y^* X^{*n})f(p).}  (more…)