# The differential of the exponential map: reductive homogeneous space

The differential of the exponential map on a symmetric space can be expanded (abusing some notation) as

$d{\rm Exp}_X=\sum_{n=0}^{\infty}\frac{({\rm ad}X)^{2n}}{(2n+1)!}.$

This is an old (1958) result of Helgason.

Question (EDITED): Is there any generalization to reductive homogeneous spaces?

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