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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?

edited Jul 23 at 16:19

asked Jul 20 at 10:55

Alex Gavrilov
658●2●8

DamienC · Accepted Answer · 2012-07-26 13:23:46Z UTC

I think the formula will be essentially the same.

The formula you wrote is valid in general for the exponential map of analytic manifolds equipped with an analytic affine connection. It is stated and proved in this paper by Helgason (see pages 6-7 of the linked .pdf): Some remarks on the exponential mapping of an affine connection. Math. Scand. 9 (l961), l29-l46.

The existence of such connections on reductive homogeneous spaces, with an additional invariance property (that you might probably need at some point) is stated and proved in K.Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65.

The differential of the exponential map: reductive homogeneous space

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