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Helgason’s formula II
*November 7, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: analytic manifolds, exponential map, Lie bracket, Sigurdur Helgason

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Tags: analytic manifolds, exponential map, Lie bracket, Sigurdur Helgason

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Ok, recall our goal was to prove Helgason’s formula,

and that we have already shown

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Helgason’s formula for the differential of the exponential map
*November 6, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: analytic manifolds, exponential map, Sigurdur Helgason

6 comments

Tags: analytic manifolds, exponential map, Sigurdur Helgason

6 comments

We showed that the differential of the exponential map for a smooth manifold and is the identity at . In the case of analytic manifolds, it is possible to say somewhat more. First of all, if we’re working with real-analytic manifolds, we can say that a connection is **analytic** if is analytic for analytic vector fields . Using the real-analytic versions of the ODE theorem, it follows that is an analytic morphism.

So, make the above assumptions: analyticity of both the manifold and the connection. Now there is a small disk such that maps diffeomorphically onto a neighborhood containing . We will compute when is sufficiently small and (recall that we identify with its tangent spaces at each point). (more…)