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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Ok, recall our goal was to prove Helgason’s formula,
and that we have already shown
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
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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…)