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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…)

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The tubular neighborhood theorem
*November 5, 2009*

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

Tags: collar neighborhood theorem, connections, exponential map, tubular neighborhood theorem

4 comments

Tags: collar neighborhood theorem, connections, exponential map, tubular neighborhood theorem

4 comments

If is a manifold and a compact submanifold, then a **tubular neighborhood** of consists of an open set diffeomorphic to a neighborhood of the zero section in some vector bundle over , by which corresponds to the zero section.

Theorem 1Hypotheses as above, has a tubular neighborhood. (more…)

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Geodesics and the exponential map
*November 4, 2009*

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

Tags: connections, exponential map, geodesics, ordinary differential equations

5 comments

Tags: connections, exponential map, geodesics, ordinary differential equations

5 comments

Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space at one point to the manifold which is a local isomorphism. This is interesting because it gives a way of saying, “start at point and go five units in the direction of the tangent vector ,” in a rigorous sense, and will be useful in proofs of things like the tubular neighborhood theorem—which I’ll get to shortly.

Anyway, first I need to talk about geodesics. A **geodesic** is a curve such that the vector field along created by the derivative is parallel. In local coordinates , here’s what this means. Let the Christoffel symbols be . Then using the local formula for covariant differentiation along a curve, we get

so being a geodesic is equivalent to the system of differential equations