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Helgason’s formula II November 7, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, recall our goal was to prove Helgason’s formula,

\displaystyle \boxed{ (d \exp)_{tX}(Y) = \left( \frac{ 1 - e^{\theta( - tX^* )}}{\theta(tX^*)} (Y^*) \right)_{\exp(tX)}.}  

and that we have already shown

\displaystyle {(d \exp)_{tX}(Y) f = \sum_{n=0}^{\infty} \frac{t^n}{(n+1)!} ( X^{*n} Y^* + X^{*(n-1)} Y^* X^* + \dots + Y^* X^{*n})f(p).}  (more…)

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

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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We showed that the differential of the exponential map {\exp_p: T_p(M) \rightarrow M} for {M} a smooth manifold and {p \in M} is the identity at {0 \in T_p(M)}. 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 {\nabla} is analytic if {\nabla_XY} is analytic for analytic vector fields {X,Y}. Using the real-analytic versions of the ODE theorem, it follows that {\exp_p} is an analytic morphism.

So, make the above assumptions: analyticity of both the manifold and the connection. Now there is a small disk {V_p \subset T_p(M)} such that {\exp_p} maps {V_p} diffeomorphically onto a neighborhood {U \subset M} containing {p}. We will compute {d(\exp_p)_{X}(Y)} when {X \in V_p} is sufficiently small and {Y \in T_p(M)} (recall that we identify {T_p(M)} with its tangent spaces at each point). (more…)

The tubular neighborhood theorem November 5, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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If {M} is a manifold and {N} a compact submanifold, then a tubular neighborhood of {N} consists of an open set {U \supset N} diffeomorphic to a neighborhood of the zero section in some vector bundle {E} over {N}, by which N corresponds to the zero section.

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

Geodesics and the exponential map November 4, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space {T_p(M)} at one point to the manifold {M} which is a local isomorphism. This is interesting because it gives a way of saying, “start at point {p} and go five units in the direction of the tangent vector {v},” 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 {c} such that the vector field along {c=(c_1, \dots, c_n)} created by the derivative {c'} is parallel. In local coordinates {x_1, \dots, x_n}, here’s what this means. Let the Christoffel symbols be {\Gamma^k_{ij}}. Then using the local formula for covariant differentiation along a curve, we get

\displaystyle D(c')(t) = \sum_j \left( c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) \right) \partial_j,

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

\displaystyle c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) = 0, \ 1 \leq j \leq n. (more…)

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