Helgason’s formula IINovember 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 mapNovember 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 theoremNovember 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 mapNovember 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…)