exponential map « Delta Epsilons

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,

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

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.
Tags: analytic manifolds, exponential map, Sigurdur Helgason
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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.
Tags: collar neighborhood theorem, connections, exponential map, tubular neighborhood theorem
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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.
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 ${T_p(M)}$ 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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Delta Epsilons

Helgason’s formula II November 7, 2009

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

The tubular neighborhood theorem November 5, 2009

Geodesics and the exponential map November 4, 2009

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