geodesics | Delta Epsilons

Hopf-Rinow II and an application November 15, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: geodesic completeness, geodesics, homotopy, Hopf-Rinow theorem, Riemannian manifolds
2 comments

Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold ${(M,g)}$ which is a metric space ${d}$ , the existence of arbitrary geodesics from ${p}$ implies that ${M}$ is complete with respect to ${d}$ . Actually, this is slightly stronger than what H-R states: geodesic completeness at one point ${p}$ implies completeness.

The first thing to notice is that ${\exp: T_p(M) \rightarrow M}$ is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from ${p}$ exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a ${d}$ -Cauchy sequence ${q_n \in M}$ . We will show that it converges. Draw minimal geodesics ${\gamma_n}$ travelling at unit speed with

$\displaystyle \gamma_n(0)=p, \quad \gamma_n( d(p,q_n)) = q_n.$ (more…)

The Hopf-Rinow theorems and geodesic completeness November 14, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem, Riemannian manifolds
10 comments

Ok, yesterday I covered the basic fact that given a Riemannian manifold ${(M,g)}$ , the geodesics on ${M}$ (with respect to the Levi-Civita connection) locally minimize length. Today I will talk about the phenomenon of “geodesic completeness.”

Henceforth, all manifolds are assumed connected.

The first basic remark to make is the following. If ${c: I \rightarrow M}$ is a piecewise ${C^1}$ -path between ${p,q}$ and has the smallest length among piecewise ${C^1}$ paths, then ${c}$ is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick ${a,b \in I}$ very close to each other, so that ${c([a,b])}$ is contained in a neighborhood of ${c\left( \frac{a+b}{2}\right)}$ satisfying the conditions of yesterday’s theorem; then ${c|_{[a,b]}}$ must be length-minimizing, so it is a geodesic. We thus see that ${c}$ is locally a geodesic, hence globally.

Say that ${M}$ is geodesically complete if ${\exp}$ can be defined on all of ${TM}$ ; in other words, a geodesic ${\gamma}$ can be continued to ${(-\infty,\infty)}$ . The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)

The following are equivalent:

${M}$ is geodesically complete.

In the metric ${d}$ on ${M}$ induced by ${g}$ (see here), ${M}$ is a complete metric space (more…)

Geodesics are locally length-minimizing November 13, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: Gauss lemma, geodesics, Riemannian manifolds
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Fix a Riemannian manifold with metric ${g}$ and Levi-Civita connection ${\nabla}$ . Then we can talk about geodesics on ${M}$ with respect to ${\nabla}$ . We can also talk about the length of a piecewise smooth curve ${c: I \rightarrow M}$ as

$\displaystyle l(c) := \int g(c'(t),c'(t))^{1/2} dt .$

Our main goal today is:

Theorem 1 Given ${p \in M}$ , there is a neighborhood ${U}$ containing ${p}$ such that geodesics from ${p}$ to every point of ${U}$ exist and also such that given a path ${c}$ inside ${U}$ from ${p}$ to ${q}$ , we have

$\displaystyle l(\gamma_{pq}) \leq l(c)$

with equality holding if and only if ${c}$ is a reparametrization of ${\gamma_{pq}}$ .

In other words, geodesics are locally path-minimizing. Not necessarily globally–a great circle is a geodesic on a sphere with the Riemannian metric coming from the embedding in $\mathbb{R}^3$ , but it need not be the shortest path between two points. (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

Hopf-Rinow II and an application November 15, 2009

The Hopf-Rinow theorems and geodesic completeness November 14, 2009

Geodesics are locally length-minimizing November 13, 2009

Geodesics and the exponential map November 4, 2009

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