maximal geodesic

Let $M$ be a differentiable manifold, let $\Del$ be an affine connection on $M$, and let $p$ be a point in $M$. Given a tangent vector $x$ at $p$, there is are several geodesics $\gamma$ on $M$ tangent to $x$ at $p$. Note that $\gamma$ is formally a differentiable map to $M$ from some interval ${]a,b[}$, in particular a partial differentiable map to $M$ from the real line $\mathbb{R}$. (Here, $a$ is an upper real number so may be $-\infty$; and $b$ is a lower real number so may be $\infty$.)

If $\gamma\colon {]a,b[} \to M$ and $\gamma'\colon {]a',b'[} \to M$ are geodesics (on $M$ tangent to $x$ at $p$), then they agree on their common domain ${]max(a,a'),min(b,b')[}$. Accordingly, If ${]a,b[} \subseteq {]a',b'[}$, then there is at most one extension of $\gamma$ to ${]a',b'[}$ that remains a geodesic. If $M$ is complete? (and perhaps in any case), then $\gamma$ may be extended (uniquely) to all of $\mathbb{R} = {]-\infty,\infty[}$. Regardless, there is a unique *maximal* geodesic (on $M$ tangent to $x$ at $p$). This is *the* **maximal geodesic on $M$ tangent to $x$ at $p$**.

Last revised on April 30, 2013 at 18:05:38. See the history of this page for a list of all contributions to it.