---- Let $(M,g)$ be a [[Riemannian manifold]]. > [!definition]+ Definition. ([[arc length of a path]]) > The **(arc) length** of a [[parameterized curve|curve]] $\alpha:[0,L] \to M$ [[integral|is]] $L(\alpha):=\int_{0}^{L} \|\dot{\alpha}(t)\|_{g} \, dt .$ > > Arc lengths define a [[metric]] $d$ on $(M,g)$, called the **Riemannian distance function**, as $d(p,q):= \inf \{ L(\alpha): \alpha \text{ is a curve from }p \text{ to }q \}.$ (By convention, $d(p,q)=\infty$ if $p,q$ lie in distinct [[path-connected component|path components]].)[^1] > The **geodesic ball of radius $r$ about $p \in M$** is $B_{r}(p)=\{ q: d(p,q) \leq r \}.$ The **geodesic sphere of radius $r$ about $p \in M$** is $S_{r}(p)=\{ q: d(p,q) \}=r.$ As the names would suggest, these agree with the [[geodesic polar coordinates|geodesic ball/sphere]] defined here when relevant (I want to say more). > [!basicexample]+ > Consider the helix $\boldsymbol{\alpha}(t)=(\sin t, \cos t, t), t \in (-\infty, \infty)$ of pitch $2\pi$. The [[velocity vector of a parameterized curve|tangent vector]] to the trace of $\boldsymbol{\alpha}$ is $\boldsymbol{\alpha}'(t)=(\cos t, -\sin t, 1)$ and consequently we have parameterization speed $\|\boldsymbol{\alpha}'(t)\|_{2}=\sqrt{ 2 }$. Thus the [[arc length of a path|arc length]] function starting at $t_{0}=0$ is $\int_{0}^{t} \sqrt{ 2 } \, dt= t\sqrt{ 2 }$. > > We can obtain a [[parameterization by arc length|(re)parameterization]] of $\boldsymbol{\alpha}$ by [[arc length of a path|arc length]] as follows: define $\begin{align} \Phi : & (-\infty, \infty) \to (-\infty, \infty) \\ s \mapsto & \Phi(s) := \frac{t}{\sqrt{ 2 }} ;\end{align}$ then we can verify that $\tilde{\alpha}:= \boldsymbol{\alpha} \circ \Phi$ is a [[parameterization by arc length]] thus: >$\begin{align} \|\tilde{\alpha}'(t)\|_{2}= & \| D\boldsymbol{\alpha} |_{\Phi(t)} D\Phi(t) \|_{2} \\ = & \|(\cos \Phi(t), \sin \phi(t), 1) \frac{1}{\sqrt{ 2 }}\|_{2} \\ = & 1. \end{align}$ ^13e6ff [^1]: As one would hope, this [[metric]] [[metric topology|induces]] the usual [[topological space|topology]] [[metrizable|on]] $M$ (nestle the opens). [[Normal (geodesic) coordinates about a point on a Riemannian manifold|Normal neighborhoods]] are a [[basis for a topology|basis]] [[topology generated by a basis|for this topology]]. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```