---- > [!definition] Definition. ([[geodesic polar coordinates]]) > Let $(M,g)$ be a [[Riemannian manifold]], $D$ the [[Levi-Civita connection]] on $M$. The [[inner product]] on $T_{p}M$ gives a canonical identification with $(\mathbb{R}^{n},\text{eucl.})$, and allows us to view any element $a \in T_{p}M$ as $(r, \boldsymbol v)$ for some 'magnitude $r >0$ and (normalized) direction $\boldsymbol v \in \mathbb{S}^{n-1}. More precisely, we have a '[[polar coordinates]] [[diffeomorphism]] ' $\begin{align} > (0, \infty)\times \mathbb{S}^{n-1} & \xrightarrow{g} T_{p} M \\ > (r, \boldsymbol v) & \mapsto r \boldsymbol v > \end{align}$ > > For $\varepsilon>0$ small enough, the [[Riemannian exponential map|exponential map]] $\exp_{p}:B_{g}(0, \varepsilon ) \to M$ is defined; we can 'parameterize it with polar coordinates' via precomposition with (the restriction to $0<r<\varepsilon$ of) the above change of variables $g$: $\begin{align} > (0, \varepsilon) \times \mathbb{S}^{n-1} \xrightarrow[]{g} T_{p}M-\{ 0 \} \xrightarrow{\exp_{p}} M. > \end{align}$ > For small enough $r_{0}>0$, $\exp_{p}$ restricts further to a [[diffeomorphism]] $B_{g}(0, r_{0}) \to U$ for some $U \subset M$, by the [[inverse function theorem]]. This yields [[normal (geodesic) coordinates about a point on a Riemannian manifold|normal coordinates]] on $M$ about $p$. Precomposing with (the restriction to $0<r<r_{0}$ of) $g$ as above gives a [[coordinate patch]] $\alpha:(0, r_{0}) \times \mathbb{S}^{n-1} \to U \subset M$, $\alpha:=(0, r_{0}) \times \mathbb{S}^{n-1} \xrightarrow[\cong]{g} B_{g}(0, r_{0}) \xrightarrow[\cong]{\exp_{p}} U \subset M.$ > These coordinates $\alpha(r, \boldsymbol v)=\exp_{p}(r \boldsymbol v)$ are called **geodesic polar coordinates about $p \in M$**. The image $\Sigma_{r}$ of $\{ r \} \times \mathbb{S}^{n-1}$ under $\alpha$, which equals the image of the radius-$r$ sphere $r\mathbb{S}^{n-1}$ under $\exp_{p}$, is called the **geodesic sphere of radius $r$ about $p$**, an [[embedded submanifold|embedded hypersurface]] in $M$. > [!basicproperties] > - [[Gauss's Lemma (Riemannian geometry)|Gauss's lemma]] ^properties ---- #### [[geodesic on a Riemannian manifold|geodesic]] ---- #### 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 > ```