---- > [!definition] Definition. ([[normal (geodesic) coordinates about a point on a Riemannian manifold]]) > Let $(M,g)$ be a [[Riemannian manifold]] and $D$ the [[Levi-Civita connection]] on $M$. Let $p \in M$. We will liberally identify the [[tangent space at a point of a smooth manifold|tangent space]] $T_{p}M$ with $\mathbb{R}^{n}$ as needed, sometimes implicitly. > > > For small enough $r_{0}>0$, the the [[Riemannian exponential map|exponential map]] $\exp_{p}:T_{p}M \to M$ restricts to a [[diffeomorphism]] $B_{g}(0, r_{0}) \to U$ for some $U \subset M$, by the [[inverse function theorem]]. Thus: > > - $\exp_{p} |_{B_{g}(0, r_{0})}:B_{g}(0, r_{0}) \to U$ locally defines a [[coordinate patch]] on $M$ about $p$; > - The inverse $\log_{p} |_{U}:U \to B_{g}(0, r_{0})$ locally defines a [[coordinate chart]] on $M$ about $p$. > The resulting coordinates are called the **normal coordinates on $M$ about $p$**, or the **geodesic coordinates on $M$ about $p$**. Can imagine $B_{g}(0, r_{0})$ as a blanket which $\exp_{p}$ drapes onto $M$ about $p$. Picture below tried but failed to express this intuition. > > ![[Pasted image 20250521122223.png|500]] > > As a consequence of the homogeneous symmetry $\gamma_{p}(\lambda t,a)=\gamma_{p}(t, \lambda a)$ exhibited by geodesics,[^1] we have $\log_{p} \big( \gamma_{p}(t, a) \big)=\log_{p} \exp_{p} (ta)=ta$ > for all $a \in T_{p}M$ and $|t|$ small. Thus, [[ray|rays]] (which are geodesics in Euclidean space) in $T_{p}M$ emanating from the origin map to earnest geodesics emanating from $p$ (purple in picture) in geodesic coordinates. A **radial geodesic on $M$** is one which is a ray in geodesic coordinates about $q$, for some $q \in M$.[^2] > > ---- #### [^1]: Explicitly: $\gamma_{p}(t, a)=\gamma_{p}(1, ta)=\exp_{p}(ta)$ on the neighborhood where $\exp_{p}$ is invertible. Then applying $\log_{p}$ cancels with $\exp_{p}$ and we are left with just the ray $t \mapsto ta$ emanating from the origin in Euclidean space. [^2]: Not every geodesic is radial; e.g. consider 'long' geodesics that don't land in any single normal neighborhood at all. [[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 > ```