---- > [!definition] Definition. ([[Riemannian exponential map]]) > Let $(M,g)$ be a [[Riemannian manifold]], $p \in M$ and $D$ the [[Levi-Civita connection]] on $M$. Let $p \in M$. > The map $\begin{align} \exp_{p}: T_{p}M &\to M \\ a & \mapsto \gamma_{p}(1, a) \end{align}$ is called the **(Riemannian) exponential map at $p \in M$**. It is [[well-defined]] and smooth on $B_{g}(0, \varepsilon) \subset T_{p}M$ for $\varepsilon>0$ small enough. >![[Pasted image 20250521111321.png|500]] [^1] > [!definition] Definition. (Logarithmic map) > Since $\exp_{p}$ is smooth around the origin (viewing $T_{p}M$ as $(\mathbb{R}^{n}, g)$), can look at the [[differential of a smooth map between smooth manifolds|differential]] $(d \exp_{p}) |_{a=0}:T_{0}(T_{p}M) \to T_{p}M$ there. After canonically identifying $T_{0}(T_{p}M)$ with $T_{p}M$, one can calculate that $(d \exp_{p}) |_{a=0}$ is just the [[identity map]] on $T_{p}M$. By the [[inverse function theorem]], there exists a ball of radius (say) $r_{0}$ about $0 \in T_{p}M$ such that $\exp_{p}:B_{g}(0, r_{0}) \to M$ is a [[diffeomorphism]] onto its image $U \subset M$. The inverse map $\log_{p}=\exp_{p} ^{-1}:U \to \mathbb{R}^{n}$ is called the **logarithmic map around $p$**. ^definition ---- #### [^1]: Image taken from: https://www.researchgate.net/figure/The-exponential-map-allow-us-to-locally-carry-the-Euclidean-structure-of-the-tangent_fig5_311807546 ---- #### 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 > ```