---- > [!definition] Definition. ([[pullback of a Riemannian metric]]) > Let $F:M \to N$ be a [[smooth maps between manifolds|smooth map]] between [[smooth manifold|smooth manifolds]] $M$ and $N$. Given a [[Riemannian manifold|Riemannian metric]] $h$ on $N$, the [[differential of a smooth map between smooth manifolds|formula]] $(F^{*}h)_{p}(X_{p}, Y_{p}):= h_{F(p)} \big( dF_{p}X_{p}, dF_{p}Y_{p} \big)$ > for $p \in M$ and $X_{p}, Y_{p} \in T_{p}M$ defines a smooth [[symmetric multilinear map|symmetric]] positive-semidefinite $(0,2)$-[[tensor field]] on $M$ > It is in fact a [[Riemannian manifold|Riemannian metric]] (i.e. is [[inner product|positive-definite]]) if and only if $F$ is a [[smooth immersion]].[^1] In this case we call the [[Riemannian manifold|metric]] $F^{*}h$ the **pullback of $h$ by $F$**. [^1]: Suppose $F$ is a [[smooth immersion]], so that $F_{*}=dF$ is [[injection|injective]]. Since $h$ is positive-definite, $h_{F(p)}\big( F_{*}X_{p}, F_{*}X_{p} \big)=0$ if and only if $F_{*}X_{p}=0$. Since $F_{*}$ is injective, this happens if and only if $X_{p}=0$. Conversely suppose $F^{*}h$ is positive-definite. Suppose $F_{*}X_{p}=0$ for some $X_{p} \in T_{p}M$. Then $F^{*}h(X_{p}, X_{p})=0$. Hence $X_{p}=0$. ---- #### ---- #### 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 > ```