---- > [!definition] Definition. ([[Riemannian manifold]]) > Let $M$ be a [[smooth manifold]]. A [[metric tensor|metric (tensor)]] $g$ on $M$ is called a **Riemannian metric** if it is not only [[nondegenerate bilinear form|nondegenerate]], but [[positive definite bilinear form|positive definite]]. > A **Riemannian manifold $(M,g)$** is a [[smooth manifold]] $M$ together with a Riemannian metric $g$. > > > As a (smooth) section of the [[symmetric power|symmetric square]] $\mathbb{S}^{2} T^{*} M \subset T^{*}M \otimes T^{*}M$, in a [[coordinate chart|coordinate neighborhood]] $\big(U, (x^{i})\big)$ $g$ looks like[^2] $g=g _{ij} \ dx^{i} \otimes dx^{j}$ > [[matrix of a bilinear form|where]] $g_{ij}=g\left( \frac{ \partial }{ \partial x^{i} }, \frac{ \partial }{ \partial x^{j} } \right) \in C^{\infty}(U)$. > > > If $X=X^{i} \frac{ \partial }{ \partial x^{i} }$ and $Y=Y^{j} \frac{ \partial }{ \partial x^{j} }$ are [[vector field|vector fields]] on $M$, locally [[the space of differential 1-forms is dual to that of vector fields|one has]] $\begin{align} > g(X,Y)&=(g_{ij} dx^{i} \otimes dx^{j})\left( X^{i} \frac{ \partial }{ \partial x^{i} } , Y^{j} \frac{ \partial }{ \partial x^{j} } \right) \\ > &= g_{ij} X^{i} Y^{j} \in C^{\infty}(U) > \end{align}$ ---- #### [^2]: Commonly the $\otimes$ is dropped notationally; we write $dx^{i} dx^{j}$ instead of $dx^{i} \otimes dx^{j}$. But never lose track of the fact that this object is a [[symmetric power|symmetric tensor]], *not* a 2-form (not skew-symmetric). In particular we will have $dx^{2}$ type terms. ---- #### 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 > ```