---- Let $(M,g)$ be a [[Riemannian manifold|Riemannian]] [[smooth manifold|manifold]] of dimension $n$ that is [[orientable manifold|oriented]] via a never-zero $n$-[[differential form|form]] $\varepsilon \in \Omega^{n}(M)$. Let $\big( U, (x^{i}) \big)$ be a [[coordinate chart|coordinate neighborhood]]. > [!definition] Definition. (The inner product on $\Lambda^{p}T_{x}^{*}M$) > The metric $g$ allows us to apply [[Gram-Schmidt Procedure|Gram-Schmidt]] to the [[frame]] [[vector field|field]] $(\frac{ \partial }{ \partial x^{i} })$ to obtain an [[frame]] [[vector field|field]] $e_{1},\dots,e_{n}$ on $U$ that is [[orthonormal|orthonormal]] wrt $g$ positively [[orientation of a vector space|oriented]]. Call the [[dual basis|dual]] field $e^{1},\dots,e^{n} \in \Gamma(T^{*}U)$. > For each $x \in U$, the [[inner product]] $g_{x}$ on $T_{x}M$ dualizes to an [[inner product]] on $T_{x}M$ via $\langle e^{i}, e^{j} \rangle_{g}:=g(e_{i}, e_{j})=g_{ij}=\delta _{ij}$. This in turn determines an [[inner product]] $\langle -,- \rangle_{g}$ on each [[exterior power]] $\Lambda^{p}T^{*}_{x}M$ via[^1] $\langle e^{I} , e^{J}\rangle_{g} := \delta^{IJ}.$ Then one (optionally) shows $\langle \alpha^{1} \wedge \dots \wedge \alpha^{p} , \beta^{1} \wedge \dots \wedge \beta^{p} \rangle =\det (\langle \alpha^{i}, \beta^{j} \rangle_{g} )^{i,j=1,\dots,n}$ for $\alpha^{i}, \beta^{i} \in T_{x}^{*}M$. > > Note that the top-degree form $e^{1} \wedge\dots \wedge e^{n}$ must equal $a \varepsilon$ for some $a \in C^{\infty}(U)$ with $a>0$. ^definition > [!definition] Definition. ([[Riemannian volume form]]) > Let $p=n$. If $(\widetilde{e}^{i})$ were another positively-oriented orthonormal coframe field, then general argument analogous [[orientation of a smooth manifold|to here]] says $\widetilde{e}^{1} \wedge \dots \wedge \widetilde{e}^{n}=(\det \Phi)e^{1} \wedge\dots \wedge e^{n}$, where $\Phi \in C^{\infty}(U; \mathbb{R}^{n \times n})$ is the change of basis matrix between two coframes. [[special orthogonal group|But]] $\Phi \in \text{SO}(n)$, since the coframes are [[orthonormal basis|orthonormal]] and positively-oriented. So $\det \Phi=1$. Thus local $n$-forms glue into a global [[orientation of a vector bundle|orientation form]] $\omega \in \Omega^{n}(M)$, called the **Riemannian volume form of $(M,g)$ determined by the orientation**. It depends (only) on the metric and orientation. Sometimes the notations $\omega=\operatorname{vol}_{g}$ and $\omega=\d V$ are used. > The **volume** of a [[compact]] Riemannian manifold is $\operatorname{vol}(M):=\int _{M} \operatorname{vol}_{g}$. The **integral of a compactly supported scalar field $f \in C^{\infty}(M)$** on a Riemannian manifold is $\int_{M} f :=\int _{M} f \, \d V.$ [[surface integral]] - [ ] coordinate formula in terms of $\sqrt{ \det G}$ - [ ] familiar curves/surfaces Aexamples from math 396 ---- #### [^1]: Usual multi-indexing notation: $I$ indexes tuples $1 \leq i_{1}<\dots<i_{p} \leq n$, $J$ indexes tuples $1 \leq j_{1} < \dots < j_{p} \leq n$. ---- #### 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 > ```