---- Recall that for any (say, $n$-dimensional) [[vector space]] $V$, one can speak of the [[exterior algebra]] $\Lambda^{\bullet}(V)$. For the [[dual vector space|dual space]] $V^{*}$ of $1$-[[multilinear map|forms]] in particular, there is a canonical identification $\Lambda^{\bullet}(V^{*}) \cong \text{Alt}^{\bullet}(V)$ with the [[algebra of alternating multilinear forms]] (also called alternating tensors) on $V$. Also recall that $\Lambda^{r}(V^{*})$ is $n \choose r$-dimensional: if $V$ has [[basis]] $e_{1},\dots,e_{n}$, so that if $V^{*}$ has [[dual basis]] $e^{1},\dots,e^{n}$, then $\Lambda^{r}(V^{*})$ has basis $\{ e^{I}: I \in \text{ASC}_{r,n} \}= \{ e^{i_{1}} \wedge \dots \wedge e^{i_{r}} : 1 \leq i_{1} < \dots < i_{r} \leq n \}.$ > [!definition] Definition. ([[space and algebra of alternating tensors at a point on a manifold]]) > Now let $M$ be a [[smooth manifold]] of dimension $n$ and let $p \in M$. Let $T_{p}M$ be the [[tangent space at a point of a smooth manifold|tangent space]] to $M$ at $p$ and let $T_{p}^{*}M$ be the [[cotangent space]]. > The **space of alternating $r$-tensors at $p$** is the [[exterior power]] $\Lambda^{r}(T_{p}^{*}M) \cong \text{Alt}^{r}(T_{p}^{}M),$whose elements are viewed as [[alternating multilinear map|alternating multilinear forms]] $T_{p}^{}M \times \dots \times T_{p}^{}M \to \mathbb{R}.$ (By convention, $\Lambda^{0}=\mathbb{R}$.) > Fixing [[coordinate chart|local coordinates]] $(x^{i})$ around $p$ induces a [[tangent space at a point of a smooth manifold|coordinate]] [[basis]] $\left( \frac{ \partial }{ \partial x^{i} } |_{p} \right)$ of $T_{p}M$ and [[dual basis]] $(dx^{i} |_{p})$ of $T_{p}^{*}M$. In turn, we have a [[basis]] $\{ dx^{I} |_{p} : I \in \text{ASC}_{k,n}\}= \{ dx^{i_{1}} |_{p} \wedge \dots \wedge dx^{i_{r}} |_{p}: 1 \leq i_{1} < \dots < i_{r} \leq n \}$ for $\Lambda^{r}(T_{p}^{*}M)$, where $\wedge$ is the [[algebra of alternating multilinear forms|wedge product of alternating forms]]. > > The **algebra of alternating tensors at $p$** is the [[exterior algebra]] $\Lambda^{\bullet}(T_{p}^{*}M)\cong \text{Alt}^{\bullet}(T_{p}M).$ ^definition ---- #### ---- #### 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 > ```