---- > [!definition] Definition. ([[Laplace-Beltrami operator]]) > Let $(M,g)$ be an [[orientation of a vector bundle|oriented]] [[Riemannian manifold|Riemannian]] [[smooth manifold|manifold]] of dimension $n$. The **Laplace-Beltrami operator**, or **(Hodge) Laplacian**, on $M$ is the second-order linear differential map $\Delta:=\underbrace{ \delta d }_{ =:\Delta^{\text{up}} } + \underbrace{ d \delta }_{=: \Delta ^{\text{down}} } : \Omega^{p}(M) \to\Omega^{p}(M),$ > where $d:\Omega^{p} (M) \to \Omega^{p+1}(M)$ denotes the [[exterior derivative]] and $\delta:\Omega^{p+1}(M) \to \Omega^{p}(M)$ denotes the [[Hodge star|Hodge codifferential]], $\delta=d^{*}$. Elements of $\operatorname{ker }\Delta=\{ \alpha \in \Omega^{p}(M): \Delta \alpha=0 \}$ are called **harmonic forms**. The [[vector space|space]] of harmonic forms is denoted $\mathcal{H}^{p}(M)$. ^definition > [!equivalence] Equivalence. > A form $\alpha \in \Omega^{p}(M)$ is harmonic if and only if it is [[closed form|closed]] and [[closed form|coclosed]]: $\Delta \alpha = 0 \iff d\alpha =0 \text{ and } \delta \alpha=0.$ > > > [!proposition] In grading zero. > > If $p=0$, so we are working with functions $f \in C^{\infty}(M)$, then $df=0 \iff f \text{ is locally constant}$. So: $f \in \Omega^{0}(M) \text{ is harmonic } \iff f \text{ is locally constant}.$ > Or if $M$ is [[connected]]: $f \in \Omega^{0}(M) \text{ is harmonic } \iff f \text{ is constant}.$ > This is what we would expect.[^1] It is a special case of the [[Hodge decomposition|Hodge theorem]]. > ^proposition > > > [!basicproperties] > $\Delta$ is [[positive semidefinite operator|PSD]] > ^properties ---- #### [^1]: This fails easily without [[compact|compactness]], e.g., any linear [[polynomial 4|polynomial]] on $(\mathbb{R}, \text{eucl.})$ is harmonic but not constant. ---- #### 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 > ```