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A Riemannian metric $g$ on a complex manifold is a Kähler metric when its complex-linear extension is a Hermitian form on each tangent space and its fundamental two-formis closed. Equivalently, $\omega$ is a positive real closed $(1,1)$-form, making $X$ a Kähler manifold.

The Lefschetz operator of a Kähler manifold and its adjoint areBecause $d\omega =0$ and $\omega$ has type $(1,1)$, both $\partial \omega$ and $\bar{\partial }\omega$ vanish. The graded Leibniz rule therefore gives $[L,\partial ]=[L,\bar{\partial }]=0$.

Writing formal adjoints with stars, define the three Laplacians byThe supplied Kähler identities identity $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$ gives, by complex conjugation and taking adjoints,Expanding these commutators and using $\partial \bar{\partial }+\bar{\partial }\partial =0$ shows that the mixed terms in $\Delta$ vanish and that ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Since $d=\partial +\bar{\partial }$, it follows that

The adjoint of $[L,\partial ]=0$ gives $[\Lambda ,{\partial }^{\ast }]=0$. Hencewhere the last equality is the adjoint of $\partial \bar{\partial }+\bar{\partial }\partial =0$. Thus $\Lambda$, and therefore every power ${\Lambda }^k$, commutes with ${\Delta }_{\partial }$. It follows that ${\Lambda }^k$ sends every $\partial$-harmonic $(p,q)$-form to a $\partial$-harmonic $(p-k,q-k)$-form whenever $0\le k\le \min (p,q)$.

For the Poincaré metric on ${\mathrm{AdS}}_6$,The massless Klein-Gordon equation, evaluated with the Laplace-Beltrami operator, isorA boundary power $\varphi \sim z^{\alpha }$ obeys the indicial equationThusThe constant branch is the nonnormalizable source and the $z^5$ branch is the normalizable response. Since $m^2=\Delta (\Delta -d)=0$ with $d=5$, the roots are ${\Delta }_{-}=0$ and ${\Delta }_{+}=5$. Only standard quantization is unitary here: the putative alternative operator of dimension zero would be the identity, whereas the field is a nontrivial fluctuating operator. The permissible conformally invariant source-free boundary condition is therefore Dirichlet,More generally one may prescribe $J$ as an external source. The holographic dictionary gives

For an arbitrary smooth variation $u_t=u+t\phi$, differentiation under the integral givesSince $M$ is compact without boundary, integration by parts turns this intoThe fundamental lemma of the calculus of variations therefore gives the Euler-Lagrange equationor equivalently $-{\Delta }_{g}u=f$ for the Laplace-Beltrami operator.

An orientation selects the positive ordered bases in each tangent space. On an oriented $n$-dimensional Riemannian manifold, the Riemannian volume form ${\omega }_g$ is the unique smooth $n$-form satisfyingfor every positively oriented orthonormal frame. In positively oriented local coordinates,

The metric induces an inner product on the bundle ${\Lambda }^p{T}^{\ast }M$ of $p$-forms. The Hodge star operator is the unique linear mapsuch thatfor all $p$-forms $\alpha ,\beta$. With the codifferential $\delta$, the Laplace-Beltrami operator on differential forms is

The Hodge decomposition theorem says that on a compact oriented Riemannian manifold,an $L^2$-orthogonal direct sum, where ${\mathcal{H}}^p(M)=\ker \Delta$ is the finite-dimensional space of harmonic $p$-forms. Every de Rham cohomology class has exactly one harmonic representative.

In ${\mathbb{R}}^{d+1}$, $g=d{r}^2+r^2{g}_{S^d}$ and the volume density is $r^d$. With the nonnegative Laplacian convention, $\tilde{\Delta }=-{\partial }_r^2-(d/r){\partial }_r+r^{-2}{\Delta }_{S^d}$. The identity separates radial homogeneity from the angular Laplace-Beltrami operator and derives the spectrum of the Laplacian on a sphere.

The positive convention for the Laplace-Beltrami operator is ${\Delta }_{+}f=\delta df$, where $\delta$ is the codifferential. It is the restriction of the Hodge Laplacian to functions. In coordinates,On a compact boundaryless Riemannian manifold, or for compactly supported functions, $⟨f,{\Delta }_{+}f⟩=\Vert df{\Vert }^2\ge 0$. The opposite $\mathrm{div}\mathrm{grad}$ convention is also standard; formulas involving the sign must specify which is used.

For a function $f$ and one-form $\alpha$, the codifferential obeys $\delta (f\alpha )=f\delta \alpha -⟨df,\alpha {⟩}_g$. This follows from the graded Leibniz rule and the definition of the Hodge star operator. Applying it to $d(fh)=fdh+hdf$ gives the product formula. In particular ${\Delta }_{+}(f^2)=2f{\Delta }_{+}f-2|df{|}_g^2$. The cross-term sign reverses for the opposite Laplace-Beltrami operator convention.

A smooth local basis of vector fields whose inner products under a Riemannian metric are the identity matrix. Tracing the Riemannian Hessian in such a frame gives the Laplace-Beltrami operator, with the chosen sign.

For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.

Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.

With the nonnegative Laplace-Beltrami operator, a frequency $w$ in the dual lattice gives the eigenfunction $e^{2\pi i⟨w,x⟩}$ and eigenvalue $4{\pi }^2|w{|}^2$. Its multiplicity is the number of dual vectors with that norm. Fourier series prove completeness. In dimension two the shortest vector, shortest independent vector and covolume determine a reduced Gram matrix, proving spectral rigidity.

For the unit $S^d$ and the nonnegative Laplace-Beltrami operator, the eigenvalues are $\ell (\ell +d-1)$, with multiplicities $\left(\genfrac{}{}{0ex}{}{\ell +d}{d}\right)-\left(\genfrac{}{}{0ex}{}{\ell +d-2}{d}\right)$ for $\ell \ge 0$. The eigenfunctions are restrictions of degree-$\ell$ harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.

For a smooth real function on the unit two-sphere, integration of the covariant-derivative commutation formula gives the displayed identity. Explicitly, ${\nabla }^a{\nabla }_a{\nabla }_{b}w={\nabla }_b{\Delta }_{S}w+{\mathrm{Ric}}_b{}^c{\nabla }_{c}w$, and the unit sphere has $\mathrm{Ric}=g$. Integrating by parts once and then again proves the formula. It controls all angular second derivatives by the Laplace-Beltrami operator without singular estimates at coordinate poles.

Suppose a finite group $G$ acts by Riemannian isometries on a compact Riemannian manifold $M$, and Gassmann equivalent subgroups $H_1,H_2$ act freely. The quotient manifolds $H_i\backslash M$ have equal Laplace-Beltrami operator spectra. For an eigenfunction space $E$, the quotient multiplicity is $|H_i{|}^{-1}{\sum }_{h\in H_i}\mathrm{tr}(h|E)$, and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside $G$ alone does not guarantee that the quotients are nonisometric.

The surface Laplacian is the Laplace-Beltrami operator of a surface with its induced Riemannian metric. It is ${\Delta }_{s}f={\nabla }_s\cdot {\nabla }_{s}f$, using the surface gradient and surface divergence. On a sphere of radius $a$, a degree-$l$ spherical harmonic has eigenvalue $-l(l+1)/a^2$.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.