In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.
Connection Laplacian The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator. It is defined as the trace of the second covariant derivative:
Δ T = tr ∇ 2 T , {\displaystyle \Delta T={\text{tr}}\;\nabla ^{2}T,}
where T is any tensor, ∇ {\displaystyle \nabla } is the Levi-Civita connection associated to the metric, and the trace is taken with respect to the metric. Recall that the second covariant derivative of T is defined as
∇ X , Y 2 T = ∇ X ∇ Y T − ∇ ∇ X Y T . {\displaystyle \nabla _{X,Y}^{2}T=\nabla _{X}\nabla _{Y}T-\nabla _{\nabla _{X}Y}T.}
Note that with this definition, the connection Laplacian has negative spectrum. On functions, it agrees with the operator given as the divergence of the gradient. If the connection of interest is the Levi-Civita connection one can find a convenient formula for the Laplacian of a scalar function in terms of partial derivatives with respect to a coordinate system:
Δ ϕ = | g | − 1 / 2 ∂ μ ( | g | 1 / 2 g μ ν ∂ ν ϕ ) {\displaystyle \Delta \phi =|g|^{-1/2}\partial _{\mu }\left(|g|^{1/2}g^{\mu \nu }\partial _{\nu }\phi \right)}
where ϕ {\displaystyle \phi } is a scalar function, | g | {\displaystyle |g|} is absolute value of the determinant of the metric (absolute value is necessary in the pseudo-Riemannian case, e.g. in General Relativity) and g μ ν {\displaystyle g^{\mu \nu }} denotes the inverse of the metric tensor.
Hodge Laplacian The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
Δ = d δ + δ d = ( d + δ ) 2 , {\displaystyle \Delta =\mathrm {d} \delta +\delta \mathrm {d} =(\mathrm {d} +\delta )^{2},\;}
where d {\displaystyle \mathrm {d} } is the exterior derivative or differential and δ {\displaystyle \delta } is the codifferential. The Hodge Laplacian on a compact manifold has nonnegative spectrum. The connection Laplacian may also be taken to act on differential forms by restricting it to act on skew-symmetric tensors. The connection Laplacian differs from the Hodge Laplacian by means of a Weitzenböck identity.
Bochner Laplacian The Bochner Laplacian is defined differently from the connection Laplacian, but the two will turn out to differ only by a sign, whenever the former is defined. Let M be a compact, oriented manifold equipped with a metric. Let E be a vector bundle over M equipped with a fiber metric and a compatible connection, ∇ {\displaystyle \nabla } . This connection gives rise to a differential operator
∇ : Γ ( E ) → Γ ( T ∗ M ⊗ E ) {\displaystyle \nabla :\Gamma (E)\rightarrow \Gamma (T^{*}M\otimes E)}
where Γ ( E ) {\displaystyle \Gamma (E)} denotes smooth sections of E, and T*M is the cotangent bundle of M. It is possible to take the L 2 {\displaystyle L^{2}} -adjoint of ∇ {\displaystyle \nabla } , giving a differential operator
∇ ∗ : Γ ( T ∗ M ⊗ E ) → Γ ( E ) . {\displaystyle \nabla ^{*}:\Gamma (T^{*}M\otimes E)\rightarrow \Gamma (E).}
The Bochner Laplacian is given by
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