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In [[differential geometry]] there are a number of second-order, linear, [[elliptic operator|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 metric|Riemannian]]- or [[Pseudo-Riemannian manifold|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:

:<math>\Delta T= -\text{tr}\;\nabla^2 T,</math>
where ''T'' is any tensor, <math>\nabla</math> 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

:<math>\nabla^2_{X,Y} T = -\left(\nabla_X \nabla_Y T - \nabla_{\nabla_X Y} T\right).</math>

Note that with this definition, the connection Laplacian has negative [[Spectrum of an operator|spectrum]].  On functions, it agrees with
the operator given as the divergence of the gradient.

If connection of interest is [[Levi-Civita connection]] one can find convenient formula for Laplacian of scalar function in terms of partial derivatives with respect to chosen coordinates:

:<math>\Delta \phi = |g|^{-1/2} \partial_\mu\left( |g|^{1/2} g^{\mu\nu} \partial_\nu\right)\phi </math>

where <math>\phi</math> is scalar function, <math>|g|</math> is absolute value of determinant of metric (the use of absolute value is necessary in [[Pseudo-Riemannian manifold|Pseudo Riemmanian case]], for example in [[General Relativity]]) and <math>g^{\mu \nu}</math> denotes [[Metric tensor#Inverse_metric|inverse of the metric tensor]]

== Hodge Laplacian ==
The '''Hodge Laplacian''', also known as the '''Laplace–de Rham operator''', is differential operator on 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 metric|Riemannian]]- or [[Pseudo-Riemannian manifold|pseudo-Riemannian]] metric.

:<math>\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;</math>

where d is the [[exterior derivative]] or differential and δ is the [[codifferential]].  The Hodge Laplacian on a compact manifold has nonnegative [[Spectrum of an operator|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 a fiber metric and a compatible connection, <math>\nabla</math>.  This connection gives rise to a differential operator
::<math>\nabla:\Gamma(E)\rightarrow \Gamma(T^*M\otimes E)</math>
where <math>\Gamma(E)</math> denotes smooth sections of ''E'', and ''T''<sup>*</sup>M is the [[cotangent bundle]] of ''M''.  It is possible to take the <math>L^2</math>-adjoint of <math>\nabla</math>, giving a differential operator
::<math>\nabla^*:\Gamma(T^*M\otimes E)\rightarrow \Gamma(E).</math>
The '''Bochner Laplacian''' is given by 
::<math>\Delta=\nabla^*\nabla</math>
which is a second order operator acting on sections of the vector bundle ''E''.  Note that the connection Laplacian and Bochner Laplacian differ only by a sign:
::<math> \nabla^* \nabla =  - \text{tr}\, \nabla^2</math>

== Lichnerowicz Laplacian ==
The '''Lichnerowicz Laplacian'''<ref>{{Citation | last1=Chow | first1=Bennett | last2=Lu | first2=Peng | last3=Ni | first3=Lei | title=Hamilton's Ricci flow | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-4231-7; 978-0-8218-4231-7 | id={{MathSciNet | id = 2274812}} | year=2006 | volume=77}}
</ref> is defined on symmetric tensors by taking <math>\nabla : \Gamma(\operatorname{Sym}^k(TM))\to \Gamma(\operatorname{Sym}^{k+1}(TM))</math> to be the symmetrized covariant derivative.  The Lichnerowicz Laplacian is then defined by <math>\Delta_L = \nabla^*\nabla</math>, where <math>\nabla^*</math> is the formal adjoint.  The Lichnerowicz Laplacian differs from the usual tensor Laplacian by a [[Weitzenbock formula]] involving the [[Riemann curvature tensor]], and has natural applications in the study of [[Ricci flow]] and the [[prescribed Ricci curvature problem]].

== Conformal Laplacian ==
On a [[Riemannian manifold]], one can define the '''conformal Laplacian''' as an operator on smooth functions; it differs from the Laplace–Beltrami operator by a term involving the [[scalar curvature]] of the underlying metric.  In dimension ''n''&nbsp;&ge;&nbsp;3, the conformal Laplacian, denoted ''L'', acts on a smooth function ''u'' by

:<math>Lu = -4\frac{n-1}{n-2} \Delta u + Ru,</math>

where &Delta; is the Laplace-Beltrami operator (of negative spectrum), and ''R'' is the scalar curvature.  This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric.  If ''n''&nbsp;&ge;&nbsp;3 and ''g'' is a metric and ''u'' is a smooth, positive function, then the [[Conformal map|conformal]] metric

: <math>\tilde g = u^{4/(n-2)} g \, </math>

has scalar curvature given by

:<math>\tilde R = u^{-(n+2)/(n-2)} L u. \, </math>

==See also==
*[[Weitzenböck identity]]

==References==
<references/>

[[Category:Differential operators]]
[[Category:Differential geometry]]