- Laplacian operators in differential geometry
In
differential geometry there are a number of second-order, linear, ellipticdifferential operators bearing the name Laplacian. This article provides an overview of some of them.Connection Laplacian
The connection Laplacian is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemmanian- or pseudo-Riemannian metric. When applied to functions (i.e, tensors of rank 0), the connectionLaplacian is often called the
Laplace-Beltrami operator . It is defined as the trace of the second covariant derivative::Delta T= ext{tr}; abla^2 T,where "T" is any tensor, abla is the
Levi-Civita connection associated to the metric, and the trace is taken with respect tothe metric. Recall that the second covariant derivative of "T" is defined as:abla^2_{X,Y} T = abla_X abla_Y T - abla_{ abla_X Y} T.
Note that with this definition, the connection Laplacian has negative spectrum. On functions, it agrees withthe operator given as the divergence of the gradient.
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 thecotangent bundle .) This operator is defined on any manifold equipped witha Riemmanian- or pseudo-Riemannian metric.:Delta= mathrm{d}delta+deltamathrm{d} = (mathrm{d}+delta)^2,;
where d is the
exterior derivative or differential and δ is thecodifferential . The Hodge Laplacian has positive spectrum.The connection Laplacian may also be taken to act on differential forms by restricting in 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, abla. This connection gives rise to a differential operator::abla:Gamma(E) ightarrow Gamma(T^*Motimes E)where Gamma(E) denotes smooth sections of "E", and "T"*M is the
cotangent bundle of "M". It is possible to take the L^2-adjoint of abla, giving a differential operator::abla^*:Gamma(T^*Motimes E) ightarrow Gamma(E).The Bochner Laplacian is given by ::Delta= abla^* ablawhich 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:::abla^* abla = - ext{tr} abla^2Lichnerowicz Laplacian
The Lichnerowicz Laplacian Fact|date=May 2008 is defined only on symmetric tensors of rank 2, and makes sense on manifolds equipped with a metric. It differsfrom the connection Laplacian by terms involving the
Riemann curvature tensor , and has natural applications in the study ofRicci flow and thePrescribed 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 thescalar curvature of the underlying metric. In dimension n geq 3, the conformal Laplacian, denoted "L",acts on a smooth function "u" by:Lu = -4frac{n-1}{n-2} Delta u + Ru,
where Delta is the Laplace-Beltrami operator (of negative spectrum), and "R" is the scalar curvature. This operator oftenmakes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric. If n geq 3 and "g" is a metric and "u" is a smooth, positive function, then the conformal metric ilde g = u^frac{4}{n-2} g has scalar curvature given by
:ilde R = u^{-frac{n+2}{n-2 L u.
ee also
*
Weitzenböck identity
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