Hodge, "The theory and application of harmonic integrals", Cambridge Univ. Symbol of an operator) is equal to the quadratic form on the cotangent bundle which is dual to $ g $. The Laplace operator of a Riemannian metric $ g $Ĭan also be defined as the real symmetric second-order linear partial differential operator which annihilates the constant functions and for which the principal symbol (cf. In this chapter we will generalize the Laplacian on Euclidean space to an oper. It is again extended pointwise to forms on complex manifolds with a Hermitian metric. The point is that ( 26 ) gives the divergence in any coordinates and any curved space. structed from the curvature of a Riemannian metric. In this case the Hodge $ \star $-operator is defined relative to this inner product and this orientation. quantum chaos, geodesic flows and Anosov flows on (negatively curved). Provides an inner product on $ E ^ \prime $ our three-dimensional (Euclidean) space the Laplace operator (or just Laplacian). Then the fundamental 2-form associated to $ h $, I am trying to understand the Weitzenbock equality that for a curved-space spinors Dirac operator D / and the associated Laplacian satisfy. Examples are provided by analysis on curved space-times in. The Hodge star operator on an oriented Riemannian manifold $ M $īe a complex vector space of (complex) dimension $ n $īe the underlying $ 2n $-dimensional real vector space. ential equations on singular and non-compact spaces, which are more and more.
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