Laplacian of a 2-form

  1. haushofer

    haushofer 980
    Science Advisor

    Hi,

    According to eg Nakahara's conventions the Laplacian on a form K is given by

    [tex]
    \Delta K = (dd^{\dagger} + d^{\dagger}d)K
    [/tex]

    In my case K is a two form living in R^3. I've calculated the Laplacian and arrive at

    [tex]
    \Delta K = \Bigl( \frac{1}{3!}\epsilon^{klm}\epsilon^n_{\ ij}\partial_k \partial_n K_{lm} - \frac{1}{4}\partial_{i}\partial^k K_{jk} \Bigr) dx^i \wedge dx^j
    [/tex]

    However, the answer seems a little odd to me. Does anyone have a reference where Laplacians of two forms are evaluated, or some comment? Thanks in forward! :)
     
  2. jcsd
  3. haushofer

    haushofer 980
    Science Advisor

    And another question: is on a compact manifold a harmonic form neccesarily constant, or is this only true for 0-forms (functions)?
     
  4. How would you go about defining a constant form?

    The obvious answer to this would be that you can't define it in a coordinate-invariant way. The generalization of the idea of constant functions to forms is with the definition: a constant function is d and d* closed.
     
  5. For your first question, if the codifferential is defined with a sign (-1)^n(p+1), it can be shown that in euclidean space in general laplace beltrami of a form is the same as the usual old laplacian of every function factor (times the wedged covectors). It is a lengthy calculation and took me some hours, but it works ;)
     
  6. haushofer

    haushofer 980
    Science Advisor

    Hi,

    but that would mean that my calculation isn't right?

    About this "constant form": that should be zero, ofcourse; if a tensor is zero in one coordinate system, it's zero in every coordinate system. My bad ;)
     
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