wedge product

facenian

I have this problem(from Tensor Analysis on Manyfolds by Bishop and Goldberg): prove that
[itex]e_1^ e_2 + e_3^e_4[/itex] is not decomposable when the dimension of the vector space is greater than 3 and e_i are basis vectors.
I solved it by mounting a set of 6 equations with 8 unknows and studying the different posibilities cheking that each one is not solvable.
Is there any nicer way to tackle this problem? if so please let me know

tiny-tim

hi facenian!

(use "\wedge" in latex )
you need to prove that it cannot equal [itex]a\wedge b[/itex] where a and b are 1-forms …

so express a and b in terms of the basis

facenian

helo tiny-tim, thanks for your prompt response and yes I did what you suggested and it led me to what I explained

tiny-tim

how about [itex]a\wedge (e_1\wedge e_2 + e_3\wedge e_4)[/itex] ?

facenian

you mean, let [itex]a=\sum_{i<j} x_{ij} e_i\wedge e_j[/itex] and then conclude tha [itex]a[/itex] must be null? Please let me know if that's what you meant and/or if I'm correct

tiny-tim

hi facenian!

no, i'm using the same a as before (in a∧b, which you're trying to prove it isn't)

so let a = ∑_i x_ie_i

facenian

I'm sorry I did not explained it correctly I should have said:

you mean, let [itex]a=\sum_i x_{i} e_i[/itex] and then conclude tha [itex]a[/itex] must be null because we are left with a linear conbination of basic vectors of the form [itex] \sum x_i e_i\wedge e_j\wedge e_k=0[/itex] .Please let me know if that's what you meant and/or if I'm correct

tiny-tim

… which has to be 0, because a ∧ (a ∧ b) = 0

yes

facenian

thank you very much tiny-tim your method is much better than mine!