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Thread: graded Ideal and differential forms

  1. #1
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    graded Ideal and differential forms

    Hello,

    i have a proof of a statement, but i don't understand it very well. I have concrete questions about it and it would be very nice, when someone can help me.

    Let E be a subbundle of TM, we define the (graded) ideal $\displaystyle J=\bigoplus_{k=1}^n J^k \; in \; \Omega(M)$ as follows $\displaystyle w \in J^k \;<=> w(X_1,...,X_k)=0$ for any sections $\displaystyle X_i$ of E

    Claim: J is locally generated by q linearly independent 1-forms:

    Pf: Choose a local frame$\displaystyle X_1,...,X_n$ of TM, s.t.$\displaystyle X_1,...,X_{n-q}$ form a frame of E.
    There is the dual frame of differential 1-forms $\displaystyle w_1,...,w_n$ of TM* and the linearly independent 1-forms $\displaystyle w_{n-q+1},..,w_n$ clearly generate the ideal J.

    Why do they generate the Ideal J????
    What does it mean "generate"? the $\displaystyle w_i$ are 1-forms, i.e. sections: $\displaystyle M->TM^*, w_i (p)\in T^{*}_p M$.
    and J is the direct sum of k-forms, for k=1,...,n?

    Can you please explain it for me?

    Regards
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  2. #2
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    means any element in J can be expressed in terms of these q forms.
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