Operators on Sections of Vector Bundles











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Let $(E,M,pi)$ and $(F,M,pi')$ be $C^{infty}$ vector bundles. An operator $alpha:Gamma(E)to Gamma(F)$ is said to be local if whenever $sin Gamma(E)$ vanishes on an open set $Usubset M$ then $alpha(s)$ vanishes as well on $U$. This is the definition of the locality of a single variable operator between sections of vector bundles over the same manifold. How can I define the locality of an operator $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ where $E,E',F$ are vector bundles over the same manifold?
Does it make sense to the define the locality as follows: $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ is local if whenever either $sin Gamma(E)$ or $s'in Gamma(E')$ vanishes on an open set $Usubset M$ then $alpha(s,s')$ vanishes on $U$ ?.



I appreciate any help. Thanks in advance.






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    Let $(E,M,pi)$ and $(F,M,pi')$ be $C^{infty}$ vector bundles. An operator $alpha:Gamma(E)to Gamma(F)$ is said to be local if whenever $sin Gamma(E)$ vanishes on an open set $Usubset M$ then $alpha(s)$ vanishes as well on $U$. This is the definition of the locality of a single variable operator between sections of vector bundles over the same manifold. How can I define the locality of an operator $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ where $E,E',F$ are vector bundles over the same manifold?
    Does it make sense to the define the locality as follows: $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ is local if whenever either $sin Gamma(E)$ or $s'in Gamma(E')$ vanishes on an open set $Usubset M$ then $alpha(s,s')$ vanishes on $U$ ?.



    I appreciate any help. Thanks in advance.






    vector-bundles







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      up vote
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      down vote

      favorite











      Let $(E,M,pi)$ and $(F,M,pi')$ be $C^{infty}$ vector bundles. An operator $alpha:Gamma(E)to Gamma(F)$ is said to be local if whenever $sin Gamma(E)$ vanishes on an open set $Usubset M$ then $alpha(s)$ vanishes as well on $U$. This is the definition of the locality of a single variable operator between sections of vector bundles over the same manifold. How can I define the locality of an operator $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ where $E,E',F$ are vector bundles over the same manifold?
      Does it make sense to the define the locality as follows: $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ is local if whenever either $sin Gamma(E)$ or $s'in Gamma(E')$ vanishes on an open set $Usubset M$ then $alpha(s,s')$ vanishes on $U$ ?.



      I appreciate any help. Thanks in advance.






      vector-bundles







      share|cite|improve this question















      Let $(E,M,pi)$ and $(F,M,pi')$ be $C^{infty}$ vector bundles. An operator $alpha:Gamma(E)to Gamma(F)$ is said to be local if whenever $sin Gamma(E)$ vanishes on an open set $Usubset M$ then $alpha(s)$ vanishes as well on $U$. This is the definition of the locality of a single variable operator between sections of vector bundles over the same manifold. How can I define the locality of an operator $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ where $E,E',F$ are vector bundles over the same manifold?
      Does it make sense to the define the locality as follows: $alpha:Gamma(E)times Gamma(E')to Gamma(F)$ is local if whenever either $sin Gamma(E)$ or $s'in Gamma(E')$ vanishes on an open set $Usubset M$ then $alpha(s,s')$ vanishes on $U$ ?.



      I appreciate any help. Thanks in advance.






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      vector-bundles






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      edited Nov 22 at 17:02

























      asked Nov 22 at 14:53









      Hussein Eid

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