I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $D_1 : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1)$ which can easily be lifted to the Fredholm differential operator $D_2 : \Gamma(M, E_1 \oplus E_2) \rightarrow \Gamma(M, E_1 \oplus E_2)$. What can be said about their indices $\mathrm{Ind}(D_1)$ and $\mathrm{Ind}(D_2)$?
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$\begingroup$ could you please more explain how do you deduce $D_2$ from $D_1$? Moreover is $\lambda$ the obvious embedding?Or you fix an arbitrary embedding?Finally can you explain what is the motivation for this question? $\endgroup$– Ali TaghaviFeb 1, 2020 at 14:02
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