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Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset \mathbb{R}^n$ and let $h_i:E|_{U_i} \to V_i \times \mathbb{C}^N$ be a collection of trivialisations (composed with $\varphi_i \times id$ in order to land in the product $V_i \times \mathbb{C}^N$). Let $h_{i*}$ be a induced map $h_{i*}\xi=h_i \circ \xi \circ \varphi_i^{-1}$ acting from $\Gamma^{\infty}(U_i,E|{U_i})$ to $\Gamma^{\infty}(V_i,\mathbb{C}^N)$ which can be indentified with $[C^{\infty}(V_i)]^N$. Let $(f_i)_i$ be a smooth partition of unity subordinate to $(U_i)_i$ and put $$||\xi||_{s,E}:=\Sigma_i||h_{i*}(f_i\xi)||_{s,\mathbb{R}^n}$$ where on the right hand side we have usual Sobolev $s$-norm given by $\Big(\int_{\mathbb{R}^n}|\hat{f}(y)|^2(1+|y|^2)^sdy\Big)^{\frac{1}{2}}$. In this definition we made three non canonical choices: of coordinates, of trivialisations and of partition of unity. I would therefore like to ask

Why these norms are equivalent for different choices of coordinates/trivialisations/partition of unity?

Moreover, for $s=0$ we can consider the scalar product $$(\xi,\eta):=\int_X\langle\xi(x),\eta(x) \rangle d\mu(x)$$ as an integral with respect to the Riemannian density $d\mu$ and $\langle \cdot,- \rangle$ is a Hermitian metric on $E$.

Why the norm coming from this calar product is equivalent to the norm introduced above with $s=0$?

This question is motivated by the considerations in Chapter IV of Wells' book "Differential Analysis on Complex Manifolds" where details are left to the reader. However I don't see why these questions have to be easy.

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1 Answer 1

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Hint: For the first question, first show that the norms induced by different (finite) partitions of unity are equivalent. Then, if you have two different choices of charts and trivializations, you find another couple refining them both. This reduces your problem to showing that if you have a refinement of a system of charts and trivializations, together with a change of coordinates, then the induced norms are equivalent. Now just write down the formulae under a change of coordinates, and use compactness to get your finite coefficients to show equivalence.

The second question is trivial, just write down the definitions.

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  • $\begingroup$ Well, I'm functional analyst, my knowledge in geometry is rather scattered so I would be very grateful if you could expand your answer. These are my first steps in the attempt to finally learn Atiyah-Singer index theorem. $\endgroup$
    – truebaran
    Mar 14, 2016 at 10:33
  • $\begingroup$ If you are still interested in discussing this problem, I offered a bounty on it. $\endgroup$
    – truebaran
    Mar 27, 2016 at 22:40

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