I'm reading Terence Tao's blog and he says
Once one has a connection on a bundle $V$, one automatically can define a connection on the dual bundle $V^*$.
Can anyone tell me how?
$\endgroup$1 Answer
$\begingroup$I assume you are talking about linear connections ("covariant derivatives").
Assume that $V$ is a vector bundle over $M$. The dual connection is uniquely defined by requiring that it satisfies the product rule with respect to the natural pairing between $V$ and $V^*$. Let $\xi \in \Gamma(V)$ and $\varphi \in \Gamma(V^*)$, and denote by $$ \left< \varphi, \xi \right> = \varphi(\xi ) \in C^\infty(M) $$ the natural pairing between the bundles. That is, at each point $p \in M$, you apply the functional $\varphi_p$ to the vector $\xi_p$ to get a number $\varphi_p(\xi_p)=(\varphi(\xi))(p)$. Then, the dual connection satisfies the "product" rule $$ \nabla_X \left< \varphi, \xi \right> = X \left< \varphi, \xi \right> = \left< \nabla_X \varphi, \xi \right> + \left< \varphi, \nabla_X \xi \right> $$ for all $X \in \Gamma(TM)$.
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