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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?

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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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    @PtF: It doesn't have to satisfy it, this is merely a definition but a reasonable one if you think about the situation in $\mathbb{R}^n$. Regarding the left hand side, usually when $f$ is a function, the notations $\nabla_X f$, $\mathcal{L}_X(f)$, $Xf$ and $df(X)$ are all denoting the same thing so you can write it also as \mathcal{L}_X( \left< \varphi, \xi \right>) if you wish.2016-09-26