 Pullback connection

If a connection on $E\to B$ has connection matrix $A=A_i(y)d{y}^i$ in a local frame and $f:M\to B$, its pullback connection has matrixIt is characterized by ${\nabla }_X^{f^{\ast }A}(f^{\ast }s)=f^{\ast }({\nabla }_{df(X)}^{A}s)$.