Endomorphism-induced tensor derivation
= Endomorphism-induced tensor derivation
{title2=$D_A$}
A smooth <endomorphism> $A$ of the <tangent bundle> defines the <tensor derivation> $D_A$ by $D_Af=0$ and $D_AY=A(Y)$. On a <differential one-form>, $D_A\omega=-\omega\circ A$. On a general <tensor field>, it acts by $A$ in each vector factor and by the negative dual action in each covector factor. The two actions cancel in each contracted pairing, proving compatibility with <tensor contraction>.