  Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach
space, $p>1,$ and $(X,\sigma)$ be a space with a finite measure. We prove that
$E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if
$E$ is isometric to a subspace of $F.$ Moreover, every isometry $T$ from $E$
into $L_p(X;F)$ has the form $Te(x)=h(x)U(x)e, e\in E,$ where $h:X\rightarrow
R$ is a measurable function and, for every $x\in X,$ $U(x)$ is an isometry from
$E$ to $F.$
